IRMA Lectures in Mathematics and Theoretical Physics 22 Edited by Christian Kassel and Vladimir G. Turaev
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IRMA Lectures in Mathematics and Theoretical Physics 22 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 6 Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature 7 Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) 8 AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) 9 Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) 10 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)

Handbook of Hilbert Geometry Athanase Papadopoulos Marc Troyanov Editors

Editors: Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 Rue René Descartes 67084 Strasbourg Cedex France

Marc Troyanov Section de mathématiques École Polytechnique Fédérale de Lausanne SMA-Station 8 1015 Lausanne Switzerland

2010 Mathematics Subject Classification: 01A55, 01-99, 35Q53, 37D25, 37D20, 37D40, 47H09, 51-00, 51-02, 51-03, 51A05, 51B20, 51F99, 51K05, 51K10, 51K99, 51M10, 52A07, 52A20, 52A99, 53A20, 53A35, 53B40, 53C22, 53C24, 53C60, 53C70, 53B40, 54H20, 57S25, 58-00, 58-02, 58-03, 58B20, 58D05, 58F07. Key words: Hilbert metric, Funk metric, non-symmetric metric, Finsler geometry, Minkowski space, Minkowski functional, convexity, Cayley-Klein-Beltrami model, projective manifold, projective volume, Busemann curvature, Busemann volume, horofunction, geodesic flow, Teichmüller space, Hilbert fourth problem, entropy, geodesic, Perron-Frobenius theory, geometric structure, holonomy homomorphism

ISBN 978-3-03719-147-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2014 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Foreword

The idea of collecting the surveys that constitute this Handbook came out of a desire to present in a single volume the foundations as well as the modern developments of Hilbert geometry. In the last two decades the subject has grown into a very active field of research. The Handbook will allow the student to learn this theory, to understand the questions and problems that it leads to, and to acquire the tools that can be used to approach them. It should also be useful to the confirmed researcher and to the specialist, for it contains an exposition and an update of the most recent developments. Thus, some chapters contain classical material, highlighting works of Beltrami, Klein, Hilbert, Berwald, Funk, Busemann, Benzécri and the other founders of the theory, and other chapters present recent developments. Hilbert geometry can be regarded from different points of view: the calculus of variations, Finsler geometry, projective geometry, dynamical systems, etc. At several places in this volume, the fruitful relations between Hilbert geometry and other subjects in mathematics are reported on. These subjects include Teichmüller spaces, convexity theory, Perron–Frobenius theory, representation theory, partial differential equations, coarse geometry, ergodic theory, algebraic groups, Coxeter groups, geometric group theory, Lie groups, and discrete group actions. All these important topics appear in one way or another in this book. We would like to take this opportunity to thank Gérard Besson who helped us at an early stage of this project. Our warm thanks go to Manfred Karbe from the European Mathematical Society who encouraged the project and to Irene Zimmermann for a very efficient collaboration and for the seriousness of her work. This work was supported in part by the French research program ANR FINSLER and by the Swiss National Science Foundation. Strasbourg and Lausanne, September 2014

Athanase Papadopoulos Marc Troyanov

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I. Minkowski, Hilbert and Funk geometries Chapter 1. Weak Minkowski spaces by Athanase Papadopoulos and Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2. From Funk to Hilbert geometry by Athanase Papadopoulos and Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter 3. Funk and Hilbert geometries from the Finslerian viewpoint by Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 4. On the Hilbert geometry of convex polytopes by Constantin Vernicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry by Cormac Walsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Chapter 6. Characterizations of hyperbolic geometry among Hilbert geometries by Ren Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Part II. Groups and dynamics in Hilbert geometry Chapter 7. The geodesic flow of Finsler and Hilbert geometries by Mickaël Crampon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Chapter 8. Around groups in Hilbert geometry by Ludovic Marquis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

viii

Contents

Chapter 9. Dynamics of Hilbert nonexpansive maps by Anders Karlsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Chapter 10. Birkhoff’s version of Hilbert’s metric and and its applications in analysis by Bas Lemmens and Roger Nussbaum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Part III. Developments and applications Chapter 11. Convex real projective structures and Hilbert metrics by Inkang Kim and Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Chapter 12. Weil–Petersson Funk metric on Teichmüller space by Hideki Miyachi, Ken’ichi Ohshika and Sumio Yamada . . . . . . . . . . . . . . . . . . . . . 339 Chapter 13. Funk and Hilbert geometries in spaces of constant curvature by Athanase Papadopoulos and Sumio Yamada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Part IV. History of the subject Chapter 14. On the origin of Hilbert geometry by Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Chapter 15. Hilbert’s fourth problem by Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Introduction

The project of editing this Handbook arose from the observation that Hilbert geometry is today a very active field of research, and that no comprehensive reference exists for it, except for results which are spread in various papers and a few classical (and very inspiring) pages in books of Busemann. We hope that this Handbook will serve as an introduction and a reference for both beginners and experts in the field. Hilbert geometry is a natural geometry defined in an arbitrary convex subset of real affine space. The notion of convex set is certainly one of the most basic notions in mathematics, and convexity is a rich theory, offering a large supply of refined concepts and deep results. Besides being interesting in themselves, convex sets are ubiquitous; they are used in a number of areas of pure and applied mathematics, such as number theory, mathematical analysis, geometry, dynamical systems and optimization. In 1894 Hilbert discovered how to associate a length to each segment in a convex set by way of an elementary geometric construction and using the cross ratio. In fact, Hilbert defined a canonical metric in the relative interior of an arbitrary convex set. Hilbert geometry is the geometric study of this canonical metric. The special case where the convex set is a ball, or more generally an ellipsoid, gives the Beltrami–Klein model of hyperbolic geometry. In this sense, Hilbert geometry is a generalization of hyperbolic geometry. Hilbert geometry gives new insights into classical questions from convexity theory, and it also provides a rich class of examples of geometries that can be studied from the point of view of metric geometry or differential geometry (in particular Finsler geometry). Let us recall Hilbert’s construction. The line joining two distinct points x and y in a bounded convex domain intersects the boundary of that domain in two other points p and q. Assuming that y lies between x and p, the Hilbert distance from x to y is the logarithm of the cross ratio of these four points: p y x

d.x; y/ D

1 jx log 2 jy

pj jy pj jx

qj : qj

q

This distance was defined in a letter to Klein written by Hilbert in 1894. It is a distance in the usual sense, and the relative interior of the convex set is a complete metric space for this distance. The Hilbert metric is invariant under projective transformations and depends in a monotonic way on the domain: a larger domain induces a smaller Hilbert distance. The Hilbert metric is projective in the sense that the straight

2

Introduction

lines are geodesics. In other words, d.x; y/ D d.x; z/ C d.z; y/ whenever z 2 Œx; y. Furthermore, if the convex domain is strictly convex, then the affine segment is the unique geodesic joining two points. The fourth Hilbert problem asks for a description of all projective metrics in a convex region, that is, metrics for which the straight lines are geodesics. At the beginning of the twentieth century, Hamel, who was a student of Hilbert, worked on this problem; he discovered new examples of projective metrics and found some deep results using differential calculus and the calculus of variations. The subject of Finsler geometry gradually emerged as an independent topic, and at the end of the 1920s, Funk and Berwald gave a differential geometric characterization of Hilbert metrics among all Finsler metrics on a domain with a smooth and strongly convex boundary. All these facts and several others which we describe below are reported on in this Handbook. The deepest and most thorough studies in Hilbert geometry during the twentieth century are due to Busemann and his students and collaborators. During the period from the 1940s to the 1990s this school investigated Hilbert geometry from the viewpoint of metric geometry. A variety of questions regarding these metrics were studied, concerning their geodesics, their convexity theory (convexity of the distance function, of the spheres, etc.), their curvature, area, asymptotic geometry, limit cycles (horocycles) and limit spheres (horospheres), and several other features. For instance, these authors gave several characterizations of the ellipsoid in terms of its Hilbert geometry. They noticed (like Hilbert did before them) that the special case of the simplex is particularly interesting and they studied it in detail. They worked on the metrical properties of the Hilbert metric as well as on the axiomatic theory, making relations with the axioms and the basic notions of Euclidean and of non-Euclidean geometries, in particular the theory of parallels. They established relations between Hilbert geometry and other fields, including the foundations of mathematics, the calculus of variations, convex geometry, Minkowski geometry, geometric group theory and projective geometry. They also developed the basics of the closely related Funk metric. Busemann formulated and initiated the study of several problems of which he (and his collaborators) gave only partial solutions, and a large amount of the research on Hilbert geometry that was done after him is directly or indirectly inspired by his work. Let us briefly mention two further important directions in which the subject developed in the last century. In the late 1950s, Birkhoff found a new proof of the classical Perron–Frobenius theorem on eigenvectors of non-negative matrices based on the Hilbert metric in the positive cone. This new proof brought a new point of view on the subject and initiated a rich generalization of Perron–Frobenius theory. During the same period, Benzécri initiated the theory of divisible convex domains, that is, convex domains admitting a discrete cocompact group of projective transformations. The quotient manifold or orbifold naturally carries a Finsler structure whose universal cover is a Hilbert geometry. The reader will find more information on twentieth

3

Introduction

century developments in Chapters 3, 10 and 15 of this Handbook. During the last fifteen years, the subject grew rapidly and a number of these recent developments are discussed in the other chapters. We now describe the content of the book. The various chapters are written by different authors and are meant to be read independently from each other. Each chapter has its own flavor, due to the variety of tastes and viewpoints of the authors. Although we tried to merge the chapters into a coherent whole, we did not unify the different notation systems, nor did we try to avoid repetitions, hopefully to the benefit of the reader. The book is divided into four parts. Part I contains surveys on Minkowski, Funk and Hilbert geometries and on the relations between them. In Chapter 1, A. Papadopoulos and M. Troyanov treat weak Minkowski spaces. A weak metric on a set is a non-negative distance function ı that satisfies the triangle inequality, but is allowed to be non-symmetric (we may have d.x; y/ 6D d.y; x/) or degenerate (we may have d.x; y/ D 0 for some x 6D y). A weak Minkowski metric on a real vector space is a weak metric that is translation-invariant and projective. The authors define the fundamental concept of weak Minkowski space and they give several examples and counterexamples. The basic results of the theory are stated and proved. Minkowski geometry shares several properties with Hilbert geometry, one of them being that the Euclidean geodesics are geodesics for that geometry. An important observation is that the infinitesimal – or tangential – geometry of a Hilbert or a Funk geometry is of Minkowski type. One of the main results of this chapter is the following: A continuous weak metric ı on Rn is a weak Minkowski metric if and only if it satisfies the midpoint property, that is, ı.p; q/ D 2ı.p; m/ D 2ı.m; q/ for any points p, q where m is the affine midpoint. Other characterizations of weak Minkowski distances are given, providing various important aspects of this geometry. The relations with Busemann’s G-spaces and Desarguesian spaces and comparisons with the Funk and the Hilbert metrics are also highlighted. Chapter 2, From Funk to Hilbert geometry, by the same authors, is devoted to the study of the distance in a convex domain introduced by P. Funk in 1929. Using the notation of the figure on page 1, the Funk distance is defined by Â

Ã

jx pj : F .x; y/ D log jy pj Observe that this distance is a non-symmetric version of the Hilbert distance. Many properties of the Hilbert distance can be obtained as consequences of similar properties of the Funk distance. In this chapter, metric balls, the topology, convexity and orthogonality properties in Funk geometry are studied. A full description of Funk geodesics is also given. In the case of smooth curves, the property can be described as follows: A smooth curve .t / in a convex domain is geodesic for the Funk metric if and only if there is a face D in @ such that the velocity vector P .t / points toward D for all t .

4

Introduction

Chapter 3, by M. Troyanov, concerns the Funk and the Hilbert metrics from the point of view of Finsler geometry. This approach dates back to works done at the end of the 1920s, by Funk and by Berwald, who gave a characterization of Hilbert geometry from the Finslerian viewpoint. Funk and Berwald proved the following theorem: A smooth Finsler metric defined on a convex bounded domain of Rn is the Hilbert metric of that domain if and only if this geometry is complete (in an appropriate sense), if its geodesics are straight lines and if its flag curvature is equal to 1. The author explains these notions in detail and he gives a complete proof of this result. At the same time, the chapter constitutes an introduction to the Finsler nature of the Funk and Hilbert metrics, where the Funk and the Hilbert Finsler structures appear respectively as the tautological and the symmetric tautological Finsler structures on . Chapter 4, On the Hilbert geometry of convex polytopes, by C. Vernicos, concerns the Hilbert geometry of an open set Rn which is a polytope. A bounded convex domain is a polytope if and only if its Hilbert metric is bi-Lipschitz equivalent to a Euclidean space. An equivalent condition is that the domain is isometrically embeddable in a finite-dimensional normed vector space. Another characterization states that the volume growth is polynomial of order equal to the dimension of the convex domain. The author discusses several other aspects of the Hilbert geometry of polytopes. The main goal of Chapter 5, by C. Walsh, is to give an explicit description of the horofunction boundary of a Hilbert geometry. This notion is based on ideas that go back to Busemann but which were formally introduced by Gromov. The results in this chapter are mainly due to Walsh. Walsh gives a sketch of how this boundary may be used to study the isometry group of these geometries. The main result in the chapter is that the group of isometries of a bounded convex polyhedron which is not a simplex coincides with the group of projective transformations leaving the given domain invariant. Let us note that the horofunction boundaries of several other spaces have been described during the last few years (mostly by Walsh), in particular, Minkowski spaces and Teichmüller spaces equipped with the Thurston and with the Teichmüller metrics. These descriptions have also been applied to the characterization of the isometry groups of the corresponding metrics spaces. Chapter 6, by R. Guo, gives a number of characterizations of hyperbolic geometry (or, equivalently, the Hilbert geometry of an ellipsoid) among Hilbert geometries. All these geometric characterizations are formulated in simple geometric terms. Part II concerns the dynamical aspects of Hilbert geometry. It consists of four chapters: Chapters 7 to 10. Chapter 7, by M. Crampon, concerns the geodesic flow of a Hilbert geometry. The study is based on a comparison of this flow with the geodesic flow of a negatively curved Finsler or Riemannian manifold. The main interest is in Hilbert geometries that have some hyperbolicity properties. Such geometries correspond to convex sets with C 1 boundary. In this case, stable and unstable manifolds exist, and there is a relation between the asymptotic behaviour of these manifolds along an orbit of the flow and the shape of the boundary at the endpoint of that orbit. The author then

Introduction

5

studies the particular case of the geodesic flow associated to a compact quotient of a strictly convex Hilbert geometry, and he shows that such a flow satisfies an Anosov property. This property implies some regularity properties at the boundary of the convex set. The author then describes the ergodic properties of the geodesic flow, and he also surveys several notions of entropy that are associated to Hilbert geometry. In Chapter 8, L. Marquis studies Hilbert geometry in the general setting of projective geometry. More precisely, the author surveys the various groups of projective transformations that appear in Hilbert geometry. In the first part of the chapter, he describes the projective automorphism group of a convex set in terms of matrices and then from a dynamical point of view. He shows the existence of convex sets with large groups of symmetries. He exhibits relations with several areas in mathematics, in particular with the theory of spherical representations of semi-simple Lie groups and with Schottky groups. He then explains how Hilbert geometry involves geometric group theory in various contexts. For instance, the Gromov hyperbolicity of a socalled divisible Hilbert metric (that is, one that admits a compact quotient action by a discrete group of isometries) is equivalent to a smoothness property of the boundary of the convex set. Note that the fact that the convex set is divisible means in some sense that it has a large group of symmetries. The other aspects of Hilbert geometry in which group theory is involved include differential geometry, convex affine geometry, real algebraic group theory, hyperbolic geometry, the theories of moduli spaces, of symmetric spaces, of Hadamard manifolds, and the theory of geometric structures on manifolds. In Chapter 9, A. Karlsson considers the dynamical aspect of the theory of nonexpansive (or Lipschitz) maps in Hilbert geometry. He makes relations with works of Birkhoff and Samelson done in the 1950s on Perron–Frobenius theory, and with a more recent work by Nussbaum and Karlsson–Noskov. He explains in particular how the theory of Busemann functions, horofunctions and horospheres in Hilbert geometries appear in the study of nonexpansive maps of these spaces and in their asymptotic theory. This is another example of the fact that a theory which is quite developed in the setting of spaces of negative curvature can be generalized and used in an efficient way in Hilbert geometry, which is not negatively curved (except in the case where the convex set is an ellipsoid). In Chapter 10, B. Lemmens and R. Nussbaum give a thorough survey of the development of the ideas of Birkhoff and Samelson on the applications of Hilbert geometry to the contraction mapping principle and to the analysis of non-linear mappings on cones and in particular to the so-called non-linear Perron–Frobenius theory. The setting is infinite-dimensional. The authors also show how this theory leads to the result that the Hilbert metric of an n-simplex is isometric to a Minkowski space whose unit ball is a polytope having n.n C 1/ facets. Part III contains extensions and generalizations of Hilbert and Funk geometries to various contexts. It consists of three chapters: Chapters 11 to 13. Chapter 11, written by I. Kim and A. Papadopoulos, concerns the projective geometry setting of Hilbert geometry. A Hilbert metric is defined on any convex subset

6

Introduction

of projective space and descends to a metric on convex projective manifolds, which are quotients of convex sets by discrete groups of projective transformations. Thus, it is natural in this Handbook to have a chapter on convex projective manifolds and to study the relations between the Hilbert metric and the other properties of such manifolds. There are several parametrizations of the space of convex projective structures on surfaces; a classical one is due to Goldman and is an analogue of the Fenchel– Nielsen parametrization of hyperbolic structures. A more recent parametrization was developed by Labourie and Loftin in terms of hyperbolic structures on a Riemann surface together with cubic differentials, making use of the Cheng–Yau classification of complete hyperbolic affine spheres. Chapter 11 contains an introduction to these parametrizations and to some related matters. Teichmüller spaces appear naturally in this setting as they are important subspaces of the deformation spaces of convex projective structures of surfaces. The authors also discuss higher-dimensional analogues, covering in particular the work of Johnson–Millson and of Benoist and Kapovich on deformations of convex projective structures on higher-dimensional manifolds. Relations with geodesic currents and topological entropy are also treated. The Hilbert metrics which appear via their length spectra in the parametrization of the deformation spaces of convex projective structures also show up in the compactifications of these spaces. Some of the subjects mentioned are only touched on; this chapter is intended to open up new perspectives. In Chapter 12, S. Yamada, K. Ohshika and H. Miyachi report on a new weak metric which Yamada defined recently on Teichmüller space and which he calls the Weil–Petersson Funk metric. This metric shares several properties of the classical Funk metric. It is defined using a similar variational formula, involving projections on hyperplanes in an ambient space that play the role of support hyperplanes, namely, they are the codimension-one strata of the Weil–Petersson completion of the space. The ambient Euclidean space of the Funk metric is replaced here by a complex which Yamada introduced in a previous work, and which he calls the Teichmüller–Coxeter complex. It is interesting that the Euclidean setting of the classical Funk metric can be adapted to a much more complex situation. In Chapter 13, A. Papadopoulos and S. Yamada survey analogues of the Funk and Hilbert geometries on convex sets in hyperbolic and in spherical geometries. These theories are developed in a way parallel with the classical Funk theory. The existence of a Funk geometry in these non-linear spaces is based on some non-Euclidean trigonometric formulae, and the fact that the analogies can be carried over is somehow surprising because the study of the classical Funk metric on convex subsets of Euclidean space involves a lot of similarity properties and the use of parallels, which do not exist in the non-Euclidean settings. The geodesics of the non-Euclidean Funk and Hilbert metrics are studied, and variational definitions are given of these metrics. These metrics are shown to be Finsler. The Hilbert metric in each of the constant curvature convex sets is also a symmetrization of the Funk metric. The Hilbert metric of a convex subset in a space of constant curvature can also be defined using a notion

Introduction

7

of a cross ratio which is adapted to that space. A relation is made with a generalized form of Hilbert’s Problem IV. Part IV consists of a two chapters which have a historical character. Chapter 14, written by M. Troyanov, contains a brief description of the historical origin of Hilbert geometry. The author presents a summary with comments of a letter of Hilbert to Klein in which Hilbert announces the discovery of that metric. Chapter 15, written by A. Papadopoulos, is a report on Hilbert’s Fourth Problem, one of the famous twentythree problems that Hilbert presented at the second ICM held in Paris in 1900. The problem asks for a characterization and a study of metrics on subsets of projective space for which the projective lines are geodesics. The Hilbert metric is one of the finest examples of metrics that satisfy the requirements of this problem. The author also reports on the relations between this problem and works done before Hilbert on metrics satisfying this requirements, in particular, by Darboux in the setting of the calculus of variations and by Beltrami in the setting of differential geometry. Making historical comments and giving information on the origin of a problem bring into perspective the motivations behind the ideas. The comments that we include also make relations between the theory that is surveyed here and other mathematical subjects. We tried to pay tribute to the founders of the theory as we feel that historical comments usually make a theory more attractive. Several chapters contain questions, conjectures and open problems, and the book also contains a special section on open problems proposed by various authors.

Part I

Minkowski, Hilbert and Funk geometries

Chapter 1

Weak Minkowski spaces Athanase Papadopoulos and Marc Troyanov

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Weak metric spaces . . . . . . . . . . . . . . . . . . . . . . 3 Weak Minkowski norms . . . . . . . . . . . . . . . . . . . 4 The midpoint property . . . . . . . . . . . . . . . . . . . . 5 Strictly and strongly convex Minkowski norms . . . . . . . 6 The synthetic viewpoint . . . . . . . . . . . . . . . . . . . 7 Analogies between Minkowski, Funk and Hilbert geometries References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 12 14 21 25 26 29 30

1 Introduction In the last decade of the 19th century, Hermann Minkowski (1864–1909) initiated new geometric methods in number theory, which culminated with the celebrated Geometrie der Zahlen1 . Minkowski’s work is referred to several times by David Hilbert in his 1900 ICM lecture [15], in particular in the introduction Hilbert declares: The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski’s work, Die Geometrie der Zahlen.

Regarding the influence of this book on the birth of metric geometry, let us mention the following from the paper [10] by Busemann and Phadke, p. 181: 1 Minkowski’s first paper on the geometry of numbers was published in 1891. A first edition of his book Geometrie der Zahlen appeared in 1896, and a more complete edition was posthumously edited by Hilbert and Speiser and published in 1910 [23]. This edition was reprinted several times and translated, and it is a major piece of the mathematical literature of the early 20th century.

12

Athanase Papadopoulos and Marc Troyanov Busemann had read the beginning of Minkowski’s Geometrie der Zahlen in 1926 which convinced him of the importance of non-Riemannian metrics.

An early result of Minkowski in that theory, related to number theory, states that any convex domain in R2 which is symmetric around the origin and has area greater than four contains at least one non-zero point with integer coordinates. One step in Minkowski’s proof amounts to considering a metric on the plane for which the unit ball at any point in R2 (in fact in Z2 ) is a translate of the initial convex domain. Such a metric is not Euclidean, it is translation-invariant and the Euclidean lines are shortest paths. The geometric study of this type of metrics is called (since Hilbert’s writings) Minkowski geometry. We refer to [4], [12], [20], [21], [27] for general expositions of the subject. Minkowski formulated the basic principles of this geometry in his 1896 paper [22], and these principles are recalled in Hilbert’s lecture [15] (Problem IV). An express description of Minkowski geometry is the following: choose a convex set in Rn that contains the origin. For points p and q in Rn , define a number ı > 0 as follows. First dilate by the factor ı and then translate the set in such a way that 0 is sent to p and q lies on the boundary of the resulting set. In other words, ı is defined by the condition q 2 @.p C ı /: (1.1) We denote by ı.p; q/ the number defined in this way. The function ı W Rn Rn ! RC is what we call a weak metric. It satisfies the triangle inequality and ı.p; p/ D 0. It is not symmetric in general and it can be degenerate in the sense that ı.p; q/ D 0 does not imply p 6D q. On the other hand, the straight lines are geodesics for this metric and ı is translation-invariant. Minkowski geometry is the study of such weak metrics. It plays an important role in convexity theory and in Finsler geometry, where Minkowski spaces play the role played by flat spaces in Riemannian geometry. There is a vast literature on Minkowski metrics, and the goal of the present chapter is to provide the reader with some of the basic definitions and facts in the theory of weak Minkowski metrics, because of their relation to Hilbert geometry, and to give some examples. We give complete proofs of most of the stated results. We end this chapter with a discussion about the relations and analogies between Minkowski geometry and Funk and Hilbert geometries.

2 Weak metric spaces We begin with the definition of a weak metric space. Definition 2.1 (Weak metric). A weak metric on a set X is a map ı W X X ! Œ0; 1 satisfying the following two properties: a) ı.x; x/ D 0 for all x in X; b) ı.x; y/ C ı.y; z/ ı.x; z/ for all x, y and z in X .

Chapter 1. Weak Minkowski spaces

13

We often require that a weak metric satisfies some additional properties. In particular one says that the weak metric ı on X is c) separating if x ¤ y implies ı.x; y/ > 0, d) weakly separating if x ¤ y implies max fı.x; y/; ı.y; x/g > 0, e) finite if ı.x; y/ < 1, f) reversible (or symmetric) if ı.y; x/ D ı.x; y/, g) quasi-reversible if ı.y; x/ C ı.x; y/ for some constant C , for all x and y in X . One sometimes says that ı is strongly separating if condition b) holds, in order to stress the distinction with condition d). Observe that for reversible metrics both notions of separation coincide. A metric in the classical sense is a reversible, finite and separating weak metric. Thus, it satisfies 0 < ı.x; y/ D ı.y; x/ < 1 for all x ¤ y in X . Definition 2.2. Let U X be a convex subset of a real vector space X . A weak metric ı in U is said to be projective (or projectively flat) if satisfies the condition ı.x; y/ C ı.y; z/ D ı.x; z/

(2.1)

whenever the three points x, y and z in U are aligned and y 2 Œx; z, the affine segment from x to z (equivalently, if y D tx C .1 t /z for some 0 t 1). The weak metric is strictly projective if it is projective and ı.x; u/ C ı.u; z/ > ı.x; z/ whenever u 62 Œx; z. Definition 2.3 (Weak Minkowski metric). A weak Minkowski metric on a real vector space X is a weak metric ı on X that is translation-invariant and projective. Example 2.4. Let X be a real vector space and ' W X ! R a linear form. Define ı' .x; y/ D maxf0; '.y x/g. Then ı is a weak Minkowski metric. It is finite, but it is neither reversible nor weakly separating. We note that in functional analysis, given a real vector space X, the collection of sets B.';x;r/ D fy 2 X j ı' .x; y/ < rg X; where x 2 X is an arbitrary point, r > 0 and ' 2 X is an arbitrary linear form generates a topology which is called the weak topology on X . This observation is a possible justification for the name “weak metric” that we give to such functions. The terminology has its origin in the work of Ribeiro who was interested around 1943 [26]

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in some generalization of the Urysohn metrization theorem for the topology associated to ı. Example 2.5 (Counterexample). Let X be a real vector space and let k k W X ! R be a norm on X. Then ı.x; y/ D maxfky xk; 1g is a metric that is translation-invariant, but it is not a Minkowski metric because it is not projective. Indeed, suppose kzk D 1, then ı.0; 2z/ D 1 < ı.0; z/ C ı.z; 2z/ D 2: In this example, the metric is “projective for small distances”, in the sense that if kz xk 1 and y 2 Œx; z, then (2.1) holds. On the other hand, large closed balls are not compact; in fact any ball of radius 1 is equal to the whole space X . Example 2.6 (Counterexample). This is a variant of the previous example. Let again k k W X ! R be a norm on the real vector space X. Then ˛ .x; y/ D ky xk˛ is a metric if and only if 0 < ˛ 1. It is clearly translation-invariant, but it is not projective if ˛ < 1, and thus it is not a Minkowski metric. Unlike the previous metric ı, the metric ˛ is not projective for small distances (if ˛ < 1). On the other hand, every closed ball is compact.

3 Weak Minkowski norms Proposition 3.1. Let ı be a weak Minkowski metric on some real vector space X and set F .x/ D ı.0; x/. Then the function F W X ! Œ0; 1 satisfies the following properties. i) F .x1 C x2 / F .x1 / C F .x2 / for all x1 ; x2 2 X. ii) F .x/ D F .x/ for all x 2 X and for all 0. Proof. The first property is a consequence of the triangle inequality together with the fact that ı is translation-invariant: F .x C y/ D ı.0; x C y/ ı.0; x/ C ı.x; x C y/ D ı.0; x/ C ı.0; y/ D F .x/ C F .y/: To prove the second property, observe for any x 2 X and any ; 0 we have ı.0; x/ C ı.x; . C /x/ D ı.0; . C /x/; because x belongs to the segment Œ0; . C /x. Since we have ı.x; . C /x/ D ı.0; x/ D F .x/, the previous identity can be written as F .x/ C F .x/ D F .. C /x/

(3.1)

Chapter 1. Weak Minkowski spaces

15

and we conclude from the next lemma that F .x/ D F .x/ for all > 0. We also have F .0 x/ D 0 F .x/ D 0 since F .0/ D ı.0; 0/ D 0. Lemma 3.2. Let f W RC ! Œ0; 1 be a function such that f . C / D f ./ C f ./ for any ; 2 RC , then f ./ D f .1/ for every > 0. Proof. We first assume f .a/ < 1 for every a 2 RC . We have by hypothesis f .k a/ D f ...k 1/ C 1/ a/ D f ..k 1/ a/ C f .a/ for any k 2 N. We thus have by induction f .k a/ D k f .a/ for any k 2 N and any a 2 RC . Using the above identity with k; m 2 N, we have

k k Df m D f .k/ D k f .1/: mf m m Dividing both sides of the identity by m, we obtain f .˛/ D ˛f .1/ for any ˛ 2 QC . Consider now 2 RC arbitrary, and choose ˛1 ; ˛2 2 QC such that ˛1 < < ˛2 . Then f ./ D f .˛1 / C f . ˛1 / > f .˛1 / D ˛1 f .1/ and f ./ D f .˛2 / f .˛2 / < f .˛2 / D ˛2 f .1/: Since ˛2 ˛1 > 0 is arbitrarily small, we deduce that f ./ D f .1/ for any > 0. So far we assumed f .a/ < 1 for any a > 0. Assume now there exists a > 0 such that f .a/ D 1. Then f ./ D 1 for any > 0. Indeed choose an integer k such that k > a. Then kf ./ D f .k/ D f .k a/ C f .a/ f .a/ D 1: Therefore f ./ D f .1/ D 1. Definition 3.3. A function F W X ! Œ0; 1 defined on a real vector space X is a weak Minkowski norm if the following two conditions hold: i) F .x1 C x2 / F .x1 / C F .x2 / for all x1 ; x2 2 X; ii) F .x/ D F .x/ for all x 2 X and for all 0. Proposition 3.1 states that a weak Minkowski metric determines a weak Minkowski norm. Conversely, a weak Minkowski norm defines a weak Minkowski metric ıF by the formula (3.2) ıF .x; y/ D F .y x/: We then naturally define a weak Minkowski norm F to be

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• separating if x ¤ 0 implies F .x/ > 0; • weakly separating if x ¤ 0 implies maxfF .x/; F .x/g > 0; • finite if F .x/ < 1; • reversible (or symmetric) if F .x/ D F .x/; for any x 2 X. Example 3.4. The function F W RN ! Œ0; 1 defined by F ..xk // D sup maxfxk ; 0g k2N

is a weak Minkowski norm which is neither finite, nor separating, nor symmetric. It is however weakly separating. A weak Minkowski norm with similar properties also exists in finite dimensions: Example 3.5. The function F W R2 ! Œ0; 1 defined by F .x1 ; x2 / D maxfx1 ; 0g if x2 D 0 and F .x1 ; x2 / D 1 if x2 ¤ 0 is a weak Minkowski norm with the same properties: it is neither finite, nor separating, nor symmetric, but it is weakly separating. Observe that in both examples F is finite on some vector subspace of R2 . This is a general fact: Proposition 3.6. Let F W X ! Œ0; 1 be a weak Minkowski norm on the real vector space X and set DF D fx 2 X j F .x/ < 1g. Then DF is a vector subspace of X. Furthermore, the restriction of F to any finite-dimensional subspace E DF is continuous. Proof. If x; y 2 DF , then F .x/ and F .y/ are finite and therefore F .x C y/ F .x/ C F .y/ < 1 and F .x/ D F .x/ < 1. Therefore x C y 2 DF and x 2 DF , which proves the first assertion. To prove the second assertion, we consider a finite-dimensional subspace E DF and we choose a basis e1 ; e2 ; : : : ; em 2 E. Define the constant C D max .F .ej / C F .ej //: 1j m

For an arbitrary vector x D

Pm

j D1

F .x/

˛j ej 2 E, we then have

m X j D1

F .˛j ej / C

m X

j˛j j:

j D1

In particular, if x ! 0, then F .x/ ! 0. More generally, if a sequence x 2 E converges to some a 2 E, then lim sup F .x / D lim sup F .a C .x a// F .a/ C lim sup F ..x a// D F .a/: !1

!1

!1

Chapter 1. Weak Minkowski spaces

17

Since F .a/ F .x / C F .a x / we also have F .a/ lim inf .F .x / C F .a x // D lim inf F .x /: !1

!1

It follows that lim sup F .x / F .a/ lim inf F .x /; !1

!1

and the continuity on E follows. Corollary 3.7. Any weak Minkowski norm on a finite-dimensional vector space X is lower semi-continuous. Proof. We need to prove that F .a/ lim inf !1 F .x / for every sequence x 2 X converging to a. If F .a/ D 1, then a belongs to the open set X n DF . It follows that x 62 DF for large enough and therefore lim inf F .x / D 1 D F .a/: !1

If F .a/ < 1, then two cases may occur. If infinitely many x belong to DF , then by the previous proposition we have F .a/ D

lim

!1;x 2DF

F .x / D lim inf F .x /: !1

If on the other hand DF contains only finitely many elements of the sequence x , then lim inf F .x / D 1 > F .a/: !1

Definition 3.8. Given a weak Minkowski norm F on a vector space X, we define the open and closed unit balls at the origin as F D fx 2 X j F .x/ < 1g

and

x F D fx 2 X j F .x/ 1g:

The set F D fx 2 X j F .x/ D 1g is called the unit sphere or the indicatrix of F . Proposition 3.9. Let F be a weak Minkowski norm on a finite-dimensional vector space X . Then the following are equivalent: (1) F is finite; (2) F is continuous; (3) F is open; (4) 0 is an interior point of F . Proof. The implication (1) ) (2) is Proposition 3.6 and the implications (2) ) (3) ) (4) are obvious. To prove (4) ) (1), we suppose that F is not finite. Then there exists a 2 X such that F .a/ D 1. Thus, F .a/ D 1 for all > 0, in particular a 62 F for all > 0 and therefore 0 is not an interior point of F .

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Proposition 3.10. Let F be a weak Minkowski norm on Rn . Then the following are equivalent: (1) F is separating (i.e. F .x/ > 0 for all x ¤ 0); (2) F is bounded below on the Euclidean unit sphere S n1 U; x F is bounded. (3) Proof. (1) ) (2): Suppose that F is not bounded below on S n1 . Then there exists a sequence xj 2 S n1 such that F .xj / ! 0. Choosing a subsequence if necessary, we may assume, by compactness of the sphere, that F .xj / < 1 for all j , i.e. xj 2 DF \ S n1 , and that xj converges to some point x0 2 DF \ S n1 . Since F is continuous on DF , we have F .x0 / D limj !1 F .xj / D 0. Since x0 ¤ 0 (it is a point on the sphere), it follows that F is not separating. (2) ) (3): Condition (2) states that there exists > 0 such that F .x/ for all x 2 S n1 . Therefore F .y/ 1 implies kyk 1 . (3) ) (1): Suppose F is non-separating. Then there exists x ¤ 0 with F .x/ D 0. x F which is therefore Therefore F .x/ D 0 for any > 0. In particular RC x unbounded. Definition 3.11. A Minkowski norm is a weak Minkowski norm that is finite and separating. It is simply called a norm if it is furthermore reversible. To a finite and separating norm is associated a well-defined topology, viz. the topology associated to the symmetrization of the weak metric defined by Equation (3.2) (which is a genuine metric). For a deeper investigation of various topological questions we refer to the book [11] by S. Cobzas. Corollary 3.12. The topology defined by the distance (3.2) associated to a Minkowski norm on Rn coincides with the Euclidean topology. Proof. Proposition 3.6 implies that F is continuous. From the compactness of the Euclidean unit sphere S n1 we thus have a constant > 0 such that F .x/ 1 for all points x on S n1 . It follows that kxk F .x/

1 kxk

(3.3)

for all x 2 Rn and therefore F induces the same topology as the Euclidean norm. The next result shows how one can reconstruct the weak Minkowski norm from its unit ball. Proposition 3.13. Let Rn be a convex set containing the origin. Define a function F W Rn ! Œ0; 1 by F .x/ D infft 0 W x 2 t g:

(3.4)

19

Chapter 1. Weak Minkowski spaces

x F , that is, Then F is a weak Minkowski norm and the closure of coincides with x D fx 2 Rn j F .x/ 1g: Moreover, if is open, then D fx 2 Rn j F .x/ < 1g. The function F defined by (3.4) is called the Minkowski functional of . Proof. We need to verify the two conditions in Definition 3.3. For > 0, we have F .x/ D inffs 0 W x 2 s g s D inffs 0 W x 2 g D infft 0 W x 2 t g .sDt /

D F .x/: Now because is convex we have for s; t > 0 s xs C t y xCy x 2 and 2 H) D s s sCt sCt

y t

2 :

Therefore F .x/ < s and F .y/ < t H) F .x C y/ < s C t; which is equivalent to F .x C y/ F .x/ C F .y/. This proves the first part of the proposition. To prove the remaining assertions, observe that F .x/ 1 means that tx 2 for x This shows that any 0 < t < 1 and thus x 2 . x fx 2 Rn j F .x/ 1g : x fx 2 Rn j F .x/ 1g follows from the lower semiThe converse inclusion continuity of F (Corollary 3.7). Finally, if is open, then F is continuous (Proposition 3.9) and therefore D fx 2 Rn j F .x/ < 1g. As a result, we have established one-to-one correspondences between weak Minkowski metrics on Rn , weak Minkowski norms and closed convex sets containing the origin. The closed convex set associated to a weak Minkowski norm F is the set x F D fx 2 X j F .x/ 1g. The associated weak metric is separating if and only if the associated convex set is bounded and the metric is finite if and only if the origin is an interior point of the convex set. Remark 3.14. These concepts have some important consequences in convex geometry. For instance one can easily prove that every unbounded convex set in Rn must contain a ray. Indeed, let Rn be unbounded and convex. One may assume that contains the origin. Then, by Proposition 3.10, its weak Minkowski functional F is not separating, that is, there exists a ¤ 0 in Rn such that F .a/ D 0; but then F .a/ D 0 for every > 0 and therefore the ray contains the ray RC a. Let us conclude this section with two important results from Minkowski geometry. A Minkowski norm on Rn is said to be Euclidean if it is associated to a scalar product.

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Proposition 3.15. Let ı be a Minkowski metric on Rn . Then ı is a Euclidean metric if and only if the ball B.a;r/ D fx 2 X j ı.a; x/ < rg Rn (for some a 2 Rn and r > 0, or, equivalently, for any a 2 Rn and r > 0) is an ellipsoid centered at a. Notice that the above proposition is false if the ellipsoid is not centered at a. Proof. Recall that, by definition, an (open) ellipsoid is a convex set in Rn that is the affine image of the open Euclidean unit ball. If the weak metric ı is Euclidean, then it is obvious that every ball is an ellipsoid. Conversely, suppose that some ball of an arbitrary Minkowski metric ı is an ellipsoid. Then the ball with the same radius centered at the origin is also an ellipsoid since ı is translation-invariant, that is, B.0;r/ D fx 2 X j F .x/ D ı.0; x/ < rg is an ellipsoid. But then 1 B.0;1/ r is also an ellipsoid. Changing coordinates if necessary, one may assume that ˚ P D x 2 Rn j i xi2 < 1 ; D B.0;1/ D

which is the Euclidean unit ball. It follows that F .x/ D infft > 0 j x 2 t g D infft > 0 j kxk < t g D kxk where k k denotes the Euclidean norm. We have the following result on the isometries of a Minkowski metric. Theorem 3.16. Let ı be a Minkowski metric on Rn . Then every isometry of ı is an affine transformation of Rn , and the group Iso.Rn ; ı/ of isometries of ı is conjugate within the affine group to a subgroup of the group E.n/ of Euclidean isometries of Rn . Furthermore Iso.Rn ; ı/ is conjugate to the full group E.n/ if and only if ı is a Euclidean metric. Proof. The first assertion is the Mazur–Ulam Theorem, see [18]. To prove the second assertion, we recall that every bounded convex set in Rn with non-empty interior contains a unique ellipsoid J of maximal volume, called the John ellipsoid of , see [1]. Let us consider the unit ball D B.ı;0;1/ of our Minkowski metric and let us denote by J its John ellipsoid and by z 2 J its center. We call J D J z the centered John ellipsoid of . Consider now an arbitrary isometry g 2 Iso.Rn ; ı/. Set g.x/ Q D g.x/ b, where b D g.0/. Then gQ is an isometry for ı fixing the origin. By construction and uniqueness, the centered John ellipsoid is invariant: g.J Q / D J .

Chapter 1. Weak Minkowski spaces

21

There exists an element A 2 GL.n; R/ such that AJ D B is the Euclidean unit ball. Let us set f WD A B gQ B A1 . Then f .B/ D B: By the Mazur–Ulam theorem, gQ is a linear map, therefore f is a linear map preserving the Euclidean unit ball, which means that f 2 O.n/. We thus obtain g.x/ D A1 .f .x/ C Ab/A where A is linear and x 7! f .x/ C Ab is a Euclidean isometry. To prove the last assertion, one may assume, changing coordinates if necessary, that Iso.Rn ; ı/ D E.n/. Then the ı-unit ball is invariant under the orthogonal group O.n/ and it is therefore a round sphere. We now conclude from Proposition 3.15 that ı is Euclidean.

4 The midpoint property Definition 4.1. A weak metric ı on the real vector space X satisfies the midpoint property if for any p; q 2 X we have ı.p; m/ D ı.m; q/ D

1 ı.p; q/ 2

where m D 12 .p C q/ is the affine midpoint of p and q. To describe the main features of this property, we shall use the notion of dyadic numbers. Definition 4.2. A dyadic number is a rational number of the type D 2k m with m; k 2 Z. We denote the set of dyadic numbers by DD

1 [

2k Z;

kD0

and the subset of non-negative dyadic numbers by DC D: Proposition 4.3. Let ı be a weak metric on the real vector space X . Then ı satisfies the midpoint property if and only if for any pair of distinct points p; q 2 X and for any ; in D with we have ı../; .// D . / ı.p; q/;

(4.1)

where .t / D tp C .1 t /q. Proof. It is obvious that if (4.1) holds, then ı satisfies the midpoint property. The proof of the other direction requires several steps. Assume that ı satisfies the midpoint

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Athanase Papadopoulos and Marc Troyanov

property. Then we have 1 ı.p; q/ and 2 By an induction argument, we then have ı.p; . 12 // D

ı.p; .2/ D 2ı.p; q/:

ı.p; .2m // D 2m ı.p; q/

(4.2)

for any k 2 N. Because p is the midpoint of .2m / and .2m /, we deduce that ı..2m /; .2m // D 2mC1 ı.p; q/:

(4.3)

Now we have, for k 2 Z, ı..k 1/; .k// D ı..k/; .k C 1// since .k/ is the midpoint of .k 1/ and .k C1/. Because ı.p; q/ D ı..0/; .1//, we deduce that ı..k/; .k C 1// D ı.p; q/; and by the triangle inequality we have ı..i/; .j // .j i /ı.p; q/

(4.4)

for any i; j 2 Z with i < j . We show that this inequality is in fact an equality. Choose m 2 N with 2m max.jij; jj j//. Then we have from (4.3) and (4.4) 2mC1 ı.p; q/ D ı..2m /; .2m // ı..2m /; .i// C ı..i /; .j // C ı..j /; .2m //: Using now (4.4) we have ı..i/; .j // .j i /ı.p; q/, but also ı..2m /; .i// .i C 2m /ı.p; q/; and ı..j /; .2m // .2m j /ı.p; q/: Since .i C 2k / C .j i/ C .2m j / D 2mC1 ; all the above inequalities must be equalities. Thus, we have established that ı..i/; .j // D .j i /ı.p; q/ for any i; j 2 Z. Let us now fix k 2 N and set qk D .2k / and k .t/ D .t 2k / D tp C .1 t /qk : Applying (4.5) to k we have ı.k .i/; k .j // D .j i/ı.p; qk / D .j i /2k ı.p; q/:

(4.5)

Chapter 1. Weak Minkowski spaces

23

The last equality can be rewritten as ı.. 2ik /; . 2jk // D . 2jk

i /ı.p; q/ 2k

for any i; j 2 Z and k 2 N, which is equivalent to (4.1) for any dyadic numbers , with . The next result is a generalization to the case of weak metrics of a characterization of Minkowski geometry due to Busemann, see §17 in [4]. Theorem 4.4. A finite weak metric ı on Rn is a weak Minkowski metric if and only if it satisfies the midpoint property and if its restriction to every affine line is continuous. More precisely, the last condition means that if a and b are two points in Rn , then for any t0 2 R we have lim ı..t /; b// D ı..t0 /; b//

t!t0

and lim ı.a; .t // D ı.a; .t0 //;

t!t0

where .t / D t a C .1 t /b. Proof. If ı is a Minkowski metric, then it is projective and since ı is finite (by hypothesis), it follows from Propositions 3.1 and 3.6 that the distance is given by ı.x; y/ D F .y x/; where F is a weak Minkowski norm. The continuity of ı follows now from Proposition 3.6 and the midpoint property follows from property (ii) in Proposition 3.1. Conversely, let us assume that the weak metric ı satisfies the midpoint property and that it is continuous on every line. We need to show that ı is projective and translation-invariant. We first observe that if a; b 2 Rn are two distinct points with ı.a; b/ ¤ 0 and if x and y are two points aligned with a and b such that .y x/ is a non-negative multiple of .b a/, then jy xj ı.x; y/ D ; (4.6) ı.a; b/ jb aj where jq pj denotes the Euclidean distance between p and q in Rn . This follows from Proposition 4.3 together with the continuity of ı on lines and the density of D in R. This immediately implies that ı.p; z/ C ı.z; q/ D ı.p; q/ whenever z 2 Œp; q, meaning that the weak metric ı is projective. To prove the translation invariance, we consider four points p, q, p 0 , q 0 with 0 .q p 0 / D .q p/. If the four points are on a line, then (4.6) implies that ı.p 0 ; q 0 / D ı.p; q/. If the four points are not on a line, then pqq 0 p 0 is a nonC the degenerate parallelogram. Assume also that 0 < ı.p; q/ < 1 and denote by Lpq

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Athanase Papadopoulos and Marc Troyanov

ray with origin p through q and by Lqq 0 the line passing through q and q 0 . Choose a C such that jyj pj ! 1 and set xj D Lp0 yj \ LC sequence yj 2 Lpq qq 0 . yj

xj

q0

q

p

p0

We then have 1

ı.p; yj / ı.p; p 0 / ı.p 0 ; yj / ı.p; p 0 / D ı.p; yj / ı.p; yj / ı.p; yj / 0 ı.p ; p/ C ı.p; yj / ı.p; yj / 0 ı.p ; p/ C 1: D ı.p; yj /

Using (4.6), we have ı.p; yj / ! 1, therefore ı.p 0 ; xj / ı.p 0 ; yj / D lim D 1: j !1 ı.p; q/ j !1 ı.p; yj / lim

Because xj ! q 0 on the line Lqq 0 , we have by hypothesis lim ı.xj ; q 0 / D lim ı.q 0 ; xj / D 0;

j !1

j !1

and since ı.p 0 ; q 0 / ı.xj ; q 0 / ı.p 0 ; xj / ı.p 0 ; q 0 / C lim ı.q 0 ; xj /; j !1

0

0

0

we have ı.p ; xj / ! ı.p ; q /. Therefore ı.p 0 ; xj / ı.p 0 ; q 0 / D lim D 1: j !1 ı.p; q/ ı.p; q/ It follows that for a non-degenerate parallelogram pqq 0 p 0 , we have ı.p 0 ; q 0 / D ı.p; q/. Suppose now that ı.p; q/ D 0. Then we also have ı.p 0 ; q 0 / D 0 for otherwise, exchanging the roles of p, q and p 0 , q 0 in the previous argument, we get a contradiction. We thus established that in all cases ı.p 0 ; q 0 / D ı.p; q/ if q 0 p 0 D q p. In other words, ı is translation-invariant. Since it is projective, this completes the proof that it is a weak Minkowski metric.

Chapter 1. Weak Minkowski spaces

25

Example 4.5 (Counterexample). Let X a be real vector space and let h W X ! R be an injective Q-linear map. Then the function ı W X X ! R defined by ı.x; y/ D jh.x/ h.y/j is a metric which is translation-invariant and satisfies the midpoint property. Yet it is in general not projective (unless h is R-linear, and thus dimR .X / D 1).

5 Strictly and strongly convex Minkowski norms Definition 5.1. (i) Let F be a (finite and separating) Minkowski norm in Rn with unit ball F . Then F is said to be strictly convex if the indicatrix @F contains no non-trivial segment, that is, if for any p; q 2 @F , we have Œp; q @ H) p D q: (ii) The function F is said to be strongly convex if F is smooth on Rn n f0g and the hypersurface @F Rn has everywhere positive Gaussian curvature. Equivalently, the Hessian 1 @2 ˇˇ F 2 .y C u1 1 C u2 2 / (5.1) gy . 1 ; 2 / D ˇ 2 @u1 @u2 u1 Du2 D0 of F 2 .y/ is positive definite for any point y 2 Rn n f0g. There are several equivalent definitions of strict convexity in Minkowski spaces, see e.g. [12], [23]. It is clear that a strongly convex Minkowski norm is strictly convex. The converse does not hold: the Lp -norm kykp D

n X

jyj jp

1=p

j D1

is an example of a smooth strictly convex norm which is not strongly convex. Proposition 5.2. Let F be a strongly convex Minkowski norm on Rn . Then F can be recovered from its Hessian via the formula q F .y/ D gy .y; y/; (5.2) where gy is defined by (5.1). This result follows by applying twice the following lemma, which is sometimes called the Euler Lemma.

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Lemma 5.3. Let W R n 0 ! R be a positively homogeneous functions of degree r. @ If is of class C k for some k 1, then the partial derivatives @y i are positively homogenous functions of degree r 1 and r

.y/ D

n X

yi

iD1 @ In particular, y i @y i D 0 if

@ : @y i

is 0-homogenous.

Recall that a function W Rn n 0 ! R is said to be positively homogenous of degree r if .y/ D r .y/ for all y 2 Rn n 0 and all > 0. Proof. This is elementary: we just differentiate the function t 7! obtain @ .ty/ y i D rt r1 .y/; @y i

.ty/ D t r .y/ to

and we set t D 1. If F is a strongly convex Minkowski norm on Rn , then Formula (5.1) defines a Riemannian metric gy on Rn n f0g. Using Lemma 5.3, on gets that gy is invariant under homothety, that is, we have gy D gy for every > 0 and y 2 Rn n f0g. Furthermore, F is determined from this metric by Equation (5.2). We conclude from these remarks the following: Proposition 5.4. There is a natural bijection between strongly convex Minkowski norms on Rn and Riemannian metrics This observation can be used as a founding stone for Minkowski geometry, see e.g. [28], and it plays a central role in Finsler geometry.

6 The synthetic viewpoint Definition 2.3 of a weak Minkowski space is based on a real vector space X as a ground space. In fact, only the affine structure of that space plays a role and we could equivalently start with a given affine space instead of a vector space. The synthetic viewpoint is to start with an abstract metric space and to try to give a list of natural conditions implying that the given metric space is Minkowskian. This question, and similar questions for other geometries, has been a central and recurrent question in the work of Busemann, and it is implicit in Hilbert’s comments on his Fourth Problem [15]. Some answers are given in Busemann’s book The Geometry of Geodesics [4]. In that book Busemann introduces the notions of G-spaces and Desarguesian spaces. The goal of this section is to give a short account on this viewpoint. We restrict ourselves to the case of ordinary metric spaces.

Chapter 1. Weak Minkowski spaces

27

Definition 6.1 (Busemann G-space). A Busemann G-space is a metric space .X; d /, satisfying the following four conditions: (1) (Menger Convexity) Given distinct points x; y 2 X, there is a point z 2 X different from x and y such that d.x; z/ C d.z; y/ D d.x; y/. (2) (Finite Compactness) Every d -bounded infinite set has an accumulation point. (3) (Local Extendibility) For every point p 2 X, there exists rp > 0, such that for any pair of distinct points x; y 2 X in the open ball B.p; rp /, there is a point z 2 B.p; rp / n fx; yg such that d.x; y/ C d.y; z/ D d.x; z/. (4) (Uniqueness of Extension) Let x; y; z1 ; z2 be four points in X such that d.x; y/C d.y; z1 / D d.x; z1 / and d.x; y/Cd.y; z2 / D d.x; z2 /. Suppose that d.y; z1 / D d.y; z2 /, then z1 D z2 . A typical example of a Busemann G-space .X; d / is a strongly convex Finsler manifold of class C 2 . (In fact class C 1;1 suffices, by a result of Pogorelov.) It follows from the definition that any pair of points in a Busemann G-space .X; d / can be joined by a minimal geodesic and that geodesics are locally unique. It is also known that every G-space is topologically homogeneous and that it is a manifold if its dimension is at most 4. We refer to [2] for further results on the topology of G-spaces. Among G-spaces, Busemann introduced the class of Desarguesian spaces. Definition 6.2 (Desarguesian space). A Desarguesian space is a metric space .X; d / satisfying the following conditions: (1) .X; d / is a Busemann G-space. (2) .X; d / is uniquely geodesic, that is, every pair of points can be joined by a unique geodesic. (3) If the topological dimension2 of X equals 2, then Desargues’ theorem holds for the family of all geodesics. (4) If the topological dimension of X is greater than 2, then any triple of points lie in a plane, that is, a two-dimensional subspace of X which is itself a G-space. The reason for assuming Desargues’ property in the two-dimensional case as an axiom is due to the well-known fact from axiomatic geometry that it is possible to construct exotic two-dimensional planes in which the axioms of real projective or affine geometry are satisfied but which are not isomorphic to RP 2 or R2 (an example of such exotic object is the Moufang plane); these objects do not satisfy the Desargues property. Similar objects do not exist in higher dimensions and Desargues’ property is a theorem in all dimensions 3. In fact, Klein showed in his paper [17] that Desargues’ theorem in the plane, although a theorem of projective geometry, cannot be proved using only 2 On page 46 in [4], regarding this definition of Desarguesian space, Busemann states that he uses the Menger– Urysohn notion of dimension, but any reasonable notion of topological dimension is equivalent for a G-space.

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two-dimensional projective geometry. Hilbert, in his Grundlagen (2nd. ed., §23), constructed a plane geometry in which all the axioms of two-dimensional projective geometry hold but where the theorem of Desargues fails. This theorem follows from the axioms of three-dimensional projective geometry. Condition (3) in the above definition could be rephrased as follows: If X is two-dimensional, then it can be isometrically embedded in a three-dimensional Desarguesian space. We refer to [4] and to Chapter 15 of this volume [24] for further discussion of Desarguesian spaces. A deep result of Busemann states that a Desarguesian space can be mapped onto a real projective space or on a convex domain in a real affine space with a projective metric. More precisely he proved the following Theorem 6.3 (Theorems 13.1 and 14.1 in [4]). Given an n-dimensional Desarguesian space .X; d /, one of the following condition holds: (1) Either all the geodesics are topological circles and there is a homeomorphism ' W X ! RP n that maps every geodesic in X onto a projective line; (2) or there is a homeomorphism from X onto a convex domain C in Rn that maps every geodesic in X onto the intersection of a straight line with C . Using the notion of Desarguesian space and following Busemann, we now give two purely intrinsic characterizations of finite-dimensional Minkowski spaces among abstract metric spaces. Note that a Minkowski space .X; d / is a G-space if and only if its unit ball is strictly convex. The first result is a converse to that statement. Theorem 6.4 ([4], Theorem 24.1). A metric space .X:d / is isometric to a Minkowski space if and only if it is a Desarguesian space in which the parallel postulate holds and the spheres are strictly convex. We refer the reader to [4], p. 141, for a discussion of the parallel postulate in this context. Observe that in a Desarguesian space there are well defined notions of lines and planes and therefore Euclid’s parallel postulate can be formulated. Using Theorem 6.3 and the parallel postulate, we obtain that .X; d / is isometric to Rn with a projectively flat metric. To prove the theorem, Busemann uses the strict convexity of spheres to establish the midpoint property. The next result involves the notion of Busemann zero curvature. Recall that a geodesic metric space is said to have zero curvature in the sense of Busemann if the distance between the midpoints of two sides of an arbitrary triangle is equal to half the length of the remaining side. Busemann formulates the following characterization: Theorem 6.5 ([4], Theorem 39.12). A simply connected finite-dimensional G-space of zero curvature is isometric to a Minkowski space. Busemann came back several times to the problem of characterizing Minkowskian and locally Minkowskian spaces. In his paper with Phadke [9], written 25 years after

Chapter 1. Weak Minkowski spaces

29

[4], he gave sufficient conditions that are more technical but weaker than those of Theorem 6.5.

7 Analogies between Minkowski, Funk and Hilbert geometries Given a Minkowski metric ı in Rn whose unit ball at the origin is open and bounded, the distance between two points is obtained by setting ı.x; x/ D 0 for all x in Rn and, for x 6D y, jx yj ı.x; y/ D j0 aC j where j j denotes the Euclidean metric and the point aC is the intersection with @ of the ray starting at the origin 0 of Rn and parallel to the ray R.x; y/ from x to y. This formula is equivalent to (1.1) and it suggest an analogy with the formula for the Funk distance in the domain (see Definition 2.1 in Chapter 2 of this volume [25]). It is also in the spirit of the following definition of Busemann ([4], Definition 17.1): A metric d.x; y/ in Rn is Minkowskian if for the Euclidean metric e.x; y/ the distances d.x; y/ and e.x; y/ are proportional on each line. Minkowski metrics share several important properties of the Funk and the Hilbert metrics, and it is interesting to compare these three classes of metrics. Let us quickly review some of the analogies. We start by recalling that in the formulation of Hilbert’s fourth problem which asks for the construction and the study of metrics on subsets of Euclidean (or of projective) space for which the Euclidean segments are geodesics, the Minkowski and Hilbert metrics appear together as the two examples that Hilbert gives (see [15] and Chapter 15 in this volume [24]). A rather simple analogy between the Minkowski and the Funk geometries is that both metrics are uniquely geodesic if and only if their associated convex sets are strictly convex. (Here, the convex set associated to a Minkowski metric is the unit ball centered at the origin (Definition 3.8). The convex set associated to a Funk metric is the set on which this metric is defined.) Another analogy between Minkowski and Hilbert geometries is the well-known fact that a Minkowski weak metric on Rn is Riemannian if and only if the associated convex set is an ellipsoid, see Proposition 3.15. This fact is (at least formally) analogous to the fact that the Hilbert geometry of an open bounded convex subset of Rn is Riemannian if and only if this convex set is an ellipsoid (see [16], Proposition 19). As a further relation between Minkowski and Hilbert geometries, let us recall a result attributed to Nussbaum, de la Harpe, Foertsch and Karlsson. Nussbaum and de la Harpe proved (independently) in [19] and [14] that if Rn is the interior of the standard n-simplex and if H denotes the associated Hilbert metric, then the metric space .; H / is isometric to a Minkowski metric space. Foertsch and Karlsson proved the converse in [13], thus completing the result saying that a bounded open

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convex subset of Rn equipped with its Hilbert metric is isometric to a Minkowski space if and only if is the interior of a simplex. It should be noted that the result (in both directions) was already known to Busemann since 1967. In their paper [8], p. 313, Busemann and Phadke write the following, concerning the simplex: The case of general dimension n is most interesting. The (unique) Hilbert geometry possessing a transitive abelian group of motions where the affine segments are the chords (motion means that both distance and chords are preserved) is given by a simplex S, ([5], p. 35). If we realize [the interior of the simplex] as the first quadrant xi > 0 of an affine coordinate system, the group is given by xi0 D ˇi xi , ˇi > 0 [...] m is a Minkowski metric because it is invariant under the translations and we can take the affine segments as chords.

We finally mention the following common characterizations of Minkowski–Funk geometries and of Minkowski–Hilbert geometries: Theorem 7.1 (Busemann [6], p. 38). Among noncompact and nonnecessarily symmetric Desarguesian spaces in which all the right and left spheres of positive radius around any point are compact, the Hilbert and Minkowski geometries are characterized by the property that any isometry between two (distinct or not) geodesics is a projectivity. Theorem 7.2 (Busemann [7]). A Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space. Acknowledgement. The first author is partially supported by the French ANR project FINSLER.

References [1]

A. Barvinok, A course in convexity. Grad. Stud. in Math. 54, Amer. Math. Soc., Providence, RI, 2002.

[2]

V. N. Berestovski˘ı, D. M. Halverson and D. Repovš, Locally G-homogeneous Busemann G-spaces. Differential Geom. Appl. 29 (2011), no. 3, 299–318.

[3]

H. Busemann, The foundations of Minkowskian geometry. Comment. Math. Helv. 24 (1950), 156–187.

[4]

H. Busemann, The geometry of geodesics. Academic Press, New York 1955; reprinted by Dover in 2005.

[5]

H. Busemann, Timelike spaces. Dissertationes Math. (Rozprawy Mat.) 53, Warsaw 1967.

[6]

H. Busemann, Recent synthetic differential geometry. Ergeb. Math. Grenzgeb. 54, Springer-Verlag, Berlin 1970.

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[7]

H. Busemann, Spaces with homothetic spheres. J. Geometry 4 (1974), 175–186.

[8]

H. Busemann and B. B. Phadke, A general version of Beltrami’s theorem in the large. Pacific J. Math. 115 (1984), 299–315.

[9]

H. Busemann and B. B. Phadke, Minkowskian geometry, convexity conditions and the parallel axiom. J. Geometry 12 (1979), no. 1, 17–33.

[10] H. Busemann and B. B. Phadke, Novel results in the geometry of geodesics. Adv. in Math. 101 (1993), 180–219. [11] S. Cobzas, Functional analysis in asymmetric normed spaces. Front. Math., Birkhäuser/Springer Basel AG, Basel 2013. [12] M. M. Day, Normed linear spaces. 3rd ed., Ergeb. Math. Grenzgeb. 21, Springer-Verlag, Berlin 1973. [13] T. Foertsch, and A. Karlsson, Hilbert metrics and Minkowski norms. J. Geometry 83 (2005), no. 1–2, 22–31. [14] P. de la Harpe, On Hilbert’s metric for simplices. In Geometric group theory (Graham A. Niblo et al., eds.,) Vol. 1, London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge 1993, 97–119. [15] D. Hilbert, Mathematische Probleme. Göttinger Nachrichten 1900, 253–297, reprinted in Archiv der Mathematik und Physik, 3d. ser., vol. 1 (1901) 44–63 and 213–237; English version, “Mathematical problems”, reprinted also in Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436. [16] D. C. Kay, The Ptolemaic inequality in Hilbert geometries. Pacific J. Math. 21 (1967), 293–301. [17] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz). Math. Ann. VI (1873), 112–145. [18] B. Nica, The Mazur–Ulam theorem. Expo. Math. 30 (2012), no. 4, 397–398. [19] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 391 (1988). [20] H. Martini, K. Swanepoel and G. Weiss, The geometry of Minkowski spaces – a survey. I. Expo. Math. 19 (2001), no. 2, 97–142. [21] H. Martini and M. Spirova, Recent results in Minkowski geometry. East-West J. Math., Special Vol. (2007), 59–101. [22] H. Minkowski, Sur les propriétés des nombres entiers qui sont dérivées de l’intuition de l’espace. Nouvelles annales de mathématiques, 3e série, 15, 1896. [23] H. Minkowski, Geometrie der Zahlen. B. G. Teubner, Leipzig and Berlin, 1896 and 1910 (several editions and translations). [24] A. Papadopoulos, Hilbert’s fourth problem. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 391–431. [25] A. Papadopoulos and M. Troyanov, From Funk to Hilbert geometry. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 33–67. [26] H. Ribeiro, Sur les espaces à métrique faible. Portugaliae Math. 4 (1943), 21–40.

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[27] A. C. Thompson, Minkowski geometry. Encyclopedia Math. Appl. 63. Cambridge University Press, Cambridge 1996. [28] O. Varga, Zur Begründung der Minkowskischen Geometrie. Acta Univ. Szeged. Sect. Sci. Math. 10 (1943), 149–163.

Chapter 2

From Funk to Hilbert geometry Athanase Papadopoulos and Marc Troyanov

Contents 1 2 3 4 5 6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Funk metric . . . . . . . . . . . . . . . . . . . . . . . . . The reverse Funk metric . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The geometry of balls in the Funk metric . . . . . . . . . . . . On the topology of the Funk metric . . . . . . . . . . . . . . . The triangle inequality and geodesics . . . . . . . . . . . . . . 7.1 On the triangle inequality . . . . . . . . . . . . . . . . . 7.2 Geodesics and convexity in Funk geometry . . . . . . . . 8 Nearest points in Funk geometry . . . . . . . . . . . . . . . . . 9 The infinitesimal Funk distance . . . . . . . . . . . . . . . . . 10 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A projective viewpoint on Funk geometry . . . . . . . . . . . . 12 Hilbert geometry . . . . . . . . . . . . . . . . . . . . . . . . . 13 Related questions . . . . . . . . . . . . . . . . . . . . . . . . . A Menelaus’ Theorem . . . . . . . . . . . . . . . . . . . . . . . B The classical proof of the triangle inequality for the Funk metric References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

33 35 41 42 43 46 48 48 51 53 55 56 57 58 60 61 64 66

1 Introduction The Funk metric associated to an open convex subset of a Euclidean space is a weak metric in the sense that it does not satisfy all the axioms of a metric: it is not symmetric, and we shall also allow the distance between two points to be zero; see Chapter 1 in this volume [19], where such metrics are introduced. Weak metrics often occur in the calculus of variations and in Finsler geometry and the study of such metrics has been revived recently in low-dimensional topology and geometry by Thurston who introduced an asymmetric metric on Teichmüller space, which became the subject of intense research. In this chapter, we shall sometimes use the expression “metric” instead of “weak metric” in order to simplify.

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The Funk weak metric F .x; y/ is associated to an open convex subset of a Euclidean space Rn . It is an important metric for the subject treated in this Handbook, because the Hilbert metric H of the convex set is the arithmetic symmetrization of its Funk metric. More precisely, for any x and y in , we have H .x; y/ D

1 .F .x; y/ C F .y; x// : 2

Independently of its relation with the Hilbert metric, the Funk metric is a nice example of a weak metric, and there are many natural questions for a given convex subset of Rn that one can ask and solve for such a metric, regarding its geodesics, its balls, its isometries, its boundary structure, and so on. Of course, the answer depends on the shape of the convex set , and it is an interesting aspect of the theory to study the influence of the properties of the boundary @ (its degree of smoothness, the fact that it is a polyhedron, a strictly convex hypersurface, etc.) on the Funk geometry of this set. The same questions can be asked for the Hilbert geometry, and they are addressed in several chapters of this volume. The Funk geometry is studied in Busemann’s book The geometry of geodesics [2], although the name “Funk metric” is not used there, but it is used by Busemann in his later papers and books, see e.g. [3], and in the memoir [28] by Zaustinsky (who was a student of Busemann). We studied some aspects of this metric in [17], following Busemann’s ideas. But a systematic study of this metric is something which seems to be still missing in the literature, and the aim of this chapter is somehow to fill this gap. Euclidean segments in are geodesics for the Funk metric, and in the case where the domain is strictly convex (that is, if there is no nonempty open Euclidean segment in @), the Euclidean segments are the unique geodesic segments. For a metric d on a subset of Euclidean space, the property of having the Euclidean segments d geodesics is the subject of Hilbert’s Problem IV (see Chapter 15 in this volume [16]). One version of this problem asks for a characterization of non-symmetric metrics on subsets of Rn for which the Euclidean segments are geodesics. The Funk metric is also a basic example of a Finsler structure, perhaps one of the most basic. In the paper [17], we introduced the notions of tautological and of reversible tautological Finsler structures of the domain . The Funk metric F is the length metric induced by the tautological weak Finsler structure, and the Hilbert metric H is the length metric induced by the reversible tautological Finsler structure of the domain . This is in fact how Funk introduced his metric in 1929, see [9], [24]. The reversible Finsler structure is obtained by the process of harmonic symmetrization at the level of the convex sets in the tangent spaces that define these structures, cf. [18]. This gives another relation between the Hilbert and the Funk metrics. The Finsler geometry of the Funk metric is studied in some detail in Chapter 3 [24] of this volume. A useful variational description of the Funk metric is studied in [27]. There are also interesting non-Euclidean versions of Funk geometry, see Chapter 13 in this volume ([20]).

Chapter 2. From Funk to Hilbert geometry

35

In the present chapter, we start by recalling the definition of the Funk metric, and we give several basic properties of this metric, some of which are new, or at least formulated in a new way. In Section 3, we introduce what we call the reverse Funk metric, that is, the metric rF defined by rF .x; y/ D F .y; x/. This metric is much less studied than the Funk metric. In Section 4, we give the formula for the Funk metric for the two main examples, namely, the cases of convex polytopes and the Euclidean unit ball. In Section 5, we study the geometry of balls in the Funk metric. Since the metric is non-symmetric, one has to distinguish between forward and backward balls. We show that any forward ball is the image of by a Euclidean homothety. This gives a rigidity property, namely, that the local geometry of the convex set determines the convex set up to a scalar factor. In Section 6, we show that the topologies defined by the Funk and the reverse Funk metrics coincide with the Euclidean topology. In Section 7, we give a proof of the triangle inequality for the Funk metric and at the same time we study its geodesics and its convexity properties. In Section 8, we study the property of the nearest point projections of a point in on a convex subset of equipped with the Funk metric, and the perpendicularity properties. In the case where is strictly convex, the nearest point projection is unique. There is a formulation of perpendicularity to hyperplanes in in terms of properties of support hyperplanes for . In Section 9, we study the infinitesimal (Finsler) structure associated to a Funk metric. In Section 10, we study the isometries of a Funk metric. In the case where is bounded and strictly convex, its isometry group coincides with the subgroup of affine transformations of Rn that leave invariant. In Section 11, we propose a projective viewpoint and a generalization of the Funk metric and in Section 12, we present some basic properties of the Hilbert metric which are direct consequences of the fact that it is a symmetrization of the Funk metric. In the last section we give some new perspectives on the Funk metric, some of which are treated in other chapters of this volume. Appendix A contains two classical theorems of Euclidean geometry (the theorems of Menelaus and Ceva). Menelaus’ Theorem is used in Appendix B to give the classical proof of the triangle inequality for the Funk metric.

2 The Funk metric In this section, is a proper convex domain in Rn , that is, is convex, open, nonx the closure of in Rn and by @ D x n empty, and ¤ Rn . We denote by the topological boundary of . In the case of an unbounded domain, it will be convenient to add points at infinity to the boundary @. To do so, we consider Rn as an affine space in the projective space RP n and we denote by H1 D RP n n Rn the hyperplane at infinity. We then denote Q D Q n H1 . z n . Observe that @ D @ z the closure of in RP n and by @ by n For any two points x ¤ y in R , we denote by Œx; y the closed affine segment joining them. We also denote by R.x; y/ the affine ray starting at x and passing

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z y/ its closure in RP n . Finally we set through y and by R.x; Q 2 RP n : z y/ \ @ a .x; y/ D R.x; a ............. .............r. ......... ... ........... . . . . . . . . .. ......... .. ........ . . . . . . .. . . . . . . . . . . . . . . yr .. .... .. .... .. . . .... . . ... ..... .. ... ... r .... .. ... x ... ..... ... ........ .... .......... ................ ................... .........

We now define the Funk metric. Definition 2.1 (The Funk metric). The Funk metric on , denoted by F , is defined for x and y in , by F .x; x/ D 0 and by jx aj F .x; y/ D log jy aj if x ¤ y, where a D a .x; y/ 2 RP n . Here jp qj is the Euclidean distance between the points q and p in Rn . It is understood in this formula that if a 2 H1 , then F .x; y/ D 0. This is consistent with the convention 1=1 D 1. Let us begin with a few basic properties of the Funk metric. Proposition 2.2. The Funk metric in a convex domain ¤ Rn satisfies the following properties: (a) F .x; y/ 0 and F .x; x/ D 0 for all x; y 2 . (b) F .x; z/ F .x; y/ C F .y; z/ for all x; y; z 2 . (c) F is projective, that is, F .x; z/ D F .x; y/ C F .y; z/ whenever z is a point on the affine segment Œx; y. (d) The weak metric F is non symmetric, that is, F .x; y/ ¤ F .y; x/ in general. (e) The weak metric F is separating, that is, x ¤ y H) F .x; y/ > 0, if and only if the domain is bounded. (f) The weak metric F is unbounded. Property (a) and (b) say that F is a weak metric. Proof. Property (a) follows from the fact that y 2 Œx; a .x; y/, therefore jx a .x; y/j 1 jy a .x; y/j

Chapter 2. From Funk to Hilbert geometry

37

and we have equality if y D x. The triangle inequality (b) is not completely obvious. The classical proof by Hilbert (who wrote it for the Hilbert metric) is given in Appendix B and a new proof is given in Section 7. To prove (c), observe that if y 2 Œx; z and x ¤ y ¤ z, then a .x; y/ D a .x; z/ D a .y; z/. Denoting this common point by a, we have F .x; y/ C F .y; z/ D log

jy aj jx aj jx aj C log D log D F .x; z/: jy aj jz aj jz aj

Property (d) is easy to check, see Section 3 for more details. Property (e) follows immediately from the definition and the fact that a convex domain in Rn is unbounded if and only if it contains a ray, see [19], Remark 3.13. To prove (f), we recall that we always assume ¤ Rn , and therefore @ ¤ ;. Let x be a point in and a a point in @ and consider the open Euclidean segment .x; a/ contained in . For any sequence xn in this segment converging to a (with respect to the Euclidean metric), we have F .x; xn / D log jxjxaj ! 1 as n ! 1. n aj It is sometimes useful to see the Funk metric from other viewpoints. If x, y and z are three aligned points in Rn with z ¤ y, then there is a unique 2 R such that x D z C .y z/. We call this number the division ratio or affine ratio of x with respect to z and y and denote it suggestively by D zx=zy. Note also that > 1 if and only if y 2 Œx; z. We extend the notion of affine ratio to the case z 2 H1 by setting zx=zy D 1 if z 2 H1 . We then have F .x; y/ D log./; where is the affine ratio of x with respect to a D a .x; y/ and y. Proposition 2.3. Let x and y be two points in the convex domain and set a D a .x; y/. Let h W Rn ! R be an arbitrary linear form such that h.y/ ¤ h.x/. Then h.a/ h.x/ F .x; y/ D log : h.a/ h.y/ Proof. We first observe that a linear function h W Rn ! R is either constant on a given ray R.x; y/, or it is injective on that ray. If a D a .x; y/ 2 H1 , then h.a/ D lim h.x C t .y x// D h.x/ C t .h.y/ h.x// D ˙1; t!1

and the proposition is true since log.1=1/ D log.1/ D 0 D F .x; y/. If a 62 H1 , then the division ratio D ax=ay > 1 and we have .a x/ D .a y/ and F .x; y/ D log./. Since h W Rn ! R is linear, then .h.a/h.x// D .h.a/h.y// and the proposition follows at once. We now recall some important notions from convex geometry. For these and other classical results on convex geometry, the reader can consult the books by Eggleston [7], Fenchel [8], Valentine [26] and Rockafellar [22].

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Definition 2.4. Let Rn be a proper convex domain and a 2 @ be a (finite) boundary point. We assume that contains the origin 0. A supporting functional at a for is a linear function h W Rn ! R such that h.a/ D 1 and h.x/ < 1 for any point x in 1 . The hyperplane H D fz j h.z/ D 1g is said to be a support hyperplane at a for . If is unbounded, then the hyperplane at infinity H1 is also considered to be a support hyperplane. A basic fact is that any boundary point of a convex domain admits one or several supporting functional(s). Let us denote by S the set of all supporting functionals of . Then p .x/ D supfh.x/ j h 2 S g is the Minkowski functional of . This is the unique weak Minkowski norm such that D fx 2 Rn j p .x/ < 1g: We have the following consequences of the previous proposition: Corollary 2.5. Let Rn be a convex domain containing the origin and let h be a supporting functional for . We then have 1 h.x/ F .x; y/ log 1 h.y/ for any x; y 2 . Furthermore we have equality if and only if either a D a .x; y/ 2 H1 and h.x/ D h.y/, or a 62 H1 and h.a/ D 1. Proof. If h.x/ D h.y/ there is nothing to prove, we thus assume that h.x/ ¤ h.y/. Note that this condition means that the line through x and y is not parallel to the supporting hyperplane H D fh D 1g. We first assume that a D a .x; y/ 62 H1 , The hypothesis h.x/ ¤ h.y/ implies jxaj h.a/ ¤ h.y/. Because the function a 7! log jyaj is strictly monotone decreasing and h.a/ 1, we have by Proposition 2.3: 1 h.x/ h.a/ h.x/ log : F .x; y/ D log h.a/ h.y/ 1 h.y/ The equality holds if and only if h.a/ D 1. Suppose now that a 2 H1 ; this means that F .x; y/ D 0 and the ray R.x; y/ is contained in . In particular, we have h.x/ C h.y x/ D h.x C .y x// < 1

for all 0;

which implies h.y/ h.x/ D h.y x/ 0. Since we assumed h.x/ ¤ h.y/, we have h.y/ < h.x/ and therefore 1 h.x/ : F .x; y/ D 0 > log 1 h.y/ 1 The reader should not mistake the notion of supporting functional for that of support function h W Rn ! R defined as h .x/ D supfhx; yi j y 2 g.

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Corollary 2.6. If Rn is a convex domain containing the origin, then ° ± 1h.x/ F .x; y/ D max 0; suph2S log 1h.y/ : Proof. We first assume that F .x; y/ > 0. Then a D a .x; y/ 62 H1 . Let us denote by ˆ.x; y/ the right hand side in the above formula. Then Proposition 2.3 implies that F .x; y/ ˆ.x; y/ and Corollary 2.5 implies the converse inequality. If F .x; y/ D 0, then R.x; y/ < 0. For any supporting function h, we then have h.x C t .y x// D h.x/ C t.h.y/ h.x// < 1 for any t > 0. This implies that h.y/ h.x/ and it follows that ˆ.x; y/ D 0. Notice that for bounded domains, the previous formula reduces to 1 h.x/ F .x; y/ D sup log : 1 h.y/ h2S This can also be reformulated as follows (compare to Theorem 1 of [27]): Corollary 2.7. The Funk metric in a bounded convex domain Rn is given by dist.x; H / F .x; y/ D sup log ; dist.y; H / H where the supremum is taken over the set of all support hyperplanes H for and dist.x; H / is the Euclidean distance from x to H . The next consequence of Proposition 2.3 is the following relation between the division ratio of three aligned points in a convex domain and the Funk distances between them. Corollary 2.8. Let x, y and z be three aligned points in the convex domain Rn . Suppose that F .x; y/ > 0 and z D x C t .y x/ for some t 0. Then eF .x;y/ .eF .x;z/ 1/ ; eF .x;z/ .eF .x;y/ 1/ F .x; z/ D F .x; y/ log eF .x;y/ C t .1 eF .x;y/ / : tD

(2.1) (2.2)

Proof. Choose a supporting functional h for at the point a D a .x; y/ D a .x; z/. Then we have, from Corollary 2.5: eF .x;y/ D Therefore eF .x;z/ 1 D eF .x;z/

1 h.x/ 1 h.y/

1 h.z/ 1 h.x/

and

eF .x;z/ D

1 h.x/ : 1 h.z/

1 h.x/ h.z/ h.x/ 1 D ; 1 h.z/ 1 h.x/

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and likewise

eF .x;y/ 1 h.y/ h.x/ : D F .x;y/ 1 h.x/ e

Thus eF .x;y/ .eF .x;z/ 1/ h.z/ h.x/ h.z x/ D D t: D F .x;z/ F .x;y/ h.y x/ h.y/ h.x/ e .e 1/ This proves Equation (2.1). To prove the equation (2.2), we now resolve eF .x;z/ 1 eF .x;y/ 1 D t eF .x;y/ eF .x;z/ for eF .x;z/ . This gives us eF .x;z/ D

eF .x;y/ ; eF .x;y/ C t 1 eF .x;y/

which is equivalent to (2.2). It is useful to observe that computing the Funk distance between two points in is a one-dimensional operation. More precisely, if S D Œa1 ; a2 Rn is a compact segment in Rn containing the points x and y in its interior with y 2 Œx; a2 , we shall write jx a2 j FS .x; y/ D log : jy a2 j Although S is not an open set, FS .x; y/ clearly corresponds to the one-dimensional Funk metric in the relative interior of S. Proposition 2.9. The Funk distance between two points x and y in is given by F .x; y/ D inf fFS .x; y/ j S is a segment in containing x and yg : Proof. We identify S with a segment in R with b < x y < a and observe that the jxaj function a 7! log jyaj is strictly monotone decreasing. This result makes an analogy between the Funk metric and the Kobayashi metric in complex geometry, see [13]. It has the following immediate consequences: Corollary 2.10. (i) If 1 2 are convex subsets of Rn , then F1 F2 with equality if and only if 1 D 2 . (ii) Let 1 and 2 be two open convex subsets of Rn . Then, for every x and y in 1 \ 2 , we have F1 \2 .x; y/ D max.F1 .x; y/; F2 .x; y//: (iii) Let be a nonempty open convex subset of Rn , let 0 be the intersection of with an affine subspace of Rn , and suppose that 0 ¤ ;. Then, F0 is the metric induced by F on 0 as a subspace of .; F /.

Chapter 2. From Funk to Hilbert geometry

41

3 The reverse Funk metric Definition 3.1. The reverse Funk metric in a proper convex domain is defined as jy bj r ; F .x; y/ D F .y; x/ D log jx bj where b D a .y; x/. ................................ ........... ... ......... . . . . .. . . . . . . ........ . . . . . ... . . . . . . . . . . . . . . . . yr ..... .. .. .... ..... ..... . .. .... .. .. ... . . r .... . ..... x ... ....... .... . . . .... . r.. . b ..................................................

Figure 1. The reverse Funk metric.

Proposition 3.2. The reverse Funk metric in a convex domain ¤ Rn is a projective weak metric. It is unbounded and non-symmetric and it is separating if and only if the domain is bounded. The proof is a direct consequence of Proposition 2.2. An important difference between the Funk metric and the reverse Funk metric is the following: Proposition 3.3. Let be a bounded convex domain in Rn and x be a point in . Then the function y ! rF .x; y/ is bounded. Proof. Define x and ı by x D inf jx bj; [email protected]

and

ı D sup ja bj: a;[email protected]

Observe that ı is the Euclidean diameter of , thus ı < 1 since is bounded. We also have x > 0. The proposition follows from the inequality ı r F .x; y/ log : x In particular the reverse Funk metric rF is not bi-Lipschitz equivalent to the Funk metric F .

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4 Examples Example 4.1 (Polytopes). An (open) convex polytope in Rn is defined to be an intersection of finitely many half-spaces: D fx 2 Rn j j .x/ < sj ; 1 j kg; where j W Rn ! R is a nontrivial linear form for all j . The Funk distance between two points in such a polytope is given by ° ± s .x/ F .x; y/ D max 0; max1j k log sjj jj .y/ : The proof is similar to that of Corollary 2.6. As a special case, let us mention that the Funk metric in RnC is given by ° ± FRnC .x; y/ D max1in max 0; log yxii : Observe that the map x D .xi / 7! u D .ui /, where ui D log.xi /, is an isometry from the space .RnC ; FRnC / to Rn with the weak Minkowski distance ı.u; v/ D max1i n max ¹0; ui vi º: Example 4.2 (The Euclidean unit ball). The following is a formula for the Funk metric in the Euclidean unit ball B Rn : ! p jy xj2 jx ^ yj2 C jxj2 hx; yi ; (4.1) FB .x; y/ D log p jy xj2 jx ^ yj2 jyj2 C hx; yi p ! where jx ^ yj D jxj2 jyj2 hx; yi2 is the area of the parallelogram with sides 0x, ! 0y. y

x

a

a2 0

a1

Proof. If x D y, there is nothing to prove, so we assume that x ¤ y. Let us set a D aB .x; y/ D R.x; y/ \ @B. Using Proposition 2.3 with the linear form h.z/ D hy x; zi we get hy x; ai hy x; xi hy x; ai C jxj2 hx; yi D log : FB .x; y/ D log hy x; ai hy x; yi hy x; ai jyj2 C hx; yi

Chapter 2. From Funk to Hilbert geometry

43

So we just need to compute hy x; ai. This is an exercise in elementary Euclidean yx and geometry. Let us set u D jyxj a1 D hu; aiu;

a2 D a a1 :

Then a D a1 C a2 and a1 is a multiple of y x while a2 is the orthogonal projection of the origin O of Rn on the line through x and y. In particular the altitude of the triangle Oxy is equal to ja2 j, therefore Area.Oxy/ D

1 1 jx ^ yj D ja2 j jy xj: 2 2

Observe now that hu; ai > 0 and jaj2 D ja1 j2 C ja2 j2 D 1, we thus have hy x; ai2 D jy xj2 hu; ai2 D jy xj2 ja1 j2 D jy xj2 .1 ja2 j2 / D jy xj2 jx ^ yj2 : The desired formula follows immediately.

5 The geometry of balls in the Funk metric Since we are dealing with non-symmetric distances, we need to distinguish between forward and backward balls. For a point x in and > 0, we set B C .x; / D fy 2 B j F .x; y/ < g

(5.1)

and we call it the forward open ball (or right open ball) centered at x of radius . In a symmetric way, we set B .x; / D fy 2 B j F .y; x/ < g

(5.2)

and we call it the backward open ball (also called the left open ball) centered at x of radius . Note that the open backward balls of the Funk metric are the open forward balls of the reverse Funk metric, and vice versa. We define closed forward and closed backward balls by replacing the inequalities in (5.1) and (5.2) by non strict inequalities, and in the same way we define forward and backward spheres by replacing the inequalities by equalities. In Funk geometry, the backward and forward balls have in general quite different shapes and different properties. Proposition 5.1. Let be a proper convex open subset of Rn equipped with its Funk metric F , let x be a point in and let be a nonnegative real number. We have:

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• The forward open ball B C .x; / is the image of by the Euclidean homothety of center x and dilation factor .1 e /. • The backward open ball B .x; / is the intersection of with the image of by the Euclidean homothety of center x and dilation factor .e 1/, followed by the Euclidean central symmetry centered at x.

r

r

A forward and a backward ball in Funk geometry. The forward ball is always relatively compact in , while the closure of the backward ball may meet the boundary @ if its radius is large enough.

Proof. Let y ¤ x be a point in . If F .x; y/ D 0, then the ray R.x; y/ is contained in , and for any z on that ray, we have F .x; z/ D 0. Therefore the ray is also contained in B.x; /. If F .x; y/ D 0, then a D a .x; y/ 62 H1 and we have the following equivalent conditions for any point y on the segment Œx; a: jx aj jy xj:

y 2 B .x; / () log

Thus, for instance, if is the interior a Euclidean ball in Rn , then any forward ball for the Funk metric B C .x0 ; ı/ is also a Euclidean ball. However, its Euclidean center is not the center for the Funk metric (unless x0 is the center of ). Considering Example 4.2, if is the Euclidean unit ball and B C .x0 ; / is the Funk ball of radius and center x0 in , then y 2 B C .x0 ; / if and only if F .x0 ; y/ . Using

Chapter 2. From Funk to Hilbert geometry

45

Formula (4.1), we compute that this is equivalent to kyk2 2e hy; x0 i C e 2 kx0 k2 .1 e /2 : This set describes a Euclidean ball with center z0 D e x0 and Euclidean radius r D .1 e /. We deduce the following “local-implies-global” property of Funk metrics. The meaning of the statement is clear, and it follows directly from Proposition 5.1. Corollary 5.2. We can reconstruct the boundary @ of from the local geometry at any point of . Corollary 5.3. For any points x and x 0 in a convex domain equipped with its Funk metric and for any two positive real numbers ı and ı 0 , the forward balls B C .x; ı/ and B C .x 0 ; ı 0 / are either homothetic or a translation of each other. Proof. This follows from Proposition 5.1 and the fact that the set of Euclidean transformations which are either homotheties or translations form a group (sometimes called the the group of dilations, see e.g. [6]). Remark 5.4. The previous corollary also holds for backward balls B .x; ı/ of small enough radii. Remarks 5.5. In the case where the convex set is unbounded, the forward and backward open balls of its Funk metric are always noncompact. If is bounded. then for any x 2 and for large enough we have B .x; / D . This follows from Proposition 3.3. In particular, the closed backwards balls are not compact for large radii. The forward open balls are geodesically convex if and only if is strictly convex. Remark 5.6. The property for a weak metric on a subset of Rn to have all the right spheres homothetic is also shared by the Minkowski weak metrics on Rn . Indeed, it is easy to see that in a Minkowski weak metric, any two right open balls are homothetic. (Any two right spheres of the same radius are translates of each other, and it is easy to see from the definition that any two spheres centered at the same point are homothetic, the center of the homothety being the center of the balls.) Thus, Minkowski weak metrics share with the Funk weak metrics the property stated in Proposition 5.1. Busemann proved that in the setting of Desarguesian spaces, these are the only examples of spaces satisfying this property (see the definition of a Desarguesian space and the statement of this result in Section 6 of Chapter 1 in this volume ([19])). We state this as the following: Theorem 5.7 (Busemann [3]). A Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space.

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6 On the topology of the Funk metric Proposition 6.1. The topology induced by the Funk or reverse Funk metric in a bounded convex domain in Rn coincides with the Euclidean topology in that domain. Proof. The proof consists in comparing the balls in the Euclidean and the Funk (or reverse Funk) geometries. Let us fix a point x in . Then there exists 0 < x ƒx < 1 such that for any 2 @ we have x j xj ƒx : If we denote by B C .x; / the forward ball with center x and radius in the Funk metric, then Proposition 5.1 implies that y 2 @B C .x; / H) .1 e /x jy xj .1 e /ƒx : In other words, if EB.x; ı/ denotes the Euclidean ball with center x and radius ı, then B.x; .1 e /x / B C .x; / EB.x; .1 e /ƒx /:

E

(6.1)

This implies that the families of balls B C .x; / and EB.x; ı/ are sub-bases for the same topology. For the backward balls B .x; /, the second part of Proposition 5.1 implies the following y 2 @B .x; / H) .e 1/x jy xj .e 1/ƒx ; provided .e 1/ 1. This implies that for log.2/ we have B.x; .e 1/x / B .x; / EB.x; .e 1/ƒx /;

E

and therefore the family of backward balls B .x; / also generates the Euclidean topology.

For general convex domains, bounded or not, we have the following weaker result on the topology: Proposition 6.2. For any convex domain in Rn , F is a continuous functions on . Proof. We first consider the case where is bounded. Suppose first that x and y are distinct points in and let xn , yn be sequences in converging to x and y respectively. Taking subsequences if necessary, we may assume that xn ¤ yn for all n. Then an D a .xn ; yn / is well defined and this sequence converges to a D a .x; y/.

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Chapter 2. From Funk to Hilbert geometry

Since a ¤ y we have

jxn an j lim F .xn ; yn / D lim log n!1 n!1 jyn an j jx aj D log jy aj D F .x; y/:

Assume now that x D y and let xn ; yn 2 be sequences converging to x such that xn ¤ yn for all n. We have jxn an j F .xn ; yn / D log jyn an j j.yn an / C .xn yn /j D log jyn an j jxn yn j log 1 C : jyn an j Since yn 2 converges to a point x in , we have ı D [email protected] jyn bj1 < 1. We then have F .xn ; yn / log .1 C ıjxn yn j/ ! 0; since jxn yn j ! 0. If is unbounded, we set R D \ EB.x; R/ where EB.x; R/ is the Euclidean ball of radius R centered at the origin. It is easy to check that FR converges uniformly to F on every compact subset of as R ! 1. The continuity of F follows therefore from the proof for bounded convex domains. For the next result we need some more definitions: Definition 6.3. Let ı be a weak metric defined on a set X . A sequence fxk g in X is forward bounded if sup ı.xk ; xm / < 1 where the supremum is taken over all pairs k, m satisfying m k. Note that this definition corresponds to the usual notion in the case of a usual (symmetric) metric space. We then say that the weak metric space .X; ı/ is forward proper, or forward boundedly compact if every forward bounded sequence has a converging subsequence. The sequence fxk g is forward Cauchy if lim sup ı.xk ; xm / D 0:

k!1 mk

The weak metric space .X; ı/ is forward complete if every forward Cauchy sequence converges. We define backward properness and backward completeness in a similar way.

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Proposition 6.4. The Funk metric in a convex domain Rn is forward proper (and in particular forward complete) if and only if is bounded. The Funk metric is never backward complete. Proof. For a convex domain, the ball inclusions (6.1) immediately imply that forward complete balls are relatively compact; this implies forward properness. If is unbounded, then it contains a ray and such a ray contains a divergent sequence fxk g such that F .xk ; xm / D 0 for any m k therefore F is not complete. To prove that the Funk metric is never backward complete, we consider an affine segment Œa; b 2 Rn with a ¤ b and such that Œa; b \ @ D fa; bg. Set xk D b C k1 .a b/. If m k, then F .xk ; xm / D log

jxm aj ; jxk aj

which converges to 0 as k ! 1. Since the sequence fxk g has no limit in , we conclude that F is not backward complete. Remark 6.5. The previous proposition also says that in a bounded convex domain, the Funk metric is forward complete and the reverse Funk metric is not.

7 The triangle inequality and geodesics In this section, we prove the triangle inequality for the Funk metric and give a necessary and sufficient condition for the equality case. We also describe all the geodesics of this metric.

7.1 On the triangle inequality Theorem 7.1. If x, y and z are three points in a proper convex domain , then the triangle inequality (7.1) F .x; y/ C F .y; z/ F .x; z/ holds. Furthermore we have equality F .x; y/ C F .y; z/ D F .x; z/ if and only if the three points Q (7.2) a .x; y/; a .y; z/; a .x; z/ 2 @ are aligned in RP n . Before proving this theorem, let us first recall a few additional definitions from convex geometry: Let Rn be a convex domain. Then it is known that its closure x is also convex. A convex subset D x is a face of x if for any x; y 2 D and any

Chapter 2. From Funk to Hilbert geometry

49

0 < < 1 we have .1 /x C y 2 D H) Œx; y D: x are also considered to be faces. A face D x is called proper if The empty set and x D ¤ and D 6D ;. A face D is said to be exposed if there is a supporting hyperplane x Recall that a support hyperplane is a hyperplane H H for such that D D H \ . that meets @ and H \ D ;. It is easy to prove that every proper face is contained in an exposed face. In fact every maximal proper face is exposed. x is an exposed point of if fxg is an exposed face, that is, if there A point x 2 @ x D fxg. If is bounded, then x is the exists a hyperplane H Rn such that H \ closure of the convex hull of its exposed points (Straszewicz’s Theorem). x is an extreme point if x n fxg is still a convex set. Such a point A point x 2 x is the convex hull of its belongs to the boundary @ and if is bounded, then extreme points (Krein–Milman’s Theorem). Every exposed point is an extreme point, but the converse does not hold in general. The following result immediately follows from the definitions: Lemma 7.2. The following are equivalent conditions for a convex domain Rn : (i) Every boundary point is a extreme point. (ii) Every boundary point is an exposed point. (iii) The boundary @ does not contain any non-trivial segment. If one of these conditions holds, then is said to be strictly convex. The following result will play an important role in the proof of Theorem 7.1: Lemma 7.3. Let be bounded convex domain and x, y, z three points in . Then the following are equivalent: Q such that (a) There exists a proper face D @ a .x; y/; a .y; z/; a .x; z/ 2 D: (b) The three points a .x; y/; a .y; z/ and a .x; z/ are aligned in RP n . Proof. Let us set a D a .x; y/, b D a .y; z/ and c D a .x; z/. If x, y and z are aligned, then a D b D c. Otherwise, a, b and c belong to the 2-plane … containing x, y, z. Therefore if a, b, c belong to a proper face D, then those three points are contained in the interval … \ D. This proves the implication (a) H) (b) The converse implication (b) H) (a) is obvious. Proof of Theorem 7.1. We first consider F .x; z/ D 0. In this case the inequality (7.1) is trivial and we have c 2 H1 . We then have equality in (7.1) if and only if F .x; y/ D F .y; z/ D 0, which is equivalent to a 2 H1 and b 2 H1 (the points a and b are as in the proof of the previous lemma). It then follows from Lemma 7.3 that a, b, c lie on some line (at infinity).

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We now consider the case F .x; z/ > 0, that is, c 62 H1 . Choose a supporting functional h at the point c (that is, h.c/ D 1). We then have from Corollary 2.5: 1 h.x/ F .x; z/ D log 1 h.z/ 1 h.y/ 1 h.x/ C log D log 1 h.y/ 1 h.z/ F .x; y/ C F .y; z/: This proves the triangle inequality. Using again Corollary 2.5, we see that we have equality if and only if 1 h.x/ F .x; y/ D log 1 h.y/ and

1 h.y/ F .y; z/ D log ; 1 h.z/

and this holds if and only if one of the following cases holds: Case 1. We have a 62 H1 and b 62 H1 . In that case, h.a/ D h.b/ D 1 D h.c/. The three points a, b, c belong to the face D D @ \ fh D 1g and we conclude by Lemma 7.3 that a, b, c lie on some line . Case 2. We have a 2 H1 and b 62 H1 . In that case h.b/ D 1 D h.c/ and h.x/ D h.y/. This implies that the line through x and y is parallel to the hyperplane fh D 1g and therefore the point Q z y/ \ H1 @ a 2 R.x; belongs to the support hyperplane fh D 1g. Since h.b/ D 1, the three points a, b, c Q \ fh D 1g and we conclude by Lemma 7.3. belong to the face D D @ Case 3. We have a 62 H1 and b 2 H1 . The argument is the same as in Case 2. To complete the proof, we need to discuss the case a 2 H1 and b 2 H1 . In this case, we would have h.x/ D h.y/ and h.y/ D h.z/ and this is not possible. Indeed, we have c D x C .z x/ for some and the equality h.z/ D h.x/ would lead to the contradiction 1 D h.c/ D h.x C .z x// D h.x/ C .h.z/ h.x// D h.x/ < 1: We thus proved in all cases that the equality holds in (7.1) if and only if the points a, b and c are aligned in RP n . Corollary 7.4. Let x and z be two points in a proper convex domain Rn . Suppose that a .x; z/ 2 @ is an exposed point. Then for any point y 62 Œx; z we have F .x; z/ < F .x; y/ C F .y; z/.

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7.2 Geodesics and convexity in Funk geometry We now describe geodesics in Funk geometry. Let us start with a few definitions. Definitions 7.1. A path in a weak metric space .X; d / is a continuous map W I ! X , where I is an interval of the real line. The length of path W Œa; b ! X is defined as Length./ D sup

N 1 X

d..ti /; .tiC1 //;

iD0

where the supremum is taken over all subdivisions a D t0 < t1 < < tN D b. Note that in the case where the weak metric d is non-symmetric, the order of the arguments is important. The path W Œa; b ! X is a geodesic if d..a/; .b// D Length. /. The weak metric space .X; d / is said to be a weak geodesic metric space if there exists a geodesic path connecting any pair of points. It is said to be uniquely geodesic if this geodesic path is unique up to reparametrization. A subset A X is said to be geodesically convex if given any two points in A, any geodesic path joining them is contained in A. Lemma 7.5. The path W Œa; b ! X is geodesic if and only if for any t1 , t2 , t3 in Œa; b satisfying t1 t2 t3 we have d..t1 /; .t3 // D d..t1 /; .t2 // C d..t2 /; .t3 //. The proof is an easy consequence of the definitions. Q Let us now consider a proper convex domain Rn and a proper face D @. For any point p 2 , we denote by x p C v/ \ D ¤ ;g: Cp .D/ D fv 2 Rn j v D 0 or R.p;

(7.3)

x p C v/ is the extended ray through p and p C v in RP . (Recall that Here R.p; the projective space RP n is considered here as a completion of the Euclidean space Rn obtained by adding a hyperplane at infinity; the completion of the ray is then its topological completion.) Observe that Cp .D/ is a cone in Rn at the origin, its translate p C Cp .D/ is the cone over D with vertex at p. We then have the following n

Theorem 7.6. Let W Œ0; 1 ! be a path in a proper convex domain of Rn . Then Q such is a geodesic for the Funk metric in if and only if there exists a face D @ that for any t1 < t2 in Œ0; 1 we have .t2 / .t1 / 2 C.t1 / .D/: In particular if a .x; y/ 2 @ is an exposed point, then there exists a unique (up to reparametrization) geodesic joining x to y, and this geodesic is a parametrization of the affine segment Œx; y. Proof. This is a direct consequence of Theorem 7.1 together with Lemma 7.3 and Lemma 7.5.

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For smooth curves we have the following Corollary 7.7. Let W Œ0; 1 ! be a C 1 path in a proper convex domain of Rn . Q such that Then is a Funk geodesic if and only if there exists a face D @ P .t / 2 C.t / .D/ for any t 2 Œa; b.

D

A typical smooth geodesic in Funk geometry: all tangents to the curve meet the same face D @.

For a subset of Rn , equipped with a (weak) metric F , we have two notions of convexity: affine convexity, saying that for every pair of points in , the affine (or Euclidean) geodesic joining them is contained in , and geodesic convexity, saying that for every pair of points in , the F -geodesic joining them is contained in From the preceding results, we have the following consequence on geodesic convexity of subsets for the Funk metric. Corollary 7.8. Let be a bounded convex domain of Rn . Then the following are equivalent: (1) is strictly convex. (2) is uniquely geodesic for the Funk metric. (3) A subset A is geodesically convex for the Funk metric if and only if A is affinely convex. (4) The forward open balls in are geodesically convex with respect to the Funk metric F . Proof. (1) H) (2) immediately follows from Theorem 7.6. (2) H) (3) is obvious and (3) H) (4) follows from Proposition 5.1.

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53

q

x p y

a

b

We now prove (4) H) (1) by contraposition. Let us assume that is not strictly convex, so that there exists a non-trivial segment Œa; b @. On can then find a supporting functional h for such that h.a/ D h.b/ D 1. Let us chose a segment Œx; y such that a .x; y/ 2 Œa; b and another segment Œp; q such that p 2 Œx; y and h.q/ D h.p/. We also assume x ¤ p ¤ y and F .p; q/ > ı WD F .p; x/ C F .p; y/. Observe that we then have h.x/ < h.p/ D h.q/ < h.y/. If a .x; q/ 2 Œa; b and a .q; y/ 2 Œa; b, then the proof is finished since in this case h.x/ h.q/ h.x/ D log D F .x; q/ C F .q; y/: F .x; y/ D log h.y/ h.q/ h.y/ Since x; y 2 B C .p; ı/ while q 62 B C .p; ı/, we conclude that the forward ball B C .p; ı/ is not geodesically convex. If a .x; q/ 62 Œa; b or a .q; y/ 62 Œa; b, we let c D a .x; y/ and we consider the Euclidean homothety f W Rn ! Rn centered at c with dilation factor < 1. Let p 0 D f .p/, q 0 D f .q/; x 0 D f .x/; y 0 D f .y/ and q 0 D f .q/. It is now clear that if > 0 is small enough, then a .x 0 ; q 0 / 2 Œa; b and a .q 0 ; y 0 / 2 Œa; b. It is also clear that one can find a number ı 0 such that x 0 ; y 0 2 B C .p 0 ; ı 0 / while q 0 62 B C .p 0 ; ı 0 /. The previous argument shows then that F .x 0 ; y 0 / D F .x 0 ; q 0 / C F .q 0 ; y 0 / and therefore B C .p 0 ; ı 0 / is not geodesically convex. Remark 7.9. Note the formal analogy between Corollary 7.8 and the corresponding result concerning the geodesic segments of a Minkowski metric on Rn : if the unit ball of a Minkowski metric is strictly convex, then the only geodesic segments of this metric are the affine segments.

8 Nearest points in Funk geometry Let Rn be a convex set equipped with its Funk metric F . Definition 8.1. Let x be a point in and let A be a subset of . A point y in A is said to be a nearest point, or a foot (in Buseman’s terminology), for x on A if F .x; y/ D F .x; A/ WD inf F .x; z/: z2A

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It is clear from the continuity of the function y 7! F .x; y/ that for any closed non-empty subset A and any x 2 , there exists a nearest point y 2 A. This point need not be unique in general. Proposition 8.2. For a proper convex domain Rn , the following properties are equivalent: a) is strictly convex. b) For any closed convex subset A and for any x 2 , there is a unique nearest point y 2 A. Proof. let x be a point in and assume r D F .x; A/ > 0. Suppose that y and z are two nearest points of A for x. For any point w on the segment Œy; z we have F .x; w/ r because the closed ball BxC .x; r/ is convex. Since A is also assumed to be convex, we have w 2 A and therefore F .x; w/ r. We conclude that F .x; w/ D r, that is, w 2 @BxC .x; r/. From Proposition 5.1, we know that if is strictly convex, then BxC .x; r/ is also strictly convex and we conclude that y D z. It follows that we have a unique nearest point on A for x. This proves (a) H) (b). To prove (b) H) (a), we assume by contraposition that is strictly convex. Again from Proposition 5.1, we know that the forward ball BxC .x; r/ is not strictly convex. In particular @B C .x; r/ contains a non trivial segment A D Œy; z. Any point in the convex set A is a nearest point to x and this completes the proof. Proposition 8.3. Let A be an affinely convex closed subset of a proper convex domain Rn and let x 2 n A. A point y 2 A is a nearest point in A for x if and only if either F .x; y/ D 0 or there exists a hyperplane … Rn which contains y, which separates A and x and which is parallel to a support hyperplane H for at a D a .x; y/. Proof. We assume F .x; y/ > 0 (otherwise, there is nothing to prove). First, suppose there exists a hyperplane … Rn containing y and separating A from x and which is parallel to a support hyperplane H for at a. Let h be the corresponding supporting functional. Then we have, from our hypothesis, h.x/ < h.y/ D inf h.z/: z2A

From Proposition 2.3 and Corollary 2.5 we then have 1 h.x/ 1 h.x/ D inf log F .x; A/; F .x; y/ D log z2A 1 h.y/ 1 h.z/ therefore y is a nearest point on A for x. To prove the converse, we assume that y is a nearest point on A for x. Set r D F .x; y/ D F .x; A/, then, by definition, the forward open ball B C .x; r/ and the set A are disjoint. Since both sets are affinely convex, there exists a hyperplane … that separates them. Note that … is then a support hyperplane at y for the ball B C .x; r/.

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We conclude from Proposition 5.1 that … is parallel to a support hyperplane H for at a. There are several possible notions of perpendicularity in metric spaces. The following definition is due to Busemann (see [2], page 103). Definition 8.4 (Perpendicularity). Let A be a subset of and p a point in A. A geodesic W I ! is said to be perpendicular to A at p if the following two properties hold: (1) p D .t0 / for some t0 2 I , (2) for every t 2 I , p is a nearest point for .t / on A. From the previous results we have the following Corollary 8.5. Let x be a point in a convex domain and a 2 @ be a boundary point. If … Rn is a hyperplane containing x, then the ray Œx; a/ is perpendicular to … \ if and only if … is parallel to a support hyperplane Ha of at a. If b 2 @ is another boundary point, then the line .a; b/ is perpendicular to …\ if and only if … \ .a; b/ ¤ ; and … is parallel to both a support hyperplane Ha at a and a support hyperplane Hb at b.

9 The infinitesimal Funk distance In this section, we consider a convex domain Rn and a point p in . We define a weak distance ˆp D ˆ;p on Rn as the limit ˆp .x; y/ D lim

t&0

F;p .p C tx; p C ty/ : t

Theorem 9.1. The weak metric ˆp at a point p in Rn is a Minkowski weak metric in Rn . Its unit ball is the translated domain p D p. Proof. Choose a supporting functional h for and set 1 h.p C tx/ t h.y x/ D log 1 C : 'h .t/ D log 1 h.p C ty/ 1 h.p/ t h.y/ The first two derivatives of this functions are .1 h.p// h.y x/ ; 'h0 .t/ D .1 h.p/ th.x// .1 h.p/ t h.y// and 'h00 .t / D

..1 h.p//.h.x/ C h.y// 2t h.x/h.y// .1 h.p// h.y x/ .1 h.p/ th.x//2 .1 h.p/ t h.y//2

:

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We have in particular 'h0 .0/ D

h.y x/ ; 1 h.p/

and the second derivative is uniformly bounded in some neighborhood of p. More precisely, given a relatively compact neighborhood U Ux of p, one can find a constant C which depends on U but not on h such that j'h00 .t/j C; for any x; y 2 U and jt j 1 and any support function h. We have, from Taylor’s formula, h.y x/ 'h .t/ D t C t 2 .x; y; h/; 1 h.p/ where j.x; y; h/j C . Using Corollary 2.6 we have F;p .p C tx; p C ty/ D sup 'h .t / D sup h

h

t h.y x/ C O.t 2 /; 1 h.p/

where the supremum is taken over the set S of all support functions for . Therefore ˆp .x; y/ D sup

h2S

h.y x/ : 1 h.p/

We then see that ˆp .x; y/ is weak Minkowski distance (see Chapter 1 in this Handbook). ˆp .0; y/ 1 () sup

h2S

h.y/ 1 1 h.p/

() h.y/ 1 h.p/ for all support functions h of () h.p C y/ 1 for all h x p; () y 2 this means that the unit ball of ˆp is the translate of by p. Remark 9.2. The Funk metric of a convex domain is in fact Finslerian, and the previous theorem means that the Finslerian unit ball at any point p coincides with the domain itself with the point p as its center. This is why the Funk metric was termed tautological in [17]. The Finslerian approach to Funk geometry is developed in [24].

10 Isometries It is clear from its definition that the Funk metric is invariant under affine transformation. Conversely, we have the following:

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57

Proposition 10.1. Let 1 and 2 be two bounded convex domains in Rn . Assume that there exists a Funk isometry f W U1 ! U2 , where Ui is an open convex subset of i , i D 1; 2. If 2 is strictly convex, then f is the restriction of a global affine map of Rn that maps 1 to 2 . Proof. Let x and y be two distinct points in U1 . Then, for any z 2 Œx; y, we have F2 .f .x/; f .y// F2 .f .x/; f .z// F2 .f .z/; f .y// D F1 .x; y/ F1 .x; z/ F1 .z; y/ D 0: Since 2 is assumed to be strictly convex, this implies that f .z/ belongs to the line through f .x/ and f .y/. We can thus define the real numbers t and s by z D x C t .y x/

and

f .z/ D f .x/ C s.f .y/ f .x//:

It now follows from Corollary 2.8 and the fact that f is an isometry that s D t . We thus have proved that for any x; y 2 U1 and any t 2 Œ0; 1 we have f .x C t .y x// D f .x/ C t .f .y/ f .x//:

(10.1)

This relation easily implies that f is the restriction of a global affine mapping. We immediately deduce the following: Corollary 10.2. The group of Funk isometries of a strictly convex bounded domains Rn coincides with the subgroup of the affine group of Rn leaving invariant. Remark 10.3. The conclusion of the corollary may fail for unbounded domains. For instance, if is the upper half plane fx2 > 0g in R2 , then F .x; y/ D maxf0; log.x2 =y2 /g and any map f W ! of the type f .x1 ; x2 / D .ax1 C b; .x2 /, where a ¤ 0 and W R ! R is arbitrary, is an isometry.

11 A projective viewpoint on Funk geometry In this section we consider the following generalization of Funk geometry: We say that a subset U RP n is convex if it does not contain any full projective line and if the intersection of any projective line L RP n with U is a connected set. If U and are connected domains in RP n with U , and if x, y are two distinct points in . We denote by a .x; y/ 2 @ and ! .x; y/ 2 @U the boundary points on the line L through x and y appearing in the order a, y, x, !.

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Definition 11.1. The relative Funk metric for U is defined by F;U .x; x/ D 0 and by jy !j jx aj ; F;U .x; y/ D log jx !j jy aj if x ¤ y. The relative Funk metric is a projective weak metric, it is invariant under projective transformations in the sense that if f W RP n ! RP n is a projective transformation, then F;U .x; y/ D Ff ./;f .U / .f .x/; f .y//: Observe also that if U Rn is a proper convex domain, then F;U .x; y/ D F .x; y/ C rFU .x; y/: Lemma 11.2. In the case U D Rn , we have F;U .x; y/ D F .x; y/: Proof. If U D Rn , then its boundary is the hyperplane at infinity H1 and thus jy!j D 1 for any x; y 2 . jx!j Recall that there is no preferred hyperplane in projective space. Therefore, the classical Funk geometry is a special case of the relative Funk geometry where the englobing domain U RP n is the complement of a hyperplane. Such a set U is sometimes called an affine patch.

12 Hilbert geometry Definition 12.1. The Hilbert metric in a proper convex domain Rn is defined as 1 .F .x; y/ C rF .y; x// ; 2 where is considered as a subset of an affine patch U RP n . H .x; y/ D

This metric is a projective weak metric. Note that for x ¤ y we have jy bj jx aj 1 ; H .x; y/ D log 2 jx bj jy aj where a D a .x; y/ and b D a .y; x/. The expression inside the logarithm is the cross ratio of the points b, x, y, a, therefore the Hilbert metric is invariant by projective transformations.

Chapter 2. From Funk to Hilbert geometry

59

a...r................... .... ............ ... .......... . . . . . . . . ... . . . . . . . . . . . . . . . . ... ...... . . . . . . . . . . . . r y .. . .. .... .. ... .... ..... . .. .... .. ... .. . ... r . ..... x .. ...... ... .......r .... . . . .......... . .. b ....................................... Figure 2. The Hilbert metric.

Note also that the Hilbert metric coincides with half of the relative Funk metric of the domain with respect to itself: 1 F; .x; y/; 2 In the case where is the Euclidean unit ball, the Hilbert distance coincides with the Klein model (also called the Beltrami–Cayley–Klein model) of hyperbolic space. We refer to Sections 2.3–2.6 in [25] for a nice introduction to Klein’s model. In the case of a convex polytope defined by the linear inequalities j .x/ < sj , 1 j k, we have si j .y/ sj j .x/ 1 log : H .x; y/ D max si j .x/ sj j .y/ 1i;j k 2 H .x; y/ D

Applying our investigation on Funk geometry immediately gives a number of results on Hilbert geometry. In particular, applying Proposition 2.2 we get: Proposition 12.2. The Hilbert metric in a convex domain ¤ Rn satisfies the following properties: (a) H .x; y/ 0 and H .x; x/ D 0 for all x; y 2 . (b) H .x; z/ H .x; y/ C H .y; z/ for all x; y; z 2 . (c) H is projective, that is, H .x; z/ D H .x; y/ C H .y; z/ whenever z is a point on the affine segment Œx; y. (d) The weak metric H is symmetric, that is, H .x; y/ D H .y; x/ for any x and y. (e) The weak metric H is separating, that is, x ¤ y H) H .x; y/ > 0, if and only if the domain does not contain any affine line. (f) The weak metric H is unbounded. The proof of this proposition easily follows from the definitions and from Proposition 2.2.

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Proposition 12.3. If the convex domain ¤ Rn does not contain any affine line, then H is a metric in the classical sense. Furthermore, it is complete. A convex domain which does not contain any affine line is called a sharp convex domain. From Theorem 7.1, we deduce the following necessary and sufficient condition for the equality case in the triangle inequality: Theorem 12.4. Let x, y and z be three points in a proper convex domain . We have F .x; y/ C F .y; z/ D F .x; z/ if and only if both triples of boundary points a .x; y/; a .y; z/; a .x; z/ and a .y; x/; a .z; y/; a .z; x/ are aligned in RP n . From Theorem 7.6, we obtain: Theorem 12.5. Let W Œ0; 1 ! be a path in a sharp convex domain of Rn . Then is a geodesic for the Hilbert metric in if and only if there exist two faces D ; D C Q such that for any t1 < t2 in Œ0; 1 we have .t2 / .t1 / 2 C.t / .D C / and @ 1 .t1 / .t2 / 2 C.t2 / .D /. Recall that Cp .D/ is the cone at p on the face D, see Equation (7.3). If is smooth, then it is geodesic if and only if .t P / 2 C.t / .D C / and P .t / 2 C.t / .D / for any t 2 Œ0; 1. We then have the following characterization of smooth geodesics in Hilbert geometry which we formulate only for bounded domains for convenience: Corollary 12.6. Let W Œ0; 1 ! be a path of class C 1 in a bounded convex domain of Rn . Then is a geodesic for the Hilbert metric in if and only if there exist two Q such that for any t , the tangent line to the curve at t meets the faces D ; D C @ boundary @ on D C [ D . Corollary 12.7 (compare [10]). Assume that is strictly convex, or more generally that all but possibly one of its proper faces are reduced to points. Then the Hilbert geometry in is uniquely geodesic.

13 Related questions In this section we briefly discuss some recent developments related to the idea of the Funk distance. Yamada recently introduced what he called the Weil–Petersson Funk metric on Teichmüller space (see [27], and see [15] in this volume). The definition is analogous to one of the definitions of the Funk metric, using in an essential way the noncompleteness of the Weil–Petersson metric on Teichmüller space. One can wonder whether there are Funk-like metrics associated to other interesting known symmetric

Chapter 2. From Funk to Hilbert geometry

61

metrics. This geometry bears some analogies with Thurston’s metric on Teichmüller space and the Funk metric. One can ask for a study of the Thurston metric which parallels the study of the Funk metric (that is, study its balls, its convexity properties, orthogonality and projections, etc.). It should also be of interest to study the geometric properties of the reverse Funk metric, that is, the metric rF on an open convex set defined by rF .x; y/ D F .y; x/. Let us recall that the reverse Funk metric is not equivalent to the Funk metric in any reasonable sense, see Remark 6.5. This is also related to the fact that the forward and backward open balls at some point can be very different, as we already noticed. We note in this respect that the reverse metrics of the Thurston weak metric and of the Weil–Petersson Funk weak metric that we mentioned are also very poorly understood. Finally, there is another symmetrization of the Funk metric, besides the Hilbert metric, namely, its max-symmetrization, defined as S.x; y/ D maxf.F .x; y/; F .y; x/g; and it should be interesting to study its properties. Note that the max-symmetrization of the Thurston metric is an important metric on Teichmüller space, known as the length spectrum metric.

A Menelaus’ Theorem An elementary proof of the triangle inequality for the Funk metric is given by Zaustinsky in [28], and it is recalled in the next appendix. This proof is based on the classical Menelaus’ Theorem.2 For the convenience of the reader, we give a statement and a proof of this result in the present appendix. To state Menelaus’ Theorem, we recall the notion of division ratio of three aligned points. Consider three points A, B, P in Rn with A ¤ B. Then P belongs to the line through A and B if and only if P D tB C .1 t /A for some uniquely defined t 2 R. The number t is called the division ratio or the affine ratio of P relative to . The division ratio is invariant under any affine B and A. We denote it by t D AP AB transformation. Note that if both A ¤ B and A ¤ P , then AP Dt AB

()

PB t 1 D : PA t

For instance P is the midpoint of ŒB; A if and only if AP D 12 or, equivalently, AB PB D 1. Note that the sign is an important component of the division ratio, and in PA 2 This theorem, in the Euclidean and in the spherical case, is quoted by Ptolemy (2nd century A.D) and it is due to Menelaus (second century A.D.) Its proof is contained in Menelaus’ Spherics. No Greek version of Menelaus’ Spherics survived, but there are Arabic versions; cf. the forthcoming English edition [21] from the Arabic original of al-HarawNı (10th century).

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fact we have jP Aj AP D˙ AB jB Aj with a minus sign if and only if A lies between B and P . Proposition A.1 (Menelaus’ Theorem). Let ABC be a non-degenerate Euclidean triangle and let A0 , B 0 , C 0 be three arbitrary points on the lines containing the sides BC , AC , AB. Assume that A0 ¤ C , B 0 ¤ A and C 0 ¤ B. Then, the points A0 , B 0 , C 0 are aligned if and only if we have A0 B B 0 C C 0 A D C1: A0 C B 0 A C 0 B

A

B C

C B

A

Proof. Although a purely geometric proof is possible, it is somewhat delicate to correctly handle the signs of the division ratios throughout the arguments. We follow below a more algebraic approach. It will be convenient to assume that A, B and C are points in Rn with n 3 and to assume that the origin 0 2 Rn does not belong to the plane containing the three points A, B, C . By hypothesis, the point C 0 lies on the line through A and B, therefore C 0 D A C .1 /B;

with

1 C 0A D : 0 C B

0

AB 1 Likewise, we have A0 D B C .1 /C and B 0 D C C .1 /A with A 0C D 1 B0C 0 0 0 and B 0 A D . Now the point C lies on the line through A and B if and only if

Chapter 2. From Funk to Hilbert geometry

63

there exists 2 R such that C 0 D A0 C .1 /B 0 . We then have C 0 D .B C .1 /C / C .1 /.C C .1 /A/ D .1 /.1 /A C B C ..1 / C .1 //C D A C .1 /B: Our hypothesis implies that A, B, C correspond to three linearly independent vectors in Rn , therefore the latter identity implies

D .1 /.1 /;

.1 / D

and

..1 / C .1 // D 0:

Thus, .1 /.1 / D D 1 1

and we conclude that C 0 is aligned with A0 and B 0 if and only if . 1/. 1/. 1/ A0 B B 0 C C 0 A D D 1: A0 C B 0 A C 0 B

Remark. Using similar arguments, we can also prove Ceva’s Theorem. Both theorems are dual to each other. Let us recall the statement: Proposition A.2 (Ceva’s Theorem). Let ABC be a non-degenerate Euclidean triangle and let A0 , B 0 , C 0 be three arbitrary points on the lines containing the sides BC , AC , AB such that A0 ¤ C , B 0 ¤ A and C 0 ¤ B. Then, the lines AA0 , BB 0 and C C 0 are concurrent or parallel if and only if we have A0 B B 0 C C 0 A D 1: A0 C B 0 A C 0 B

A B

C

B A

P

C

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Proof. Let us give the main step of the proof of Ceva’s Theorem. Suppose that the lines AA0 , BB 0 and C C 0 meet at a point P . Then we can find r; s and t in R such that P D tA C .1 t /A0 D sB C .1 s/B 0 D rC C .1 r/C 0 : We also have as before A0 D B C .1 /C , B 0 D C C .1 /A and C 0 D

A C .1 /B. This implies P D tA C .1 t /B C .1 t /.1 /C D .1 s/.1 /A C sB C .1 s/C D .1 r/ A C .1 r/.1 /B C rC: By uniqueness of the barycentric coordinates with respect to the triangle ABC , we have t D .1 s/.1 / D .1 r/ ; s D .1 t / D .1 r/.1 /; r D .1 t/.1 / D .1 s/: Therefore, we have . 1/. 1/. 1/ r t s A0 B B 0 C C 0 A D D D 1: A0 C B 0 A C 0 B

s r t We leave it to the reader to discuss the case where the lines AA0 , BB 0 and C C 0 are parallel and to prove the converse direction.

B The classical proof of the triangle inequality for the Funk metric The triangle inequality is proved in Section 7. Note that it also easily follows from Corollary 2.6 (see also [27]), and it is also a consequence of the Finslerian description of the Funk metric (see Chapter 3 ([24]) in this volume). In this appendix, we present the classical proof of the triangle inequality following Zaustinsky [28]. This proof is similar to the original proof of the triangle inequality for the Hilbert distance, as given by D. Hilbert in [11], although the proof in the case of the Hilbert distance is a bit simpler and does not use the Menelaus theorem. We now prove the triangle inequality for the Funk metric following [28] (see p. 85). Let x, y, z be three points in . In view of Property (b) in Proposition 2.2, we may assume that they are not collinear. Let a, b, c, d , e, f be the intersections with @ of the lines xz, yx and zy, using the notation of the figure concerning the order of intersections.

Chapter 2. From Funk to Hilbert geometry

65

p

a

d

y

e

a z

b

x

f

y c

b

From the invariance of the cross ratio from the perspective at p, we have jx a0 j jb 0 y 0 j jx aj jb yj D 0 jy a0 j jb 0 xj jy aj jb xj and

jy 0 a0 j jb 0 zj jy cj jd zj D : jz cj jd yj jz a0 j jb 0 y 0 j

Multiplying both sides of these two equations, we get jx a0 j jb 0 zj jb xj jd yj jx aj jy cj D : jy aj jz cj jz a0 j jb 0 xj jb yj jd zj The three points b, b 0 and d lie on the sides of the triangle xyz and are aligned, therefore we have by Menelaus’ Theorem (Theorem A.1): jb 0 zj jb xj jd yj D1 jb 0 xj jb yj jd zj and ja0 xj jd xj jf yj D 0 : jd yj jf zj ja zj This gives jx a0 j jx aj jx aj jy cj D ; 0 ja cj jz cj jz a j jz aj

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and the inequality is strict unless a D a0 . This inequality is equivalent to the triangle inequality for the Funk metric3 . Acknowledgement. The first author is partially supported by the French ANR project FINSLER.

References [1]

H. Busemann, Local metric geometry. Trans. Amer. Math. Soc. 56 (1944), 200–274.

[2]

H. Busemann, The geometry of geodesics. Academic Press, New York 1955; reprinted by Dover in 2005.

[3]

H. Busemann, Spaces with homothetic spheres. J. Geometry 4 (1974), 175–186.

[4]

H. Busemann, Recent synthetic differential geometry. Ergeb. Math. Grenzgeb. 54, Springer-Verlag, Berlin 1970.

[5]

H. Busemann and P. J. Kelly, Projective geometry and projective metrics. Academic Press, New York 1953.

[6]

H. S. M. Coxeter, Introduction to geometry. John Wiley & Sons, New York 1961.

[7]

H. G. Eggleston, Convexity. Cambridge Tracts in Math. Math. Phys. 47, Cambridge University Press, New York 1958.

[8]

W. Fenchel, Convex cones, sets and functions. Mimeographed lecture notes, Princeton University, 1956.

[9]

P. Funk, Über Geometrien, bei denen die Geraden die Kürzesten sind. Math. Ann. 101 (1929), 226–237.

[10] P. de la Harpe, On Hilbert’s metric for simplices. In Geometric group theory, Vol. 1 London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge 1993. [11] D. Hilbert, Grundlagen der Geometrie. B. G. Teubner, Stuttgart 1899, several later editions revised by the author, and several translations. [12] D. Hilbert, Mathematische Probleme. Göttinger Nachrichten 1900 (1900), 253–297; reprinted in Archiv der Mathematik und Physik (3) 1 (1901), 44–63 and 213–237; English version “Mathematical problems”, translated by M. Winston Newson, Bull. Amer. Math. Soc. 8 (1902), 437– 445 and 478–479; the English translation was also reprinted in “Mathematical developments arising from Hilbert problems”, Proc. Sympos. Pure Math. XXVII, Part 1, F. Browder (ed.), Amer. Math. Soc., Providence, RI, 1974, and in Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436. [13] S. Kobayashi, Intrinsic metrics on complex manifolds. Bull. Amer. Math. Soc. 73 (1967), 347–349 [14] R. D. Nussbaum, Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations. Differential Integral Equations 7 (1994), no. 5–6, 1649–1707. 3 Observe that the argument shows that the inequality is strict for all x, y, z unless @ contains a Euclidean segment; compare with Theorem 7.1.

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[15] K. Ohshika, H. Miyachi and S. Yamada, Weil–Petersson Funk metric on Teichmüller space. In Handbook of Hilbert geometry, European Mathematical Society, Zürich 2014, 339–352. [16] A. Papadopoulos, Hilbert’s fourth problem. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 391–431. [17] A. Papadopoulos and M. Troyanov, Weak Finsler structures and the Funk weak metric. Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 419–437. [18] A. Papadopoulos and M. Troyanov, Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry. Expo. Math. 27 (2009), no. 2, 109–124. [19] A. Papadopoulos and M. Troyanov, Weak Minkowski spaces. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich, 2014, 11–32. [20] A. Papadopoulos and S.Yamada, Funk and Hilbert geometries in spaces of constant curvature. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 353–379. [21] R. Rashed andA. Papadopoulos, Critical edition with English translation and mathematical commentary of Menelaus’ Spherics, based on the Arabic text of al-HarawNı. To appear. [22] R. T. Rockafellar, Convex analysis. Princeton Mathematical Series 28, Princeton University Press, Princeton, NJ, 1970. [23] Z. Shen, Lectures on Finsler geometry. World Scientific, Singapore 2001. [24] M. Troyanov, Funk and Hilbert geometries from the Finslerian viewpoint. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 69–110. [25] W. P. Thurston Three-dimensional geometry and topology. Vol. 1, edited by Silvio Levy, Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ, 1997. [26] F. A. Valentine, Convex sets. McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York 1964. [27] S.Yamada. Convex bodies in Euclidean and Weil–Petersson geometries. Proc. Amer. Math. Soc. 142 (2014), no. 2, 603–616. [28] E. M. Zaustinsky, Spaces with non-symmetric distance. Mem. Amer. Math. Soc. 34 (1959).

Chapter 3

Funk and Hilbert geometries from the Finslerian viewpoint Marc Troyanov

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . 3 The tautological Finsler structure . . . . . . . . . . . . . 4 The Hilbert metric . . . . . . . . . . . . . . . . . . . . . 5 The fundamental tensor . . . . . . . . . . . . . . . . . . 6 Geodesics and the exponential map . . . . . . . . . . . . 7 Projectively flat Finsler metrics . . . . . . . . . . . . . . 8 The Hilbert form . . . . . . . . . . . . . . . . . . . . . . 9 Curvature in Finsler geometry . . . . . . . . . . . . . . . 10 The curvature of projectively flat Finsler metrics . . . . . 11 The flag curvature of the Funk and the Hilbert geometries 12 The Funk–Berwald characterization of Hilbert geometries A On Finsler metrics with constant flag curvature . . . . . . B On the Schwarzian derivative . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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69 72 75 81 83 84 88 92 94 97 100 104 105 106 107

1 Introduction The hyperbolic space Hn is a complete, simply connected Riemannian manifold with constant sectional curvature K D 1. It is unique up to isometry and several concrete models are available and well known. In particular, the Beltrami–Klein model is a realization in the unit ball Bn Rn in which the hyperbolic lines are represented by the affine segments joining pairs of points on the boundary sphere @Bn and the distance between two points p and q in Bn is half of the logarithm of the cross ratio of a, p with b, q, where a and b are the intersections of the line through p and q with @Bn . We refer to [1] for a historical discussion of this model. In 1895, David Hilbert observed that the Beltrami–Klein construction defines a distance in any convex set U Rn , and this metric still has the properties that affine segments are the shortest

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curves connecting points. More precisely, if the point x belongs to the segment Œp; q, then d.p; q/ D d.p; x/ C d.x; q/: (1.1) Metrics satisfying this property are said to be projective and Hilbert’s IVth problem asks to classify and describe all projective metrics in a given domain U Rn , see Chapter 15 of this volume [42]. The Hilbert generalization of Klein’s model to an arbitrary convex domain U Rn is no longer a Riemannian metric, but it is a Finsler metric. This means that the distance between two points p and q in U is the infimum of the length of all smooth curves ˇ W Œ0; 1 ! U joining these two points, where the length is given by an integral of the type Z 1 P //dt: `.ˇ/ D F .ˇ.t /; ˇ.t (1.2) 0

Here, F W T U ! R is a sufficiently regular function called the Lagrangian. Geometrical considerations lead us to assume that F defines a norm on the tangent space at any point of U; in fact it is also useful to consider non-symmetric and possibly degenerate norms (which we call weak Minkowski norms, see [46]). General integrals of the type (1.2) are the very subject of the classical calculus of variations and Finsler geometry is really a daughter of that field. The early contributions (in the period 1900–1920) in Finsler geometry are due to mathematicians working in the calculus of variations1 , in particular Bliss, Underhill, Landsberg, Hamel, Carathéodory and his student Finsler. Funk is the author of a quite famous book on the calculus of variations that contains a rich chapter on Finsler geometry [26]. The name “Finsler geometry” has been proposed (somewhat improperly) by Élie Cartan in 1933. The 1950 article [12] by H. Busemann contains some interesting historical remarks on the early development of the subject, and the 1959 book by H. Rund [51] gives a broad overview of the development of Finsler geometry during its first 50 years; this book is rich in references and historical comments. In 1908, A. Underhill and G. Landsberg introduced a notion of curvature for twodimensional Finsler manifolds that generalizes the classical Gauss curvature of Riemannian surfaces, and in 1926 L. Berwald generalized this construction to higher dimensional Finsler spaces [7], [33], [62]. This invariant is today called the flag curvature and it generalizes the Riemannian sectional curvature. In 1929, Paul Funk proved that the flag curvature of a Hilbert geometry in a (smooth and strongly convex) domain U R2 is constant K D 1 and Berwald extended this result in all dimensions and refined Funk’s investigation in several aspects, leading to the following characterization of Hilbert geometry [8], [24]: Theorem (Funk–Berwald). Let F be a smooth and strongly convex Finsler metric defined in a bounded convex domain U Rn . Suppose that F is complete with 1Although Bernhard Riemann already considered this possible generalization of his geometry in his Habilitation Dissertation, he did not pursue the subject and Finsler geometry did not emerge as a subject before the early twentieth century.

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constant flag curvature K D 1 and that the associated distance d satisfies (1.1), then d is the Hilbert metric in U. The completeness hypothesis means that every geodesic segment can be indefinitely extended in both directions; in other words, every geodesic segment is contained in a line that is isometric to the full real line R. In fact, Funk and Berwald assumed the Finsler metric to be reversible, that is, F .p; / D F .p; /. But they also needed the (unstated) hypothesis that F is forward complete, meaning that every oriented geodesic segment can be extended as a ray. Observe that reversibility together with forward completeness implies the completeness of F , and therefore the way we characterize the Hilbert metrics in the above theorem is slightly stronger than the Funk–Berwald original statement. Note that completeness is necessary: it is possible to locally construct reversible (incomplete) Finsler metrics for which the condition (1.1) holds and which are not restrictions of Hilbert metrics. The Funk–Berwald Theorem is quite remarkable, and it has been an important landmark in Finsler geometry. Our goal in this chapter is to develop all the necessary concepts and tools to explain this statement precisely. The actual proof is given in Section 12. The rest of the chapter is organized as follows. In Section 2 we explain what a Finsler manifold is and give some basic definitions in the subject with a few elementary examples. In Section 3, we introduce a very natural example of a Finsler structure on a convex domain, discovered by Funk, and which we called the tautological Finsler structure. We also compute the distances and the geodesics for the tautological structure. In Section 4, we introduce Hilbert’s Finsler structure as the symmetrization of the tautological structure and compute its distances and geodesics. In Section 5, we introduce the fundamental tensor gij of a Finsler metric (assuming some smoothness and strong convexity hypothesis). In Section 6, we compute the geodesic equations and we introduce the notion of spray and the exponential map. In Section 7, we discuss various characterizations of projectively flat Finsler metrics, that is, Finsler metrics for which the affine lines are geodesics. In Section 8, we introduce the Hilbert differential form and the notion of Hamel potential, which is a tool to compute distances in a general projectively flat Finsler space. In Section 9, we introduce the curvature of a Finsler manifold based on the classical Riemannian curvature of some associated (osculating) Riemannian metric. The curvature of projectively flat Finsler manifolds is computed in Sections 10 and 11. The characterization of Hilbert geometries is given in Section12 and the chapter ends with two appendices: one on further developments of the subject and one on the Schwarzian derivative. We tried to make this chapter as self-contained as possible. The material we present here is essentially built on Berwald’s paper [8], but we have also used a number of more recent sources. In particular we found the books [53], [54] by Z. Shen and [15] by S.-S. Chern and Z. Chen quite useful.

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2 Finsler manifolds We start with the main definition of this chapter: Definition 2.1. 1. A weak Finsler structure on a smooth manifold M is a lower semi-continuous function F W TM ! Œ0; 1 such that for every point x 2 M , the restriction Fx D FjTx M is a weak Minkowski norm, that is, it satisfies the following properties: i) F .x; 1 C 2 / F .x; 1 / C F .x; 2 /, ii) F .x; / D F .x; / for all 0, for any x 2 M and 1 ; 2 2 Tx M . 2. If F W TM ! Œ0; 1/ is finite and continuous and if F .x; / > 0 for any ¤ 0 in the tangent space Tx M , then one says that F is a Finsler structure on M . We shall mostly be interested in Finsler structures, but extending the theory to the case of weak Finsler structures can be useful in some arguments. The notion of weak Finsler structure appears for instance in [19], [29], [44]. Another situation where weak Finsler structures appear naturally is the field of sub-Finsler geometry. The function F itself is called the Lagrangian of the Finsler structure; it is also called the (weak) metric, since it is used to measure the length of a tangent vector. The weak Finsler metric is called reversible if F .x; / is actually a norm, that is, if it satisfies F .x; / D F .x; / for all points x and any tangent vector 2 Tx M . The Finsler metric is said to be of class C k if the restriction of F to the slit tangent bundle TM 0 D f.x; / 2 TM j ¤ 0g TM is a function of class C k . If it is C 1 on TM 0 , then one says F is smooth. Some of the most elementary examples of Finsler structures are the following: Examples 2.2. i) A weak Minkowski norm F0 W Rn ! R defines a weak Finsler structure F on M D Rn by F .x; / D F0 ./: See [46] for a discussion of weak Minkowski norms. This Finsler structure is constant in x and is thus translation-invariant. Such structures are considered to be the flat spaces of Finsler geometry. ii) A Riemannian metric g on the manifold M defines a Finsler structure on M by p F .x; / D gx .; /.

iii) If g is a Riemannian metric on M and 2 1 .M / is a 1-form whose g-norm is everywhere smaller than 1, then one can define a smooth Finsler structure on M by p F .x; / D gx .; / C x ./:

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Such structures are called Randers metrics. iv) If is an arbitrary 1-form on the Riemannian manifold .M; g/, then one can define two new Finsler structures on M by p p F1 .x; / D gx .; / C jx ./j; F2 .x; / D gx .; / C maxf0; x ./g: v) If F is a Finsler structure on M , then the reverse Finsler structure F W TM ! Œ0; 1/ is defined by F .x; / D F .x; /: Additional examples will be given below. A number of concepts from Riemannian geometry naturally extend to Finsler geometry, in particular one defines the length of a smooth curve ˇ W Œ0; 1 ! M as Z 1 P F .ˇ.s/; ˇ.s//dt: `.ˇ/ D 0

We then define the distance dF .x; y/ between two points p and q to be the infimum of the length of all smooth curves ˇ W Œ0; 1 ! M joining these two points (that is, ˇ.0/ D p and ˇ.1/ D q). This distance satisfies the axioms of a weak metric, see [46]. Together with the distance comes the notion of completeness. Definition 2.3. A sequence fxi g M is a forward Cauchy sequence if for any " > 0, there exists an integer N such that d.xi ; xiCk / < " for any i N and k 0. The Finsler manifold .M; F / is said to be forward complete if every forward Cauchy sequence converges. We similarly define backward Cauchy sequence by the condition d.xi Ck ; xi / < ", and the corresponding notion of backward complete. The Finsler manifold .M; F / is a complete Finsler manifold if it is both backward and forward complete. A weak Finsler structure on the manifold M can also be seen as a “field of convex sets” sitting in the tangent bundle TM of that manifold. Indeed, given a Finsler structure F on the manifold M , we define its domain DF TM to be the set of all vectors with finite F -norm. The unit domain DF is the bundle of all tangent unit balls D f.x; / 2 TM j F .x; / < 1g: The unit domain contains the zero section in TM and its restriction x D \ Tx M to each tangent space is a bounded and convex set. It is called the tangent unit ball of F at x, while its boundary x D f 2 Tx M j F .x; / D 1g Tx M is called the indicatrix of F at x. We know from Theorem 3.12 of [46] that the Lagrangian F W TM ! R can be recovered from TM via the formula F .x; / D infft > 0 j 1t 2 g:

(2.1)

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Sometimes, a weak Finsler structure is defined by specifying its unit domain, and the Lagrangian is then obtained from (2.1). Let us give two elementary examples; more can be found in [43]: Example 2.4. Given a bounded open set 0 Rn that contains the origin, one naturally defines a Finsler structure on Rn by parallel transporting 0 . That is, T Rn D Rn Rn is defined as D f.x; / 2 Rn Rn j 2 0 g: The space Rn equipped with this Finsler structure is characterized by the property that its Lagrangian F is invariant with respect to the translations of Rn , i.e. F is independent of the point x: F .x; / D F0 ./: Such a Finsler structure on Rn is of course the same thing as a Minkowski norm, see [46]. Example 2.5. Let F be an arbitrary weak Finsler structure on a manifold M with unit domain TM . If Z W M ! TM is a continuous vector field such that F .x; Z.x// < 1, for any point x (equivalently Z.M / ), a new weak Finsler structure can be defined as Z D f.x; / 2 TM j 2 .x Z.x//g:

(2.2)

Here, x Z.x/ is the translate of x Tx M by the vector Z.x/. The corresponding Lagrangian is given by FZ .x; / D infft > 0 j 1t 2 .x Z.x//g; and for ¤ 0, it is computable from the identity

(2.3)

C Z.x/ D 1: F x; FZ .x; /

(2.4)

This weak Finsler structure FZ is called the Zermelo transform of F with respect to the vector field Z. We end this section with a word on Berwald spaces. Recall first that in Riemannian geometry, the tangent space of the manifold at every point is isometric to a fixed model, which is a Euclidean space. Furthermore, using the Levi-Civita connection, one defines parallel transport along any piecewise C 1 curve, and this parallel transport induces an isometry between the tangent spaces at any point along the curve. In a general Finsler manifold, neither of these facts holds. This motivates the following definition: Definition 2.6. A weak Finsler manifold .M; F / is said to be Berwald if there exists a torsion free linear connection r (called an associated connection) on M whose

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associated parallel transport preserves the Lagrangian F . That is, if W Œ0; 1 ! M is a smooth path connecting the point x D .0/ to y D .1/ and P W Tx M ! Ty M is the associated r-parallel transport, then F .y; P .// D F .x; / for any 2 Tx M . This definition is a slight generalization of the usual definition, compare with [15], Proposition 4.3.3. It is known that the connection associated to a smooth Berwald metric can be chosen to be the Levi-Civita connection of some Riemannian metric [37], [60].

3 The tautological Finsler structure Definition 3.1. Let us consider a proper convex set U Rn . This will be our ground manifold. The tautological weak Finsler structure Ff on U is the Finsler structure for which the unit ball at a point x 2 U is the domain U itself, but with the point x as center. The unit domain of the tautological weak Finsler structure is thus defined as D f.x; / 2 T U j 2 .U x/g T U D U Rn ; and the Lagrangian is given by

® ¯ Ff .x; / D infft > 0 j 2 t .U x/g D inf t > 0 j x C t 2 U :

Equivalently, Ff is given by Ff .x; / D 0 if the ray x C RC is contained in U, and

Ff .x; / > 0 and

xC

2 @U Ff .x; /

otherwise. The convex set U can be recovered from the Lagrangian as follows: U D fz 2 Rn j Ff .x; z x/ < 1g:

(3.1)

By construction, this formula is independent of x. The tautological weak structure has been introduced by Funk in 1929, see [24]. We will often call it the Funk weak metric on U (whence the index f in the notation Ff ). Remark 3.2. If the convex set U contains the origin, then the tautological weak structure Ff is the Zermelo transform of the Minkowski norm with unit ball U for the position vector field Zx D x.

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Example 3.3. Suppose U is the Euclidean unit ball fx 2 Rn j kxk < 1g, where k k is the Euclidean norm. Then 2 x C Ff 2 @U () x C Ff D 1: Rewriting this condition as Ff2 .1 kxk2 / 2F hx; i kk2 D 0; the non-negative root of this quadratic equation is p hx; i C hx; i2 C .1 kxk2 /kk2 ; Ff .x; / D .1 kxk2 /

(3.2)

which is the Lagrangian of the tautological Finsler structure in the Euclidean unit ball. Observe that this is a Randers metric. Example 3.4. Consider a half-space H D fx 2 Rn j h; xi < g Rn where is a non-zero vector and 2 R; here h ; i is the standard scalar product in Rn . We then have D E 2 @H () ; x C D ; xC F F which implies h; i ;0 : (3.3) Ff .x; / D max h; xi We now compute the distance between two points in the tautological Finsler structure: Theorem 3.5. The tautological distance in a proper convex domain U Rn is given by ja pj ; (3.4) %f .p; q/ D log ja qj where a is the intersection of the ray starting at p in the direction of q with the boundary @U: a D @U \ .p C RC .q p// : If the ray is contained in U, then a is considered to be a point at infinity and %f .p; q/ D 0. Definition 3.6. The distance (3.4) is called the Funk metric in U. In his paper [24], Funk introduced the tautological Finsler structure while the distance (3.4) appears as an interesting geometric object in the 1959 memoir [67] by Eugene Zaustinsky, a student of Busemann. We refer to [44], [67] and Chapter 2 of this volume [47] for expositions of the Funk geometry.

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Proof of Theorem 3.5. We follow the proof in [44]. Consider first the special case where U D H is the half-space fx 2 Rn j h; xi < g Rn for some vector ¤ 0. Then the tautological Lagrangian is given by (3.3) and the length of a curve ˇ joining p to q is Z 1 ² ³ P h; ˇ.s/i max 0; dt `.ˇ/ D s h; ˇ.s/i 0 ³ ² Z 1 . h; ˇ.s/i/0 D dt max 0; j h; ˇ.s/ij 0 ² ³ h; pi ; max 0; log h; qi P with equality if and only if h; ˇ.s/i has almost everywhere constant sign. It follows that the tautological distance in the half-space H is given by ²

%f .p; q/ D max 0; log

h; pi h; qi

³

:

(3.5)

We can rewrite this formula in a different way. Suppose that the ray LC starting at p in the direction of q meets the hyperplane @H at a point a. Then h; pi D and h; pi D h; a pi D ja pj h; i; where D

ap . japj

We also have h; qi D ja qj h; i; therefore ja pj : %f .p; q/ D log ja qj

(3.6)

If the ray LC is contained in the half-space U, then %f .p; q/ D 0 and the above formula still holds in the limit sense if we consider the point a to be at infinity. For the general case, we will need two lemmas on general tautological Finsler structures. Lemma 3.7. Let U1 and U2 be two convex domains in Rn . If F1 and F2 are the Lagrangians of the corresponding tautological structures, then U1 U2 () F1 .x; / F2 .x; / for all .x; / 2 T U. Proof. We have indeed F1 .x; / D infft > 0 j 2 t .U1 x/g infft > 0 j 2 t .U2 x/g D F2 .x; /: We also have the following reproducing formula for the tautological Lagrangian.

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Lemma 3.8. Let U be a convex domain in Rn and p; q 2 U. If q D p C t for some 2 Rn , and t 0, then Ff .p; / Ff .q; / D : (3.7) 1 tFf .p; / Proof. If D 0, then there is nothing to prove. Assume ¤ 0 and denote by LpC and C LC q the rays in the direction starting at p and q. We obviously have Lp U if and C only if Lq U. But this means that Ff .p; / D 0 () Ff .q; / D 0; so Equation (3.7) holds in this case. If on the other hand the rays are not contained in U, then we have LpC \ @U D LC q \ @U: We denote by a D a.x; / this intersection point; then by definition of F we have aDqC

DpC : Ff .q; / Ff .p; /

Since q D p C t , we have

(3.8)

1 D t ; Ff .q; / Ff .p; / from which (3.7) follows, since ¤ 0. To complete the proof of Theorem 3.5, we need to compute the tautological distance between two points p and q in a general convex domain U 2 Rn . We first compute the length of the affine segment Œp; q which we parametrize as yx ; ˇ.t / D p C t ; D jy xj where 0 t jy xj. Let us denote as above by a the point LpC \ @U where LpC is the ray with origin p in the direction q. Then Ff .p; / D

1 ; ja pj

and using (3.7) we have P // D Ff .p C t ; / D Ff .ˇ.t /; ˇ.t

1

Ff .p; / 1 japj : D D t ja pj t 1 tFf .p; / 1 japj

The length of ˇ is then Z jqpj dt 1 1 log : D log `.ˇ/ D ja pj t ja pj jq pj ja pj 0

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But q 2 Œp; a, therefore ja pj jq pj D ja qj and we finally have

`.ˇ/ D log This proves that

ja pj : ja qj

ja pj %f .p; q/ log : ja qj In fact we have equality. To see this, choose a supporting hyperplane for U at a, and let H be the corresponding half-space containing U (recall that a hyperplane in Rn is said to support the convex set U if it meets the closure of that set and U is contained in one of the half-space bounded by that hyperplane). Using Lemma 3.7 and Equation (3.6), we obtain ja pj %f .p; q/ log : ja qj The proof of Theorem 3.5 follows now from the two previous inequalities. Remark 3.9. 1. From Equation (3.8), one sees that a p D . Ff .q;/

Ff .p;/

and a q D

Therefore, the Funk distance can also be written as Ff .q; q p/ : %f .p; q/ D log Ff .p; q p/

(3.9)

2. The proof of Theorem 3.5 given here is taken from [44]. Another interesting proof is given in [66]. Proposition 3.10. The tautological distance %f in a proper convex domain U satisfies the following properties. a) The distance %f is projective, that is, for any point z 2 Œp; q we have %f .p; q/ D %f .p; z/ C %f .z; p/: b) %f is invariant under affine transformations. c) %f is forward complete. d) %f is not backward complete. The proof is easy, see also Chapter 2 of this volume [47] for more on Funk geometry. Proposition 3.11. The unit speed linear geodesic starting at p 2 U in the direction 2 Tp U is the path .1 es / ˇp; .s/ D p C (3.10) Ff .p; /

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Proof. Let us define ˇ by (3.10); we have then ap D

; F .p; /

Therefore,

and

%f .p; ˇ.s// D log

a ˇ.s/ D

es : F .p; /

ja pj 1 D log s ja ˇ.s/j e

D s:

We can also prove the proposition using the fact that (3.10) parametrizes an affine segment and is therefore a minimizer for the length. We then only need to check that the curve has unit speed. Indeed we have P ˇ.s/ D

es ; Ff .p; /

and using Equation (3.7), we thus obtain

.1 es / es F pC ; Ff .p; / Ff .p; / Ff .p; / es D s / .1e Ff .p; / 1 F .p; / Ff .p;/ f

P Ff .ˇ.s/; ˇ.s// D

D 1: The tautological Finsler structure Ff in a convex domain U is not reversible. We can thus define the reverse tautological Finsler structure Ff to be the Finsler structure whose Lagrangian is defined as Ff .p; / D Ff .p; /: We then have the following Proposition 3.12. The distance %f associated to the reverse tautological Finsler structure in a proper convex domain U Rn is given by %f .p; q/

jb qj D %f .q; p/ D log ; jb pj

(3.11)

where b is the intersection of the ray starting at q in the direction p with @U. The proof is obvious. Observe in particular that %f is also a projective metric.

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4 The Hilbert metric Definition 4.1. The Hilbert Finsler structure Fh in a convex domain U is the arithmetic symmetrisation of the tautological Finsler structure: 1 .Ff .p; / C Ff .p; //: 2

Fh .x; / D

By construction, the Hilbert Finsler structure is reversible. Its tangent unit ball at a point p 2 U is obtained from the tautological unit ball by the following procedure from convex geometry: first, take the polar dual of .U p/, then symmetrize this convex set and finally take again the polar dual of the result. This procedure is called the harmonic symmetrization of U based at p, see [45] for the details. Example 4.2. Symmetrizing the metric (3.2) we obtain the Hilbert metric in the unit ball Bn : p .1 kxk2 /kk2 C hx; i2 : (4.1) Fh .x; / D .1 kxk2 / Observe that this is a Riemannian metric. We shall prove later on that it has constant sectional curvature K D 1. This metric is the Klein model for hyperbolic geometry. Using the fact that the affine segment joining two points p and q in U has minimal length for both the Funk metric Ff and its reverse Ff , it is easy to prove the following Proposition 4.3. The Hilbert distance %h in a convex domain U Rn is obtained by symmetrizing the Funk distance (3.4) in that domain:

%h .p; q/ D

1 1 ja pj jb qj .%f .p; q/ C %f .q; p// D log : 2 2 ja qj jb pj

b p

@U q

a

Using (3.9), one can also write the Hilbert distance as

%h .p; q/ D

Ff .q; q p/ Ff .p; p q/ 1 : log 2 Ff .p; q p/ Ff .q; p q/

We also have the following properties:

(4.2)

(4.3)

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Proposition 4.4. The Hilbert distance %h in a proper convex domain U satisfies the following a) The distance %h is projective: for any point z 2 Œp; q we have %h .p; q/ D %h .p; z/ C %h .z; p/: b) %h is invariant under projective transformations. c) %h is (both forward and backward) complete. The proof is elementary. For an introduction to Hilbert geometry, we refer to Sections 28 and 50 in [14] and Section 18 in [13]. We now describe the geodesics in Hilbert geometry. Proposition 4.5. The unit speed linear geodesic starting at p 2 U in the direction 2 Tp U is the path ˇp; .s/ D p C '.s/ ; (4.4) where ' is given by '.s/ D

.es es / : Ff .p; /es C Ff .p; /es

(4.5)

Proof. To simplify notation we write F D Ff .p; / and F D Ff .p; /. From (3.8) we have a p D F and b q D F , therefore

1 1 ja pj jb ˇ.s/j 1 C F '.s/ %h .p; ˇ.s// D log D log : 2 ja ˇ.s/j jb pj 2 1 F '.s/ From (4.5), we have .F es C F es / C .F es F es / 1 C F '.s/ D 1 F '.s/ .F es C F es / .F es F es / D

.F C F / e s D e 2s ; .F C F / es

and we conclude that %h .p; ˇ.s// D s: Corollary 4.6. The metric balls in a Hilbert geometry are convex sets. Proof. We see from the previous proposition that the ball of radius r around the point p 2 U is the set of points z 2 U such that er Ff .p; z p/ C er Ff .p; p z/ .er er /; which is a convex set. Note that another proof of this proposition is given in Chapter 2 of this volume [47].

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5 The fundamental tensor Our goal in this section is to write down and study an ordinary differential equation for the geodesics in a Finsler manifold .M; F /. To this aim we need to assume the following condition: Definition 5.1. A Finsler metric F on a smooth manifold M is said to be strongly convex if it is smooth on the slit tangent bundle TM 0 and if the vertical Hessian of F 2 at a point .p; / 2 T 0 M 1 @2 ˇˇ gp; . 1 ; 2 / D F 2 .p; C u1 1 C u2 2 / (5.1) ˇ 2 @u1 @u2 u1 Du2 D0 is positive definite. The bilinear form (5.1) is called the fundamental tensor of the Finsler metric. This condition is called the Legendre–Clebsch condition in the calculus of variations. Geometrically it means that the indicatrix at any point is a hypersurface of strictly positive Gaussian curvature. Classical Finsler geometry has sometimes the reputation of being an “impenetrable forest of tensors”,2 and we shall need to venture a few steps into this wilderness. It will be convenient to work with local coordinates on TM ; more precisely, if U M is the domain of some coordinate system x 1 ; x 2 ; : : : ; x n , then any vector 2 T U can be written as D y i @[email protected] i . The 2n functions on T U given by x 1 ; : : : ; x n ; y 1 ; : : : ; y n are called natural coordinates on T U . The restriction of the Lagrangian on T U is thus given by a function of 2n variables F .x; y/. The Legendre-Clebsch condition states that F .x; y/ is smooth (although C 3 would suffice) on fy ¤ 0g and that the matrix given by 1 @2 F 2 .x; y/ (5.2) gij .x; y/ D 2 @y i @y j is positive definite for any y ¤ 0. The fundamental tensor shares some formal properties with a Riemannian metric, but it is important to remember that it is not defined on U (nor on the manifold M ), but on the slit tangent bundle T 0 U . Manipulating tensors in Finsler geometry needs to be done with care. In general, a tensor on a Finsler manifold .M; F / is a field of multilinear maps on TM which smoothly depends on a point .x; y/ 2 TM 0 (and not only on a point x 2 M as in Riemannian geometry). In the case of the fundamental tensor, note that3 gy .u; v/ D gx;y .u; v/ D gij .x; y/ui v j ; where u D ui .x/ @[email protected] i and v D vj .x/ @[email protected] are elements in Tx M . 2 This comment on the subject goes back to the paper [12] by Busemann. The first sentence of this nice paper is “The term Finsler space evokes in most mathematicians the picture of an impenetrable forest whose entire vegetation consists of tensors.” A quick glance at [51] will probably convince the reader. 3 We use the summation convention: terms with repeated indices are summed from 1 to n.

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Given a point .x; y/ 2 TM 0 , we have a canonical element in Tx M , namely the vector y itself, and we can evaluate a given tensor on that canonical vector. In particular we have the following basic fact. Lemma 5.2. The Lagrangian and the fundamental tensor are related by q q F .x; y/ D gy .y; y/ D gij .x; y/y i y j : Proof. Recall that if f W Rn n f0g ! R is a smooth positively homogenous function of degree r on Rn , that is, f .y/ D r f .y/ for 0, then its partial derivatives are @f positively homogenous functions of degree .r 1/ and r f .y/ D y i @y i (see [46]).

Applying this fact twice to the function y 7! 12 F 2 .x; y/ proves the lemma.

Observe that the same argument also shows that the fundamental tensor gij .x; y/ is 0-homogenous with respect to y. This type of argument plays a central role in Finsler geometry and anyone venturing into the subject will soon become an expert in recognizing how homogeneity is being used (sometimes in a hidden way) in calculations, or she will quit the subject.

6 Geodesics and the exponential map We now consider a curve ˇ W Œa; b ! M of class C 1 in the Finsler manifold .M; F /. Recall that its length is defined as Z b P `.ˇ/ D F .ˇ.s/; ˇ.s//ds: (6.1) a

We also define the energy of the curve ˇ by Z b P E.ˇ/ D F 2 .ˇ.s/; ˇ.s//ds:

(6.2)

a

The following basic inequality between these functionals holds: Lemma 6.1. For any curve ˇ W Œa; b ! M of class C 1 , we have `.ˇ/2 .b a/ E.ˇ/; P with equality if and only if ˇ has constant speed, i.e. t 7! F .ˇ.s/; ˇ.s// is constant. P Proof. Let us denote by f .s/ D F .ˇ.s/; ˇ.s// the speed of the curve. We have by the Cauchy–Schwarz inequality Z b 1=2 Z b 1=2 p Z b p 1 f .s/ds 12 ds v.t /2 ds D .b a/ E.ˇ/: `.ˇ/ D a

a

a

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Furthermore, the equality holds if and only if f .s/ and 1 are collinear as elements of L2 .a; b/, that is, if and only if the speed f .s/ is constant. Corollary 6.2. The curve ˇ W Œa; b ! M is a minimal curve for the energy if and only if it minimizes the length and has constant speed. Definition 6.3. The curve ˇ W Œa; b ! M is a geodesic if it is a critical point of the energy functional. Sometimes the terminology varies, and the term geodesic also designates a critical or a minimal curve for the length functional. Note that our notion imposes the restriction that ˇ has constant speed. From the point of view of calculations, this is an advantage since the length is invariant under forward reparametrization. We now seek to derive the equation satisfied by the geodesics. By the classical theory of the calculus of variations, a curve s 7! ˇ.s/ in some coordinate domain U M is a critical point of the energy functional (6.2) if and only if the following Euler–Lagrange equations @F 2 d @F 2 D ds @y @x

. D 1; : : : ; n/

(6.3)

P hold, where .x.s/; y.s// D .ˇ.s/; ˇ.s//. Using the fundamental tensor, one writes F 2 .x; y/ D gij .x; y/y i y j , therefore @gij i j @F 2 D y y @x @x

(6.4)

and @gij i j @F 2 D y y C gi y i C gj y j : @y @y Observe that the sums gi y i and gj y j coincide. Using the homogeneity of F 2 in y and the fact that F is of class C 3 on y ¤ 0, we obtain @3 F 2 @3 F 2 @gij i i y D y D y i D 0: @y @y @y i @y j @y i @y @y j Therefore, @F 2 D 2gi y i : @y Differentiating this expression with respect to s, we get @gi i dx j dy i @gi i dy j d @F 2 D 2 y y C 2g C 2 : i ds @y @x j ds ds @y j ds

(6.5)

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We have, as before, xP i D y i , we obtain

@gi @y j

y i D 0 (because gi is 0-homogenous), therefore using

d @F 2 @gi D 2 j y i y j C 2gi yP i : ds @y @x

(6.6)

From Equations (6.3), (6.5) and (6.6), we obtain for D 1; : : : ; n, X 1 X @gij @gi i j gi yP i D 2 y y : 2 @x @x j i

i;j

Multiplying this identity by g k .x; y/, where g k gi D ıik , and summing over , one gets 1 X k @gij @gi i j k g 2 j y y : (6.7) yP D 2 @x @x i;j;

Proposition 6.4. The C 2 curve ˇ.s/ 2 .M; F / is a geodesic for the Finsler metric F if only if in local coordinates, we have xR k C ijk xP i xP j D 0;

(6.8)

where ˇ.s/ D .x 1 .s/; : : : ; x n .s// is a local coordinate expression for the curve and

1 @gj @gij @gi D g k C ; 2 @x j @x i @x are the formal Christoffel symbols of F . ijk .x; y/

Proof. A direct calculation shows that Equation (6.8) is equivalent to Equation (6.7) with y k D xP k . Another way to write the geodesic equation is to introduce the functions 1 k .x; y/ y i y j 2 ij 1 @gi @gij D g k 2 j y i y j 4 @x @x @F 1 k @2 F j y ; D g @y @x j @x 2

G k .x; y/ D

so the geodesic equation can be written as yP k C 2G k .x; y/ D 0;

xP k D y k

.k D 1; : : : n/:

(6.9)

The vector field on T U defined by G D yk

@ @ 2G k .x; y/ k k @x @y

(6.10)

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87

is in fact independent on the choice of the coordinates x j on U and is therefore globally defined on the tangent bundle TM . This vector field is called the spray of the Finsler manifold. A significant part of a Finsler geometry is contained in the behavior of its spray (see [53]). Observe that a curve ˇ.s/ 2 M is geodesic if and only if its lift P .ˇ.s/; ˇ.s// 2 TM is an integral curve for the spray G. Observe the rather surprising analogy between the equation (6.8) and the Riemannian geodesic equation. This is due to the fact that the 0-homogeneity of gi;j .x; y/ in y implies that all the non-Riemannian terms in the calculation of the Euler–Lagrange equation end up vanishing. The important difference between the Finslerian and the Riemannian cases lies in the fact that the formal Christoffel symbols ijk are not functions of the coordinates x i only. Equivalently, the spray coefficients are generally not quadratic polynomials in the coordinates y i . In fact, it is known that the spray coefficients G k .x; y/ of a Finsler metric F are quadratic polynomials in the coordinates y i if and only if F is Berwald. On a smooth manifold M with a strongly convex Finsler metric, one can define an exponential map as it is done in Riemannian geometry. Because the coefficients G k .x; y/ of the geodesic equation are Lipschitz continuous, for any point p 2 M and any vector 2 Tp M , there is locally a unique solution to the geodesics equation s 7! .s/ 2 M;

< s < ;

with initial conditions .0/ D p, P .0/ D . Observe that .s/ D .s/;

0;

so if is small enough, then .1/ is well defined and we denote it by expp ./ D .1/: We then have the following Theorem 6.5. The set Op of vectors 2 Tp M for which expp ./ is defined is a neighborhood of 0 2 Tp M . The map expp W Op ! M is of class C 1 in the interior of Op and its differential at 0 is the identity. In particular, the exponential is a diffeomorphism from a neighborhood of 0 2 Tp M to a neighborhood of p in M . If .M; F / is connected and forward complete, then expp is defined on all of Tp M and is a surjective map expp W Tp M ! M: The coefficients G k .x; y/ in the geodesic equation are in general not C 1 at y D 0, therefore the proof of this theorem is more delicate than its Riemannian counterpart. See [6] for the details. Finally note that expp has no reason to be of class C 2 . In fact a result of AkbarZadeh states that expp is a C 2 map near the origin if and only if .M; F / is Berwald.

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7 Projectively flat Finsler metrics Definition 7.1. 1. A Finsler structure F on a convex set U Rn is projectively flat if every affine segment Œp; q U can be parametrized as a geodesic. 2. A Finsler manifold .M; F / is locally projectively flat if each point admits a neighborhood that is isometric to a convex region in Rn with a projectively flat Finsler structure. Examples 7.2. Basic examples are the following. a) Minkowski metrics are obviously projectively flat. b) The Funk and the Hilbert metrics are projectively flat Finsler metrics. In particular the Klein metric (4.1) in Bn is a projectively flat Riemannian metric. c) The canonical metric on the sphere S n RnC1 is locally projectively flat. Indeed, the central projection of the half-sphere S n \fxn < 0g on the hyperplane fxn D 1g with center the origin maps the great circles in S n on affine lines on that hyperplane (in cartography, this map is called the gnomonic representation of the sphere). In formulas, the map Rn ! S n is given by x 7! p.x;1/ 2 , and 1Ckxk

the metric can be written as

p .1 C kxk2 /kk2 hx; i2 : Fx .x; / D .1 C kxk2 /

(7.1)

This is the projective model for the spherical metric. A classic result, due to E. Beltrami, says that a Riemannian manifold is locally projectively flat if and only if it has constant sectional curvature, see Theorem 10.6 below. In Finsler geometry, there are more examples and the Finsler version of Hilbert’s IVth problem is to determine and study all conformally flat (complete) Finsler metrics in a convex domain. A first result on projectively flat Finsler metrics is given in the next proposition: Proposition 7.3. A smooth and strongly convex Finsler metric F on a convex domain U Rn is projectively flat if only if its spray coefficients satisfy G k .x; y/ D P .x; y/ y k for some scalar function P W T U ! R. Proof. The proof is standard, see e.g. [15]. Suppose that the affine segments are geodesics. This means that ˇ.s/ D p C '.s/ satisfy the geodesic equation (6.9) for any p 2 U , 2 U and some (unknown) function '.s/. Along ˇ we have k y k D xP k D '.s/ P ;

k yP k D '.s/ R ;

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89

therefore Equation (6.9) can be written as k C 2'P 2 .s/G k .ˇ.s/; / D 0; '.s/ R

which implies that G k .x; y/ D P .x; y/ y k with P .x; / D

'.0/ R : 2'.0/ P 2

(7.2)

The function P .x; / in the previous proposition is called the projective factor. It can be computed from (7.2) if the geodesics are explicitly known. For instance we have the Proposition 7.4. The projective factor of the tautological Finsler structure Ff in the convex domain U is given by 1 Ff .x; y/: 2 For the Hilbert Finsler structure Fh , we have Pf .x; y/ D

Ph .x; y/ D

1 .Ff .x; y/ Ff .x; y//: 2

Proof. The geodesics for Ff are given by ˇ.s/ D p C '.s/y with '.s/ D

1 .1 es /: Ff .p; /

Therefore, Pf .x; y/ D

'.0/ R 1 D Ff .p; /: 2 2'.0/ P 2

The geodesics for the Hilbert metric Fh .x; y/ D curves ˇ.s/ D p C '.s/y with '.s/ D

1 .F 2 f

.x; y/ C Ff .x; y// are the

.es es / : Ff .p; /es C Ff .p; /es

The derivative of this function is '.s/ P D and '.s/ R D 4

2.Ff .p; / C Ff .p; // ; .Ff .p; /es C Ff .p; /es /2

.Ff .p; / C Ff .p; //.Ff .p; /es Ff .p; /es / : .Ff .p; /es C Ff .p; /es /3

Therefore, Ph .x; y/ D

'.0/ R 1 D .Ff .x; y/ Ff .x; y//: 2 2'.0/ P 2

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It is clear that the distance associated to a projectively flat metric is projective in the sense of Definition 2.1 in [46]. It is also clear that the sum of two projective (weak) distances is again projective. This suggests that one can write down a linear condition on F which is equivalent to projective flatness. This is the content of the next proposition which goes back to the work [31] of G. Hamel. Proposition 7.5. Let F W T U ! R be a smooth Finsler metric on the convex domain U. The following conditions are equivalent. (a) F is projective flat. @F @2 F (b) y k k m m D 0, for 1 m n. @x @x @y 2 @2 F @ F D , for 1 j; m n. (c) @x j @y m @y j @x m @2 F @2 F (d) y k m k D y k k m : @x @y @x @y Proof. The Euler–Lagrange equation for the length functional (6.1) is 0D

d @F @F @2 F @2 F @F k D x P yP k ; @x m dt @y m @x m @x k @y m @y k @y m

and this can be written as @F @2 F @2 F k y P D yk : @x m @x k @y m @y k @y m Now F is projectively flat if and only if x.t / D p C t is a solution (recall that the length is invariant under reparametrization), which is equivalent to yP k D 0. The equivalence (a) , (b) follows. To prove (b) ) (c), we differentiate (b) with respect to y j . This gives Aj m D y k and thus 0D

@2 F @2 F @3 F C D 0; m j @x @y @y j @x m @y j @x k @y m

@2 F @2 F 1 .Aj m Amj / D j m j m ; 2 @x @y @y @x

which is equivalent to (c). (c) ) (d) is obvious. Finally, to prove (d) ) (b), we use the homogeneity of F in y. We have F .x; y/ D @F y k @y k , therefore condition (d) implies 2 2 @F k @ F k @ F D y ; D y @x m @x m @y k @x k @y m

which is equivalent to (b).

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Remark 7.6. In [50] A. Rapcsàk gave a generalization of the previous conditions for the case of a pair of projectively equivalent Finsler metrics, that is, a pair of metrics having the same geodesics up to reparametrization. Example 7.7. We know that the tautological (Funk) Finsler metric Ff in a convex domain U is projectively flat. Using the previous proposition we have more examples: i) the reverse Funk metric Ff .x; / D Ff .x; /, ii) the Hilbert metric Fh .x; / D 12 .Ff .x; / C Ff .x; //, and

iii) the metric Ff .x; / C F0 ./, where F0 is an arbitrary (constant) Minkowski norm are projectively flat. More generally if F1 , F2 are projectively flat, then so is the sum F1 C F2 . Assuming either F1 or F2 to be (forward) complete, the sum is also a (forward) complete solution to Hilbert’s IVth problem. The following result gives a general formula computing the projective factor of a projectively flat Finsler metric: Lemma 7.8. Let F .x; y/ be a smooth projectively flat Finsler metric on some domain in Rn . Then the following equations hold: 2FP D y k and

@F @x k

(7.3)

@F @P @F D P m C F m; m @x @y @y

(7.4)

where P .x; y/ is the projective factor. Proof. Along a geodesic .x.s/; y.s//, we have yP k D 2G k .x; y/ D 2P .x; y/y k ;

xP k D y k :

Since the Lagrangian F .x.s/; y.s// is constant in s, we obtain the first equation 0D

@F @F @F @F @F dF D k y k C k yP k D k y k 2P k y k D k y k 2PF: ds @x @y @x @y @x

Differentiating this equation and using (b) in Proposition 7.5, we obtain

@F @P 2 P m CF m @y @y

@ @F D m yk k @y @x

D

2 @F @F k @ F C y D 2 m: m k m @x @x @x @y

A first consequence is the following description of Minkowski metrics: Corollary 7.9. A strongly convex projectively flat Finsler metric on some domain in Rn is the restriction of a Minkowski metric if and only if the associated projective factor P .x; y/ vanishes identically.

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@F Proof. Suppose F .x; y/ is locally Minkowski, then @x m D 0 and therefore the spray k k coefficient satisfies P .x; y/y D G .x; y/ D 0. Conversely, if P .x; y/ 0, then @F the second equation in the lemma implies that @x m D 0.

Another consequence is the following result about the projective factor of a reversible Finsler metric. Corollary 7.10. Let F .x; y/ be a strongly convex projectively flat Finsler metric. Suppose F is reversible, then its projective factor satisfies P .x; y/ D P .x; y/: Proof. This is obvious from the first equation in Lemma 7.8.

8 The Hilbert form We now introduce the Hilbert 1-form of a Finsler manifold and show its relation with Hamel’s condition: Definition 8.1. The Hilbert 1-form on a smooth Finsler manifold .M; F / is the 1-form on TM 0 defined in natural coordinates as @F ! D j dx j D gij .x; y/y i dx j : @y From the homogeneity of F , we have F .x; / D !.x; /. Note also that !. / D 0 for any vector 2 T TM 0 that is tangent to some level set F .x; y/ D const: These two conditions characterize the Hilbert form which is therefore independent of the choice of coordinates, see also [18]. Observe also that the length of a smooth non-singular curve ˇ W Œa; b ! M is Z b Z Q D `.ˇ/ D F .ˇ/ !; a

ˇQ

where ˇQ W Œa; b ! TM 0 is the natural lift of ˇ. We then have the Proposition 8.2 ([4], [16]). The smooth Finsler metric F W T U ! R on the convex domain U is projectively flat if and only if the Hilbert form is dx -closed, that is, if we have @2 F dx ! D i j dx i ^ dx j D 0: @x @y Equivalently, F is projectively flat if and only if the Hilbert form is dx -exact. This means that there exists a function h W T U0 ! R such that ! D dx h D

@h dx j : @x j

Chapter 3. Funk and Hilbert geometries from the Finslerian viewpoint

93

Proof. The first assertion is a mere reformulation of the Hamel Condition (c) in Proposition7.5. The second assertion is proved using the same argument as in the proof of the Poincaré Lemma. Indeed, using Condition (c) from Proposition 7.5, we compute:

d @F d t !.tx;y/ D t i .tx; y/dx i dt dt @y @F @2 F D i .tx; y/dx i C t k i x k dx i @y @x @y @F @2 F D i .tx; y/dx i C t i k x k dx i @y @x @y @F D dx x k k .tx; y/ D dx !.tx;y/ .x/ : @y Suppose now that 0 2 U and set Z 1 Z h.x; y/ D !.tx;y/ .x/dt D 0

0

then by the previous calculation we have Z 1 Z dx h D dx .!.tx;y/ /dt D 0

1

0

1

xk

@F .tx; y/ dt; @y k

d .t !.tx;y/ /dt D !.x;y/ : dt

We shall call such a function h a Hamel potential of the projective Finsler metric F . Observe that a Hamel potential is well defined up to adding a function of y. The above proof shows that one can chose h.x; y/ to be 0-homogenous in y. This potential allows us to compute distances. Corollary 8.3. The distance dF associated to the projective Finsler metric F is given by dF .p; q/ D h.q; q p/ h.p; q p/; where h.x; y/ is a Hamel potential for F . Proof. Let ˇ.t / D p C t .q p/, (t 2 Œ0; 1). Then Z 1 Z P dF .p; q/ D F .ˇ; ˇ/dt D ! D h.q; q p/ h.p; q p/: 0

ˇQ

Example 8.4. If Ff is the tautological Finsler structure in U, then from the previous corollary and Equation (3.9) we deduce that a Hamel potential is given by h.x; y/ D log.Ff .x; y//: If one prefers a 0-homogenous potential, then a suitable choice is

Ff .p; y/ h.x; y/ D log ; Ff .x; y/

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where p 2 U is some fixed point. Remark 8.5. A consequence of this example is that the tautological Finsler structure in a domain U satisfies the following equation: @Ff @Ff D Ff j : j @x @y

(8.1)

Indeed, since h.x; y/ D log.Ff .x; y// is a Hamel potential, the Hilbert form is @Ff 1 @Ff dx j D ! D dx h D dx j ; @y j Ff @x j from which (8.1) follows immediately. This equation plays an important role in the study of projectively flat metrics with constant curvature, see e.g. [9]. An intrinsic discussion of geodesics based on the Hilbert form is given in [3], [16], [17], [18]. Remark 8.6. J. C. Álvarez Paiva and G. Berck gave a nice generalization of Proposition 8.2 above to a Hamel-type characterization of higher dimensional Lagrangians whose minimal submanifolds are k-flats, see Theorem 4.1 in [4].

9 Curvature in Finsler geometry A notion of curvature for Finsler surfaces already appeared in the beginning of the last century. This notion was extended in all dimensions by L. Berwald in 1926 [7]. This curvature is an analogue of the sectional curvature in Riemannian geometry and it is best explained using the notion of osculating Riemannian metric introduced by Varga [64]. See also Auslander [5] and the book of Rund [51], p. 84. Definition 9.1. 1. Let .M; F / be a smooth manifold with a strongly convex Finsler metric and let .x0 ; y0 / 2 TM 0 . A vector field V defined in some neighborhood O M of the point x0 is said to be a geodesic extension of y0 if Vx0 D y0 and if the integral curves of V are geodesics of the Finsler metric F (in particular V does not vanish throughout the neighborhood O). 2. The osculating Riemannian metric gV of F in the direction of V is the Riemannian metric on O defined by the Fundamental tensor at the point .x; y/ D .x; Vx / 2 T O 0 . In local coordinates we have gV D gij .x/dx i dx j D gij .x; V .x//dx i dx j D

1 @2 F .x; Vx / i j dx dx : 2 @y i @y j

Let us fix a point .x0 ; y0 / 2 TM 0 and a geodesic extension of y0 . We shall denote by RiemV the .1; 3/ Riemann curvature tensor of the osculating metric gV . Recall

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Chapter 3. Funk and Hilbert geometries from the Finslerian viewpoint

that RiemV .X; Y /Z D .rX rY rY rX rŒX;Y /Z; where r is the Levi-Civita connection of gV . Definition 9.2. The Riemann curvature of gV is the field of endomorphisms ((1,1)tensor) RV W TM ! TM defined as RV .W / D RiemV .W; V /V: A basic fact of Finsler geometry is the following result: Proposition 9.3. Let .M; F / be a smooth manifold with a strongly convex Finsler metric and let .x; y/ be a point in TM 0 . Then the Riemann curvature RV at .x; y/ 2 TM 0 is well defined independently of the choice of a geodesic extension V of y. Proof. Choose some local coordinate system and let us write Ri k .x; y/ for the components of the tensor RV : RV D Ri k .x; y/ dx k ˝

@ : @x i

Then we have the formula Ri k D 2

2 i @G i @2 G i j @G i @G j j @ G y C 2G ; @y j @y k @x k @x j @y k @y j @y k

(9.1)

where the G i D G i .x; y/ are the spray coefficients of F . We refer to Lemma 6.1.1 and Proposition 6.2.2 in [54] for a proof. Formula (9.1) is also obtained in [15], p. 43, where it is seen as a consequence of the structure equations for the Chern connection, see also [53], Proposition 8.4.3. Since the spray coefficients G i .x; y/ only depend on the fundamental tensor and its partial derivatives, it follows that Ri k depends only on .x; y/ D .x; Vx / 2 TM 0 and not on the choice of a geodesic field extending y. This proposition implies that we can write Ry D RV D Ri k .x; y/ dx k ˝

@ @x i

for the Riemann curvature at a point .x; y/ 2 TM 0 , where V is an arbitrary geodesic extension of y. We then have the following important Corollary 9.4. Let Tx M be a 2-plane containing the non-zero vector y 2 Tx M . Choose a local geodesic extension V of y, then the sectional curvature K gV . / of for the osculating Riemannian metric gV is independent of the choice of V . Proof. By definition of the sectional curvature in Riemannian geometry, we have K gV . / D

gV .RV .W /; W / ; gV .y; y/ gV .W; W / gV .y; W /2

(9.2)

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where W 2 and y D Vx are linearly independent (and thus D spanfV; W g), and RV is the Riemann curvature of gV . This quantity is independent of the geodesic extension V , by the previous proposition. Definition 9.5. The pair .y; / with 0 ¤ y 2 is called a flag in M , and KgV . / is called the flag curvature of .y; /, and denoted by K.y; /. The vector y 2 is sometimes suggestively called the flagpole of the flag. In local coordinates, the flag curvature can be written as K.y; / D

Rmk .x; y/ w k w m ; F 2 .x; y/gij .x; y/w i w j .grs .x; y/w r y s /2

(9.3)

where W D w k @[email protected] 2 and y are linearly independent and Rmk D gim Ri k . Example 9.6. The flag curvature of a Minkowski metric is zero. Indeed, the fundamental tensor is constant and coincides with any osculating metric which is thus flat. Note that conversely, there are many examples of Finsler metrics with vanishing flag curvature which are not locally isometric to a Minkowski metric. The first example has been given in Section 7 of [8]. We also have a notion of Ricci curvature: Definition 9.7. The Ricci curvature of the Finsler metric F at .x; y/ 2 TM 0 is defined as n X K.y; i / Ric.x; y/ D Trace.Ry / D F 2 .x; y/ iD2

where e1 ; e2 ; : : : ; en 2 Tx M is an orthonormal basis relative to the inner product gy y such that e1 D F .x;y/ . The geometric meaning of the Finslerian flag curvature presents both similarities and striking differences with its Riemannian counterpart. The Riemann curvature Ry plays an important role in the Finsler literature. It appears naturally in the second variation formula for the length of geodesics and in the theory of Jacobi fields (see [54], Lemma 6.1.1, or [51], Chapters IV.4 and IV.5). This leads to natural formulations of comparison theorems in Finsler geometry which are similar to their Riemannian counterparts. In particular we have • In 1952, L.Auslander proved a Finsler version of the Cartan–Hadamard Theorem. If the Finsler manifold .M; F / is forward complete and has non-positive flag curvature, then the exponential map is a covering map [5], [39]. • Auslander also proved a Bonnet–Myers Theorem: A forward complete Finsler manifold .M; F / with Ricci curvature Ric.x; y/ .n 1/F 2 .x; y/ for all .x; y/ 2 TM 0 is compact with diameter , see [5], [39].

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97

• In 2004, H.-B. Rademacher proved a sphere theorem: a compact, simply-connected Finsler manifold of dimension n 3 such that F .p; / F .p; / for any .p; / 2 TM and with flag curvature satisfying

2 1 0.

References [1]

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[10] J.-P. Benzécri, Sur les variétés localement affines et projectives. Bull. Soc. Math. France 88 (1960), 229–332. [11] R. Bryant, Projectively flat Finsler 2-spheres of constant curvature. Selecta Math. (N.S.) 3 (1997), no. 2, 161–203. [12] H. Busemann, The geometry of Finsler spaces. Bull. Amer. Math. Soc. 56, (1950), 5–16. [13] H. Busemann, The geometry of geodesics. Academic Press, New York 1955; reprinted by Dover in 2005. [14] H. Busemann and P. J. Kelly, Projective geometry and projective metrics. Academic Press, New York 1953. [15] S.-S. Chern and Z. Shen, Riemann–Finsler geometry. Nankai Tracts in Mathematics 6, World Scientific Publishing, Hackensack, NJ, 2005. [16] M. Crampin, Some remarks on the Finslerian version of Hilbert’s fourth problem. Houston J. Math. 37 (2011), no. 2, 369–391. [17] M. Crampin, T. Mestdag and D.J. Saunders, Hilbert forms for a Finsler metrizable projective class of sprays. Differential Geom. Appl. 31 (2013), no. 1, 63–79. [18] M. Crampon, The geodesic flow of Finsler and Hilbert geometries. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 161–206. [19] G. De Cecco and G. Palmieri, LIP manifolds: from metric to Finslerian structure. Math. Z. 218 (1995), 223–237. [20] P. Dazord, Variétés finslériennes de dimension paire ı-pincées. C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A496–A498. [21] D. Egloff, Some new developments in Finsler geometry. Ph.D. Thesis, University of Fribourg, 1995. [22] D. Egloff, Uniform Finsler Hadamard manifolds. Ann. Inst. H. Poincaré Phys. Théor. 66 (1997), no. 3, 323–357. [23] P. Finsler, Über Kurven und Flächen in allgemeinen Räumen. O. Füssli, Zürich 1918. [24] P. Funk, Über Geometrien, bei denen die Geraden die Kürzesten sind. Math. Ann. 101 (1929), 226–237. [25] P. Funk, Über zweidimensionale Finslersche Räume, insbesondere über solche mit geradlinigen Extremalen und positiver konstanter Krümmung. Math. Z. 40 (1936), no. 1, 86–93. [26] P. Funk, Variationsrechnung und ihre Anwendung in Physik und Technik. Grundlehren Math. Wiss. 94 Springer-Verlag, Berlin 1970. [27] E. Guo, X. Mo and X. Wang, On projectively flat Finsler metrics. In Differential geometry: Under the influence of S.-S. Chern, Adv. Lect. Math. (ALM) 22, Higher Education Press, Beijing, International Press, Somerville, MA, 2012, 77–88. [28] R. Guo, Characterizations of hyperbolic geometry among Hilbert geometries. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 147–158. [29] S. V. Ivanov, Volumes and areas of Lipschitz metrics. St. Petersburg Math. J. 20 (2009), no. 3, 381–405.

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Chapter 5

The horofunction boundary and isometry group of the Hilbert geometry Cormac Walsh

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The horofunction boundary . . . . . . . . . . . . . . . . . . . 2.1 Busemann points . . . . . . . . . . . . . . . . . . . . . . 2.2 Horoballs . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The horofunction boundary of the Hilbert geometry . . . . . . . 3.1 The horofunction boundary of the reverse-Funk geometry 3.2 The horofunction boundary of the Funk geometry . . . . . 3.3 The horofunction boundary of the Hilbert geometry . . . . 4 Isometries of the Hilbert metric . . . . . . . . . . . . . . . . . 4.1 Simplicial geometries . . . . . . . . . . . . . . . . . . . 4.2 Polyhedral geometries . . . . . . . . . . . . . . . . . . . 4.3 The general case . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

127 129 130 132 135 135 136 138 140 140 142 142 145

1 Introduction Building on the work of Busemann, Gromov defined in [12] a certain compactification of a metric space. Recall that the Busemann function of a geodesic ray is the limiting function B ./ WD lim d.; .t // d.b; .t //; t!1

where b is an arbitrary base-point. Gromov’s idea was to generalise this notion by considering all possible limits of d.; z/d.b; z/ as z heads off to infinity in the metric space, not necessarily along a geodesic ray. Such a limit is called a horofunction, and may be thought of as a “point at infinity” of the metric space. By adjoining the horofunctions to the metric space, under mild conditions, one compactifies it. The set of horofunctions thus forms a “boundary at infinity” of the metric space.

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The attractiveness of this boundary arises in part from its generality: whereas the definition of other boundaries, such as the hyperbolic boundary of a Gromov hyperbolic space, or the ideal boundary of a CAT(0) space, require the metric space to have certain properties, the horofunction boundary exists for any metric space. The horofunction boundary is often closely related to these other boundaries however. For example, in the CAT(0) case it is actually the same [2], and in the Gromov hyperbolic case, the hyperbolic boundary can be recovered from the horofunction boundary by quotienting out by a certain equivalence relation [8], [30], [23]. The horofunction boundary has found diverse applications. Rieffel [21] has used it in his work on quantum metric spaces. It also appears to be the right general setting for Patterson–Sullivan measures [5]. In this chapter, we will consider the horofunction boundary of the Hilbert geometry. Let D be a bounded open convex subset of a finite-dimensional real linear space. Given a pair of distinct points x and y in D, let w and z be points in the usual boundary of D such that w, x, y, and z lie in this order along a straight line; see Figure 1. The Hilbert distance between x and y is defined to be the logarithm of the cross ratio of these four points: Hil.x; y/ WD log

jzxj jwyj ; jzyj jwxj

where j j denotes any norm on the linear space. (Be aware that some authors use a different convention here and divide by a factor of 2.)

w

x

y z

Figure 1. Definition of the Hilbert distance.

The horofunction boundary of this metric was first studied by Karlsson, Metz, and Noskov [15], who determined it in the case where D is an open simplex. The general case was considered in [25]. In that paper, use was made of the fact that the Hilbert metric is the symmetrisation of the Funk metric: Hil.x; y/ D Funk.x; y/ C RFunk.x; y/; where Funk.x; y/ WD log

jzxj ; jzyj

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 129

and RFunk.x; y/ WD Funk.y; x/ D log

jwyj : jwxj

The horofunction boundaries of the Funk and reverse-Funk geometries were studied separately, and the results were combined to find the horofunction boundary of the Hilbert geometry. It turns out that the horofunction boundary of the reverse-Funk geometry is just the usual boundary of the convex domain D. That of the Funk geometry, however, is more closely related to the usual boundary of the polar body D B of the domain. We describe these boundaries in detail in Section 3, and say how they combine to form the Hilbert-geometry horofunction boundary. Observe that, since every collineation preserves cross-ratios, any collineation that leaves the domain D invariant is an isometry of the Hilbert metric on D. There are examples known, however, [9], [18], [17], [4] where there exist other isometries as well, namely, cross-sections of non-Lorentzian symmetric cones. For these domains, the projective action of Vinberg’s -map, or equivalently, the inverse map in the associated Jordan algebra, is an isometry but not a collineation, and the isometry group is generated by this map and the collineations. It was proved in [29] that there are no other examples. The horofunction boundary is useful for studying isometry groups. This is because such groups have a natural action by homeomorphisms on the boundary, and often this action is easier to understand than the action on the space itself. The horofunction boundary was used in this way in [28] to determine the group of isometries of Teichmüller space with Thurston’s metric. It was also used in [17] to determine the isometry group of the Hilbert geometry when the domain D is polyhedral. In Section 4, we discuss in more detail the isometry group of the Hilbert geometry and how the horofunction boundary may be used to study it. Before that, however, we recall in the next section the definition of the horofunction boundary and some of its properties. To visualize horofunctions, it is often useful to consider their horoballs, that is, their sublevel sets. Intuitively, a horoball looks like a large ball of the metric whose center is at infinity. We will make this precise by showing that a sequence of points converges to a horofunction if and only if sequences of closed balls of the right size with those points as centers converge to horoballs of the horofunction, in a certain topology.

2 The horofunction boundary Since we will be dealing with non-reversible metrics, we will discuss the horofunction boundary in this setting. The development will be similar to that in [28]. Let .X; d / be a non-reversible metric space, for example the Funk or reverseFunk geometry. We consider X to be equipped with the topology coming from the

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symmetrised metric dsym .x; y/ WD d.x; y/ C d.y; x/, which for the two examples just mentioned is the Hilbert metric. Associate to each point z 2 X the function z W X ! R, z .x/

WD d.x; z/ d.b; z/;

where b 2 X is an arbitrary fixed base-point. One can show that the map W X ! C.X /; z 7! z is injective and continuous. Here, C.X / is the space of continuous real-valued functions on X, with the topology of uniform convergence on bounded sets of dsym . We define the horofunction boundary of .X; d / to be X.1/ WD cl .X / n .X /; where cl is the topological closure operator. The elements of this set are the horofunctions of .X; d /. Although this definition appears to depend on the choice of base-point, one may verify that horofunction boundaries coming from different base-points are homeomorphic, and that corresponding horofunctions differ by only an additive constant. Recall that a geodesic in a non-reversible metric space .X; d / is a map from a closed interval of R to X such that d..s/; .t // D t s, for all s and t in the domain satisfying s < t . We make the following assumptions. (I) the metric dsym is proper, that is, its closed balls are compact; (II) there exists a geodesic in .X; d / between any two given points; (III) for any point x and sequence xn in X , we have d.xn ; x/ ! 0 if and only if d.x; xn / ! 0. Both the Funk and reverse-Funk metrics satisfy these assumptions. Indeed, for (II), one may take the straight line segment between the two points, parameterised appropriately. Assumption (I) implies that uniform convergence on bounded sets is equivalent to uniform convergence on compact sets. Moreover, for elements of cl .X /, the latter is equivalent to pointwise convergence, by the Ascoli–Arzelà theorem, since elements of this set are equi-Lipschitzian with respect to dsym . Again from the Ascoli–Arzelà theorem, the set cl .X/ is compact. We call it the horofunction compactification. Under assumptions (I), (II), and (III), one may show that is an embedding of X into C.X /, in other words, that it is a homeomorphism from X to its image in C.X /. From now on we identify X with its image.

2.1 Busemann points Define an almost-geodesic to be a path W T ! X such that T RC contains 0 and does not contain sup T , and such that, for any > 0, ˇ ˇ ˇd..0/; .s// C d..s/; .t // t ˇ < ; for s; t 2 T large enough, with s t. This definition has been modified from the original in [21], where T was required to be an unbounded subset of RC containing 0.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 131

We must relax the condition that T be unbounded because, in the reverse-Funk metric, one may approach the boundary along a path of bounded length. See Figure 2 for an illustration. This modification is very minor.

b

xn

Figure 2. Finite distance to the boundary. Funk.xn ; b/ and RFunk.b; xn / are bounded above, even though the sequence xn approaches the boundary.

Rieffel proved that every almost-geodesic converges. In his setting, it will be to a horofunction, but in ours it may be to a point of X . Every point of X is trivially the limit of an almost-geodesic under our definition. We say that a horofunction is a Busemann point if it is the limit of an almost-geodesic. Non-Busemann points of the horofunction boundary have been found for normed spaces [24] and various finitely-generated groups with their word metrics [26], [27], [31]. We define the detour cost for any two horofunctions and in X.1/ to be H.; / D sup inf d.b; x/ C .x/ ; W 3 x2W \X

where the supremum is taken over all neighbourhoods W of in X [ X.1/. This concept appears in [1]. An equivalent definition is H.; / D inf lim inf d.b; .t // C ..t // ; t!sup T

where the infimum is taken over all paths W T ! X converging to . The proof of the following proposition is easy. Proposition 2.1. For all horofunctions , , and , (i) H.; / 0; (ii) H.; / H.; / C H.; /. Proposition 2.2. Assume that .X; d / satisfies assumptions (I), (II), and (III), and let be a horofunction. Then, H.; / D 0 if and only if is Busemann. The proof of this proposition may be found in [28]. There, it was assumed that d.b; xn / converges to infinity for a sequence xn in X whenever dsym .b; xn / does.

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Although this is not true for the reverse-Funk metric, it was only needed to ensure that the almost-geodesic constructed is defined on all of RC . Since the definition of almost-geodesic has been modified in the current setting, this is no longer necessary. Propositions 2.1 and 2.2 together say that, on the set of Busemann points, the detour cost is a weak metric, sometimes also called an extended pseudo quasi-metric. By symmetrising, one may obtain a genuine metric on the set of Busemann points XB .1/. For Busemann points and in XB .1/, define ı.; / WD H.; / C H.; /: This construction first appeared in [1], Remark 5.2. See [28] for a proof of the following proposition. Proposition 2.3. The function ı W XB .1/ XB .1/ ! Œ0; 1 is a (possibly 1-valued) metric. We call ı the detour metric. Note that we can partition XB .1/ into disjoint subsets such that ı.; / is finite for each pair of Busemann points and lying in the same subset. We call these subsets the parts of the horofunction boundary of .X; d /, and ı is a genuine metric on each one. Consider an isometry g from one metric space .X; d / to another .X 0 ; d 0 /. We can extend g continuously to the horofunction boundary X.1/ of X as follows: g./.x 0 / WD .g 1 .x 0 // .g 1 .b 0 //; 0

0

0

(2.1)

0

for all 2 X.1/ and x 2 X . Here b is the base-point of X .

2.2 Horoballs To visualize horofunctions, it is often useful to consider their horoballs, that is, their sublevel sets. We denote the sublevel set of a function f at height ˛ by slv.f; ˛/ WD fx 2 X j f .x/ ˛g. Intuitively, a horoball looks like a large ball of the metric whose center is at infinity. To make this precise, we define a topology on the set of closed subsets of X , called the Painlevé–Kuratowski topology. In this topology, a sequence of closed sets .Cn /n2N is said to converge to a closed set C if the upper and lower closed limits (also known as the limit superior and limit inferior) of the sequence both equal C . These limits are defined to be, respectively, \ [ cl Ci Ls Cn WD n0

and Li Cn WD

\

i>n

cl

[ i0

Cni j .ni /i2N is an increasing sequence in N :

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 133

An alternative characterisation of convergence is that .Cn /n2N converges to C if and only if each of the following hold: • for each x 2 C , there exists xn 2 Cn for all n large enough, such that .xn /n converges to x; • if .Cnk /k2N is a subsequence of the sequence of sets and xk 2 Cnk for each k 2 N, then convergence of .xk /k2N to x implies that x 2 C . See the book [3] for more about this and other topologies on sets of closed sets. For each z 2 X and ˛ 2 R, we denote by B.z; ˛/ WD fx 2 X j d.x; z/ ˛g the right closed ball about z of radius ˛. In [15], it was shown that in a geodesic metric space each horoball slv.; ˛/ of a horofunction that is the limit of a sequence zn takes the form Ls B.zn ; d.b; zn / C ˛/. Here we improve this result by establishing a necessary and sufficient condition for convergence to a horofunction in terms of convergence of balls in the Painlevé–Kuratowski topology. Lemma 2.4. Assume .X; d / satisfies (I), (II), and (III). Let x 2 X, and let 2 X.1/ be a horofunction. Then, for all 0 small enough, there exists y 2 X such that .x/ .y/ D d.x; y/ D . Proof. Choose a sequence .zn / in X converging to . Since no subsequence of .zn / converges to x, we have that .zn / eventually remains outside any sufficiently small open ball fz 2 X j d.x; z/ < g about x. So, for n large enough, we may choose a point yn on a geodesic from x to zn such that d.x; zn / d.yn ; zn / D d.x; yn / D . But some subsequence of .yn / converges to a point y 2 X, and, taking limits, we get the result. Lemma 2.5. Assume .X; d / satisfies (I), (II), and (III). Let 2 X.1/ be a horofunction. Then, slv.; / is continuous in the Painlevé–Kuratowski topology. Proof. Let ˛ 2 R, and let .˛n / be a sequence in R converging to ˛. Let .nk / be an increasing sequence in N, and let .xk / be a sequence of points in X converging to some point x 2 X , such that xk 2 slv.; ˛nk / for all k 2 N. So, .x/ D lim .xk / lim ˛nk D ˛: k!1

k!1

Therefore, x 2 slv.; ˛/. Now let y 2 slv.; ˛/. For any c 2 R, we use the notation c C WD max.c; 0/. By Lemma 2.4, for n large enough, there exists yn 2 X such that .y/ .yn / D d.y; yn / D .˛ ˛n /C : Observe that .˛ ˛n /C is a non-negative sequence converging to zero. It follows that .yn / converges to y, and that yn 2 slv.; ˛n / for all n large enough. The Painlevé–Kuratowski topology can be used to define a topology on the space of lower-semicontinuous functions on X as follows. Recall that the epigraph of a

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function f on X is the set f.x; ˛/ 2 X R j f .x/ ˛g. A sequence of lowersemicontinuous functions is declared to be convergent in the epigraph topology if the associated epigraphs converge in the Painlevé–Kuratowski topology on X R. For proper metric spaces the epigraph topology is identical to another topology called the Attouch–Wets topology. Lemma 2.6. Make assumption (I), which is that .X; dsym / is a proper metric space. Let be a real-valued lower-semicontinuous function on X, and let n be a sequence of such functions that is equi-Lipschitzian with respect to dsym . Then, n converges to in the epigraph topology if and only if it converges to uniformly on bounded sets. Proof. This is a consequence of Lemma 7.1.2 of [3] and Proposition 7.1.3 of [3], and the fact that, for proper metric spaces, Attouch–Wets convergence is equivalent to epigraph convergence. We will also need the following result relating convergence of functions to convergence of their sublevel sets. Recall that a proper metric space is always separable. Proposition 2.7 ([3], Theorem 5.3.9). Let f be a lower-semicontinuous function in a separable metric space, and let fn be a sequence of such functions. Then, fn converges to f in the epigraph topology if and only if there exists, for all ˛ 2 R, a sequence ˛n in R converging to ˛ such that slv.fn ; ˛n / converges to slv.f; ˛/ in the Painlevé–Kuratowski topology. Now we can prove our result concerning the convergence of balls. Proposition 2.8. Let zn be a sequence in X. Then, zn converges to a point in the horofunction boundary if and only if, for each ˛ 2 R, the sequence of balls B.zn ; d.b; zn / C ˛/ converges to slv.; ˛/ in the Painlevé–Kuratowski topology. Proof. Suppose that the balls converge as stated. Observe that slv.

z ; ˛/

D B.z; d.b; z/ C ˛/

for all z 2 X and ˛ 2 R:

So, by Proposition 2.7, zn converges to in the epigraph topology, which is equivalent, by Lemma 2.6, to uniform convergence on bounded sets. Now suppose that zn converges to in the horofunction boundary, and let ˛ 2 R. Choose ˇ < ˛. From Lemma 2.6 and Proposition 2.7, we get that there exists a sequence ˇn in R converging to ˇ such that B.zn ; d.b; zn / C ˇn / converges to slv.; ˇ/. For n large enough, B.zn ; d.b; zn / C ˛/ contains B.zn ; d.b; zn / C ˇn /. Therefore, Li B.zn ; d.b; zn / C ˛/ contains slv.; ˇ/. Since this is true for all ˇ < ˛, and slv.; / is continuous by Lemma 2.5, we get Li B.zn ; d.b; zn / C ˛/ slv.; ˛/: The upper bound on the upper closed limit is proved in a similar manner.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 135

Figure 3 illustrates the convergence of balls to horoballs in the case of the Hilbert metric on the 2-simplex.

Figure 3. Sequences of balls converging to horoballs in the Hilbert geometry.

3 The horofunction boundary of the Hilbert geometry In this section, we study the horofunction boundary of the Hilbert metric on a bounded open convex set D. As mentioned before, we will consider the horofunction boundaries of the Funk and reverse-Funk geometries separately, and then combine the information to get the boundary of the Hilbert geometry. Unless otherwise stated, results come from [25].

3.1 The horofunction boundary of the reverse-Funk geometry Observe that the function RFunk.; /, defined on D D extends continuously to a function on D cl D. So, for all z 2 cl D, we may define on D the function rz ./ WD RFunk.; z/ RFunk.b; z/. The following proposition shows that the horofunction compactification of the reverse-Funk geometry is basically the same as the compactification obtained by taking the closure of the domain in the usual topology. We use @D WD cl DnD to denote the boundary of D in the usual topology. Proposition 3.1. Let D be a bounded open convex set. The set of horofunctions in the reverse-Funk geometry on D is BRF WD frx j x 2 @Dg. A sequence in D converges to rx 2 BRF if and only if it converges in the usual sense to x. Since straight line segments are geodesic in the reverse-Funk geometry, it is clear that every horofunction in this geometry is Busemann. The detour cost in this geometry was calculated in [17]. Recall that a convex subset E of a convex set D is said to be an extreme set if the endpoints of any line segment

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in D are contained in E whenever any relative interior point of the line segment is. The relative interiors of the extreme sets of a convex set D partition D. When we consider RFunk etc. on convex sets other than D, we use a subscript to specify the set. Proposition 3.2 ([17], Proposition 4.3). Let x and y be in the usual boundary @D of D. Then, the detour cost in the reverse-Funk geometry is HRF .rx ; ry / D RFunkD .b; x/ C RFunk F .x; y/ RFunkD .b; y/; if y is in the smallest extreme set F of cl D containing x, and is infinity otherwise. It follows immediately that the detour metric in this geometry is given by ıRF .rx ; ry / D HilF .x; y/; when x and y are in the relative interior of the same extreme set F of cl D, and is infinity otherwise. Thus, the parts of the horofunction boundary are just the relative interiors of the proper extreme sets of cl D.

3.2 The horofunction boundary of the Funk geometry The horofunctions of the Funk geometry can be best understood by using convex duality because they take a simpler form in the dual space. Consider first the cone over D, in other words, ˚ C WD .p; 1/ 2 RnC1 j p 2 D and > 0 : Here we are assuming that D lives within Rn . We identify D in the obvious way with a cross-section of this cone. One may extend the function FunkD .; / from D to C in the following way. For x 2 RnC1 and y 2 C , define the gauge ˚ M.x=yI C / WD inf > 0 j x C y : Here C is the partial ordering on RnC1 induced by C , that is, x C y if y x 2 cl C . One can show [16], page 29, that log M.x=yI C / D FunkD .x; y/ for all x and y in D. For each z 2 D, the map z has the following extension to the whole of RnC1 : z .x/

WD log

M.x=zI C / ; M.b=zI C /

for all x 2 RnC1 :

Define the (closed) dual cone of C : C WD fu 2 RnC1 j hu; xi 0 for all x 2 C g: Here h; i denotes the standard Euclidean inner product. Observe that, by the Hahn– Banach separation theorem, x C y if and only if hu; xi hu; yi for all u 2 C . It

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 137

follows that M.x=yI C / D

hu; xi u2C nf0g hu; yi sup

for all x 2 RnC1 and y 2 C .

(3.1)

So, we see that M.=zI C / is a convex function for fixed z. Therefore, the same is true for the following function defined on RnC1 : jC;z ./ WD

M.=zI C / D exp B M.b=zI C /

z ./:

Recall that the Legendre–Fenchel transform of a convex function f W RnC1 ! R [ f1g is the function f W RnC1 ! R [ f1g defined by f .y/ WD sup hy; xi f .x/ for all y 2 RnC1 : x2RnC1

The Legendre–Fenchel transform is a bijection from the set of proper lower-semicontinuous convex functions to itself, and is in fact a homeomorphism in the Attouch–Wets topology. Using (3.1), one may calculate without much difficulty the Legendre–Fenchel transform of jC;z . We use IE to denote the characteristic function of a set E, that is, the function taking value 0 on E and C1 everywhere else. For any open cone U in RnC1 and any point x 2 U , define ZU;x WD U \ fu 2 RnC1 j M.b=xI C /hu; xi 1g: Proposition 3.3. Let C RnC1 be an open cone. For all z 2 C , we have that jC;z is the characteristic function of the set ZC;z So the set ZC;z on which jC;z is zero is the intersection of C with a half-space bounded by a hyperplane on which h; zi is constant. This constant is the largest possible such that ZC;z lies within fu 2 RnC1 j hu; bi 1g. This is illustrated in Figure 4.

M.b=z/h; zi D 1

h; bi D 1

C . Figure 4. The support of jC;x

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As z approaches the boundary of D, the hyperplane becomes more and more tilted. In the limit, the zero set will lie entirely within the (usual) boundary of C . We consider the limit of the zero sets in the Painlevé–Kuratowski topology. Proposition 3.4. A sequence zn in D converges to a point in the horofunction boundary of the Funk geometry if and only if ZC;zn converges in the Painlevé–Kuratowski topology. We have the following simple description of the Busemann points. Proposition 3.5. A function W D ! R is a Busemann point of the Funk geometry if and only if it is the restriction to D of a function of the form log IZE;x , where E is an open convex cone such that E is a proper extreme set of C , and x 2 E. It is clear from this proposition that in some cases there may be horofunctions that are not Busemann. This will happen for instance when there is a subset F of C that is not extreme but is the limit of extreme sets. In this case, the function log IZF;b will be, by Proposition 3.4, a horofunction, but will not be, by Proposition 3.5, a Busemann point. The next result shows that this phenomenon is the only way that non-Busemann points may arise. Proposition 3.6. All horofunctions of the Funk geometry on a domain D are Busemann if and only if the set of extreme sets of the polar of D is closed in the Painlevé– Kuratowski topology. Recall that the polar D B of D may be identified with a cross-section of C . The detour metric in the Funk geometry was calculated in [17]. Proposition 3.7. Let D log IZE;x and D log IZF;y be two Busemann points of the Funk geometry. The distance between them in the detour metric is ıF .; / D HilE .x; y/;

if the extreme sets E and F of C are equal, and is infinity otherwise. Thus, there is a part for each extreme set of the polar of D.

3.3 The horofunction boundary of the Hilbert geometry It is evident from the expression of the Hilbert metric as the symmetrisation of the Funk metric that every horofunction in the Hilbert geometry is the sum of one in the Funk geometry and one in the reverse-Funk geometry. The following proposition shows that there is a unique such decomposition of each Hilbert horofunction.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 139

Proposition 3.8. A sequence converges to a point in the Hilbert-geometry horofunction boundary if and only if it converges to a horofunction in both the Funk and reverseFunk geometries. Proof. This follows from the proof of Theorem 1.3 in [25]. Note that, combined with Proposition 3.1, this implies that every sequence that converges to a Hilbert horofunction also converges to a point in the usual boundary. This generalises a result of [14], where is was shown that every Hilbert-geometry geodesic converges to such a point. We also have the following description of the Busemann points. Theorem 3.9. Let h D rx C f be a Hilbert horofunction written as the sum of a reverse-Funk horofunction rx and a Funk horofunction f . Then, h is a Busemann point if and only if f is. Combining this with Proposition 3.6, we get the following. Corollary 3.10. All horofunctions of the Hilbert geometry on a domain D are Busemann if and only if the set of extreme sets of the polar of D is closed in the Painlevé– Kuratowski topology. It remains to determine which combinations of Funk and reverse-Funk horofunctions give Hilbert horofunctions. The answer is given in the next theorem. Recall that every point in @C is a supporting functional of C , and so defines an exposed face of C . Theorem 3.11. Let rx be a reverse-Funk horofunction, and let f WD log IZE;z be a Busemann point of the Funk geometry. Then, the function rx C f is a Busemann point of the Hilbert geometry if and only if the extreme set E of C is contained in the exposed face of C defined by x. Proof. This is just a restatement of Theorem 1.1 of [25] using the description given above of the Funk Busemann-points in the dual space. Indeed, by [25], Lemma 3.13, the tangent cone .C; x/ of C at x is dual to the exposed face of C defined by x. Also, by [25], Lemma 3.14, an open cone T can be obtained from another U by taking tangent cones if and only if T is an extreme set of U . The final ingredient is that the extreme sets of an exposed face of a closed cone are exactly the extreme sets of the cone that are contained within the exposed face. Using Propositions 3.2 and 3.7, one may find the detour metric of the Hilbert geometry.

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Theorem 3.12 ([17]). Let D rp C log IZE;x and D rq C log IZF;y be two Busemann points of the Hilbert geometry. Then, the distance between them in the detour metric of the Hilbert geometry is ıH .; / D HilG .p; q/ C HilE .x; y/;

(3.2)

when p and q are in the relative interior of the same extreme set G of cl D, and the extreme sets E and F of C are equal. Otherwise, it is infinity. So, we see that each part of the horofunction boundary of a Hilbert geometry is the direct product of two lower-dimensional Hilbert geometries with the `1 -sum distance.

4 Isometries of the Hilbert metric De la Harpe [9] was the first to consider the isometry group of the Hilbert geometry. Let P n D Rn [ P n1 be real n-dimensional projective space, and suppose that D is contained within the open cell Rn inside P n . Let Coll.D/ be the set of collineations, that is, elements of PGL.n C 1; R/ preserving D. As de la Harpe observed, each element of Coll.D/ is an isometry since collineations preserve the cross-ratios.

4.1 Simplicial geometries However, not every isometry is a collineation, as we will now see. Think of the nsimplex as being a cross-section of the positive cone RnC1 C . Hilbert’s projective metric on the interior of this cone is given by xi yi Hil.x; y/ D log max C log max : i i yi xi Define V to be the n-dimensional linear space RnC1 = , where x y if x D y C ˛.1; 1; : : : ; 1/ for some ˛ 2 R, and equip V with the variation norm: kxkvar WD max xi min xi : i

i

We see that the Hilbert metric on the simplex is isometric to the normed space V via the map ! RnC1 ; log W int RnC1 C

.xi /i 7! .log xi /i

that takes logarithms coordinate-wise. This isometry was first observed by Nussbaum [20] and de la Harpe [9]. It was shown by Foertsch and Karlsson [14] that simplices are the only Hilbert geometries isometric to normed spaces. The result is also stated in [6], p. 35, and [7], p. 313.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 141

Theorem 4.1. A finite-dimensional Hilbert geometry on a bounded open convex set is isometric to a normed space if and only if the domain is a simplex. Using this correspondence, and the Masur–Ulam theorem, the isometry group of the simplicial Hilbert geometry was determined for n D 2 in [9] and for arbitrary n in [17]. Let nC1 be the group of coordinate permutations on V , let W V ! V be the isometry given by .x/ D x for x 2 V , and identify the group of translations in V with Rn . Theorem 4.2 ([11], [17]). If D is an open n-simplex with n 2, then Coll.D/ Š Rn Ì nC1 and Isom.D/ Š Rn Ì nC1 ; where nC1 D nC1 hi. The same result was obtained independently in [11], although the authors there were not aware that the geometry they were considering is isometric to a normed space. Consequently, their proof is somewhat longer. So, we see that in the case of the simplex the isometry group differs from the collineation group. Indeed, the latter is a subgroup of index two of the former. Let us to the coordinate-wise look in more detail at the map , which corresponds on int RnC1 C reciprocal map O W int RnC1 ! int RnC1 C C ;

.xi /i 7!

1 xi

i

:

(4.1)

The projective action of this map is an isometry but not a collineation. This is shown in Figure 5 for the case of n D 2.

Figure 5. An isometry of the triangle that is not a collineation. The solid straight line on the left is mapped to the curve on the right.

As observed in [9], the vertices are blown up to edges, the edges are collapsed to vertices, and a straight line in the interior is mapped to a straight line if and only if one of its ends approaches a vertex of the triangle.

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4.2 Polyhedral geometries At the opposite extreme from the case of simplices, we have the following. Theorem 4.3 ([9]). If the closure of D is strictly convex, then Isom.D/ D Coll.D/. It was also shown in the same paper that the only two-dimensional polyhedra with non-collineation isometries are the triangles. This was generalised to arbitrary dimension in the following theorem. Theorem 4.4 ([17]). If .D; Hil/ is a polyhedral Hilbert geometry, then Coll.D/ differs from Isom.D/ if and only if D is an open n-simplex, with n 2. The proof involves studying the action of an isometry on the parts of the horofunction boundary, in particular, on the parts where one or other of the `1 factors is trivial, that is, consists of a single point. There is one such part for every vertex on the polyhedron, and one for every facet. It is shown that every isometry either maps vertex parts to vertex parts and facet parts to facet parts, or it interchanges them. Isometries of the former type are shown to extend continuously to the Euclidean boundary of the domain and to be collineations, whereas isometries of the latter type are shown to only exist on simplices. The above theorem verifies, for the case of polyhedral Hilbert geometries, some conjectures of de la Harpe, namely that the Hilbert isometry group Isom.D/ is a Lie group, and that its identity component coincides with that of Coll.D/. Observe that Coll.D/ is naturally a Lie group since it is a closed subgroup of PGL.n C 1; R/. A consequence of these conjectures is that Isom.D/ acts transitively on D if and only if Coll.D/ does, the latter happening exactly when D is the cross-section of an homogeneous cone. The only polyhedra that occur as cross-sections of such cones are the simplices. De la Harpe enumerates all possible cross-sections of homogeneous cones up to dimension four [9].

4.3 The general case Dropping the polyhedral assumption, can one find other domains with isometries that are not collineations? Consider again the reciprocal map O defined in (4.1) on the interior of the nC1-dimensional positive cone. One way of seeing that it is an isometry of Hilbert’s projective metric is to observe that it is order-reversing, homogeneous of degree 1, and involutive, meaning that it is its own inverse. The following proposition then implies that O is gauge-reversing, that is, nC1 M..x/= O .y/I O RnC1 C / D M.y=xI RC /

for all x; y 2 int RnC1 C :

Proposition 4.5 ([19], [20]). Let g W C ! C 0 be a bijection between two open cones in a finite-dimensional vector space. Then, g is gauge-reversing if and only if it is order-reversing and homogeneous of degree 1 and its inverse is order-reversing.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 143

It is clear that gauge-reversing bijections are isometries of Hilbert’s projective metric. However, gauge-reversing maps do not just exist on the positive cone, they exist on all symmetric cones, of which the positive cone is an example. Recall [10] that a proper open cone C in a finite-dimensional real vector space is called symmetric if it is homogeneous, meaning that its group of linear automorphisms acts transitively on it, and self dual, meaning that C D C ? , where C ? now denotes the open dual. One defines the characteristic function on C to be Z e hy;xi dy for all x 2 C :

.x/ D C?

This map is homogeneous of degree minus the dimension of the cone, which implies that Vinberg’s -map, C ! C ? ; x 7! x WD r log .x/; is homogeneous of degree 1. It is known [13] that on symmetric cones the map is order-reversing. In fact, it was shown in [13] that this property of the -map characterises the symmetric cones among the homogeneous cones. One may easily verify that the map O in (4.1) is the -map for the positive cone int RnC1 C . The symmetric cones have been completely classified. Each such cone is the product of one or more of the following irreducible cones: the positive definite Hermitian matrices Herm.n; E/, with n 3, where the set of entries E can be the reals R, the complex numbers C, or the quaternions H; the positive definite Hermitian 3 3 matrices Herm.3; O/ with octonion entries; the Lorentz cone (or ice-cream cone) ˚ ƒn D .x1 ; : : : ; xn / 2 Rn j x1 > 0 and x12 x22 xn2 > 0 ; for some n 2. For the reasons discussed above, the -map on a symmetric cone is an isometry of Hilbert’s projective metric. Its projective action is therefore an isometry of the Hilbert geometry on a cross-section D of the cone. This isometry is not a collineation except when the symmetric cone is a Lorentz cone. It was conjectured in [17] that Isom.D/ and Coll.D/ differ if and only if the cone generated by D is symmetric and not Lorentzian, in which case the isometry group was conjectured to be generated by the collineations and the isometry coming from the -map. This was known to be true for the cone of positive-definite Hermitian matrices [18], and has recently been established for general symmetric cones [4]. The conjecture has now been proved in general [29]. An important step in the proof is the following. Theorem 4.6 ([29]). An open convex cone in a finite-dimensional real vector space admits a gauge-reversing map if and only if it is symmetric. Establishing the homogeneity of such a cone admitting a gauge-reversing map is not difficult. One uses Rademacher’s theorem to get that the map is differentiable

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almost everywhere. At each point of differentiability, the negative of the differential is a linear automorphism of the cone. This gives a large supply of automorphisms with which to prove homogeneity. Proving the self-duality of the cone is more complicated. The first step is to use arguments similar to those in the proof of homogeneity to show there is a map g W C ! C that is gauge-reversing and involutive, and fixes the base-point b. One then wishes to define a positive-definite bilinear form with respect to which C is self-dual. This is done by exploiting the fact that certain Funk horofunctions, namely those associated to extreme rays of C , are the logarithms of linear functionals. Each of these horofunctions is a singleton part and so gets mapped by g to a singleton part of the reverse-Funk horofunction boundary, that is, a reverse-Funk horofunction of the form rx , where x is an extremal generator of C , in other words, a non-zero element of an extremal ray of cl C . We now define A.y; x/ WD M.x=bI C / exp B rx B g.y/ D M.x=g.y/I C /; for all y 2 C and extremal generators x of C . The equality holds because of the definition of rx . From what we have just seen, exp B rx B g is the restriction to C of a linear functional. This lets us extend the definition of A to all y in V in such a way that it is linear in this variable. Further work is then required to extend the definition to all x in V and verify that A has the right properties. One may also be interested in gauge-preserving maps,that is, maps g W C ! C satisfying M.g.x/=g.y/I C / D M.x=yI C / for all x and y in C . Such maps are also clearly isometries of Hilbert’s projective metric. The following proposition parallels Proposition 4.5. Proposition 4.7 ([19], [20]). Let g W C ! C 0 be a bijection between two open cones in a finite-dimensional vector space. Then, g is gauge-preserving if and only if it is order-preserving and homogeneous and its inverse is order-preserving. These maps are well understood. Theorem 4.8 ([19], [22]). Let C and C 0 be open cones in a finite-dimensional vector space, and let g W C ! C 0 be gauge-preserving bijection. Then, g is the restriction to C of a linear isomorphism. Here is a simple proof using horofunctions. Proof. Choose the base-points b 2 C and b 0 2 C 0 such that g.b/ D b 0 . Since g is gauge-preserving, it preserves the Funk metric, and therefore maps Funk horofunctions to Funk horofunctions according to (2.1). In particular, singleton horofunctions are mapped to singleton horofunctions. Recall that there is a singleton horofunction of the Funk geometry for each extremal ray of the closed dual cone; in fact the horofunction is just the logarithm of a suitably normalised element of the ray. So for

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 145

every extremal generator x of C , there is an extremal generator x 0 of .C 0 / satisfying hx; yi D hx 0 ; g.y/i, for all y 2 C . However, by choosing the right number of extremal generators of C we obtain a basis for the dual space, that is, a linear coordinate system. It follows that g is the restriction to C of a linear map. So the projective action of every gauge-preserving map is a collineation. The conjecture referred to above now follows from combining Theorems 4.6 and 4.8 with the following fact proved in Theorem 1.3 of [29]: every isometry of a Hilbert geometry arises as the projective action of either a gauge-preserving map or a gauge-reversing map. Acknowledgement. This work is partially supported by the joint RFBR-CNRS grant number 05-01-02807.

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[14] A. Karlsson and T. Foertsch, Hilbert metrics and Minkowski norms. J. Geometry 83 (2005), no. 1, 22–31. [15] A. Karlsson, V. Metz, and G. Noskov, Horoballs in simplices and Minkowski spaces. Int. J. Math. Math. Sci. (2006), Art. ID 23656. [16] B. Lemmens and R. Nussbaum, Nonlinear Perron–Frobenius theory. Cambridge Tracts in Math. 189, Cambridge University Press, Cambridge 2012. [17] B. Lemmens and C. Walsh, Isometries of polyhedral Hilbert geometries. J. Topol. Anal. 3 (2011), no. 2, 213–241. [18] L. Molnár, Thompson isometries of the space of invertible positive operators. Proc. Amer. Math. Soc. 137 (2009), no. 11, 3849–3859. [19] W. Noll and J. J. Schäffer, Orders, gauge, and distance in faceless linear cones; with examples relevant to continuum mechanics and relativity. Arch. Rational Mech. Anal. 66 (1977), no. 4, 345–377. [20] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 75 (391), 1988. [21] M. A. Rieffel, Group C -algebras as compact quantum metric spaces. Doc. Math. 7 (2002), 605–651. [22] O. S. Rothaus, Order isomorphisms of cones. Proc. Amer. Math. Soc. 17 (1966), 1284–1288. [23] P. A. Storm, The barycenter method on singular spaces. Comment. Math. Helv. 82 (2007), no. 1, 133–173. [24] C. Walsh, The horofunction boundary of finite-dimensional normed spaces. Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 3, 497–507. [25] C. Walsh, The horofunction boundary of the Hilbert geometry. Adv. Geom. 8 (2008), no. 4, 503–529. [26] C. Walsh, Busemann points of Artin groups of dihedral type. Internat. J. Algebra Comput. 19 (2009), no. 7, 891–910. [27] C. Walsh, The action of a nilpotent group on its horofunction boundary has finite orbits. Groups Geom. Dyn. 5 (2011), no. 1, 189–206. [28] C. Walsh, The horoboundary and isometry group of Thurston’s Lipschitz metric. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, European Mathematical Society, Zürich 2014, 327–353. [29] C. Walsh, Gauge-reversing maps on cones and Thompson isometries. Preprint, arXiv:1312.7871. [30] C. Webster and A. Winchester, Boundaries of hyperbolic metric spaces. Pacific J. Math. 221 (2005), no. 1, 147–158. [31] C. Webster and A. Winchester, Busemann points of infinite graphs. Trans. Amer. Math. Soc. 358 (2006), no. 9, 4209–4224.

Chapter 6

Characterizations of hyperbolic geometry among Hilbert geometries Ren Guo

Contents 1 Introduction . . . . . . 2 Reflections . . . . . . 3 Perpendicularity . . . 4 Ptolemaic inequality . 5 Curvature . . . . . . . 6 Median . . . . . . . . 7 Isometry group . . . . 8 Ideal triangles . . . . . 9 Spectrum: a conjecture References . . . . . . . . .

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1 Introduction The Hilbert metric is a canonical metric associated to an arbitrary bounded convex domain in Rn or in a projective space. It was introduced by David Hilbert in 1894 as an example of a metric for which the Euclidean straight lines are geodesics. Hilbert geometry generalizes Klein’s model of hyperbolic geometry. Let K be a bounded open convex set in Rn .n 2/. The Hilbert metric dK on K is defined as follows. For any x 2 K, let dK .x; x/ D 0. For distinct points x, y in K, assume the straight line passing through x, y intersects the boundary @K at two points a, b such that the order of these four points on the line is a, x, y, b as in Figure 1. Denote the cross-ratio of the points by Œx; y; b; a D

kb xk ka yk ; kb yk ka xk

where k k is the Euclidean norm of Rn . Then the Hilbert metric is dK .x; y/ D

1 logŒa; x; y; b: 2

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a

x

y

b

Figure 1. Hilbert metric.

The metric space .K; dK / is called a Hilbert geometry. A Euclidean straight line in K is a geodesic under the metric dK . When K is the unit open ball °

.x1 ; : : : ; xn / 2 Rn j

Pn

±

iD1

xi2 < 1 ;

.K; dK / is Klein’s model of hyperbolic geometry. Since the cross-ratio is invariant under a perspectivity P of center O 2 Rn [ f1g, .K; dK / and .P .K/; dP .K/ / are isometric as Hilbert geometries. In particular, when K is enclosed by an ellipsoid (ellipse when n D 2), i.e., °

K D .x1 ; : : : ; xn / 2 Rn j

Pn

iD1

xi2 ai2

±

2; properly means there exists a projective hyperplane which does not intersect the closure of , or, equivalently, there is an affine chart in which appears as a relatively compact set; • d is the distance on defined, for two distinct points x, y, by 1 logŒa; b; x; y; 2 where a and b are the intersection points of the line .xy/ with the boundary @ (chosen as in Figure 1) and Œa; b; x; y denotes the cross ratio of the four points: jayj=jbyj . if we identify the line .xy/ with R [ f1g, it is defined by Œa; b; x; y D jaxj=jbxj d .x; y/ D

2.2 (Regular) Finsler metrics and their geodesic flow Definition 2.1. A Finsler metric on a manifold M is a field of (non-necessarily symmetric) norms on M , that is, a function F W T M ! Œ0; C1/ such that: • F .x; u/ D F .x; u/, .x; u/ 2 T M , > 0; • F .x; u C v/ 6 F .x; u/ C F .x; v/, x 2 M , u; v 2 Tx M .

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b y x

a

Figure 1. The Hilbert distance.

A Finsler metric defines a (non-symmetric) distance dF on M : the distance between two points x and y of M is the minimal Finsler length LF .c/ of a C 1 curve c W Œ0; 1 ! M from x to y, that is, Z 1 F .c.t P // dt: dF .x; y/ D inf LF .c/ D inf c

c

0

We will say that a Finsler metric F is regular if F is C 2 and the boundaries of its unit balls B.x; 1/ D fu 2 Tx M; F .x; u/ < 1g; x 2 M have positive definite Hessian (for some (hence any) fixed Euclidean metric on Tx M ). This is the minimal assumption we have to make for local geodesics to exist: as in Riemannian geometry, those are C 1 curves defined through a second order differential equation. Geodesics have constant speed, that is, F .c.t P // is constant, and if x D c.t / and y D c.t 0 / are points on the curve which are close enough, then the curve c W Œt; t 0 ! M is the shortest path from x to y: Z t0 F .c.s// P ds: dF .x; y/ D t

We say that the Finsler metric is complete if geodesics exist for all times. This is always the case if the manifold is compact. If the Finsler metric is complete, we can define the geodesic flow of the metric as the flow of the second-order differential equation which defines geodesics. In this way, the geodesic flow is defined on the tangent bundle T M . But, since geodesics have constant speed, this flow preserves the subbundles T ˛ M D fu 2 T M; F .u/ D ˛g;

˛ > 0:

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The most natural space to study the geodesic flow is thus the unit tangent bundle T 1 M . However, this space depends on the metric, so we will prefer considering this flow as defined on the homogeneous tangent bundle HM D T M X f0g=RC , identifying it with the unit tangent bundle: a point of the homogeneous tangent bundle consists of a base point on the manifold and a tangent direction at this point that we denote by w D .x; Œ/; its image ' t .w/ by the geodesic flow is obtained by following during the time t the unit speed geodesic leaving x in the direction Œ. Geodesic flows are important examples in dynamics, especially when the manifold is not “too big”, so we can expect strong recurrence properties. For example, the geodesic flow of a compact negatively-curved Riemannian manifold has strong hyperbolic properties, such as the Anosov property. The main goal of this chapter is to study these kind of properties for the geodesic flow of compact quotients of some Finsler geometries, with an emphasis on Hilbert geometries.

2.3 Hilbert geometries and their isometries A Hilbert geometry .; d / is an example of a Finsler manifold. The Finsler norm of u 2 Tx is given by the formula 1 1 juj C ; F .x; u/ D 2 jxuC j jxu j where u˙ are the intersection points of the line generated by u. It is Riemannian if and only if is an ellipsoid [50], in which case .; d / is the Beltrami model of the hyperbolic space. Various volume forms can be associated to a Finsler metric. However, all natural volumes that could be associated to Hilbert geometries are equivalent (see L. Marquis’ contribution in this volume [46]. So let us fix the volume Vol once and for all as

u−

x

u

Figure 2. The Finsler metric.

u+

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being the Busemann–Hausdorff volume: this is the Hausdorff measure of .; d /; equivalently, it is the volume such that the unit Finsler ball has the same volume as the Euclidean unit ball. Our main interest will lie in those Hilbert geometries which have enough symmetries to admit compact quotients. The rest of this part recalls some facts about the existence of such quotients. 2.3.1 Regular Hilbert geometries. From the formula, we see that F is regular if the boundary @ of is C 2 with positive definite Hessian1 . Geodesics coincide with projective lines. Thus, it is easy to see the geodesic flow on H : we just have to follow lines…. However, apart from the case of the ellipsoid, a regular Hilbert geometry has essentially no quotients: Theorem 2.2 (É. Socié-Méthou [51]). The isometry group of a regular Hilbert geometry .; d / is compact, unless is an ellipsoid. Hence, there is no interesting space where to study their geodesic flows from a global point of view. That is why we have to consider less regular Hilbert geometries. 2.3.2 Geodesics. For a general Hilbert geometry, geodesics can be defined metrically: a geodesic segment is a metric isometry c W Œ0; T ! from R to .; d / and a geodesic is an isometry from R to . It is not difficult to see that projective lines are still geodesics but there might be others, even locally [22]. This happens as soon as the boundary of the convex set contains two nonempty open segments which are in the same 2-dimensional subspace but not in the same supporting hyperplane. 2.3.3 Isometries. The automorphism group Aut./ consisting of those projective transformations preserving is an important subgroup of isometries. Indeed, we expect Aut./ to be the full isometry group in most cases: Conjecture. Aut./ D Isom.; d / unless .; d / is symmetric, in which case Aut./ is a subgroup of index 2 of Isom.; d /. Homogeneous Hilbert geometries are those whose automorphism group acts transitively. They have been described by M. Koecher and E. B. Vinberg in the fifties and sixties [37], [52]. Among them, symmetric ones are those which are self-dual. They fall into three classes: the simplices, the hyperbolic space and the symmetric spaces of the groups SL.n; K/, with K D R; C; H and n > 3 or the exceptional group E6.26/ (see [24] for example). The last conjecture is confirmed in some cases: 1 The Hessian is computed in some affine chart equipped with a Euclidean metric; its positive definiteness does not depend on the choice of the chart and the metric. See Section 8.5 for more on this.

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• for strictly convex sets: it is a consequence of the uniqueness of geodesics (P. de la Harpe [22]); • for polytopes: Aut./ D Isom.; d / unless is a simplex, in which case Aut./ has index 2 in Isom.; d / (P. de la Harpe [22] for the 2-dimensional case, B. Lemmens and C. Walsh [40] for the general case) • for symmetric Hilbert geometries: A. Bosché [14] proved that Aut./ has index 2 in Isom.; d /. A careful and general study of Aut./ is made in Marquis’ contribution (Chapter 8, [46]). In particular, he describes and classifies automorphisms in terms of the dynamics of their action on . 2.3.4 Divisible Hilbert geometries Definition 2.3. A quotient M D = of a Hilbert geometry .; d / by a discrete subgroup of Aut./ is called a convex projective manifold (or orbifold in case has torsion). A Hilbert geometry .; d / or the convex set is said to be divisible if it admits a compact quotient by a discrete subgroup of Aut./. Remark that by Selberg’s lemma, if < Aut./ has finite type, in particular when acts cocompactly on , then has a finite index subgroup without torsion. In this case, the quotient M D = is a smooth manifold and the group is isomorphic to the fundamental group 1 .M / of M via a faithful linear representation 1 .M / ! SL.n C 1; R/. Socié-Méthou’s theorem asserts that the hyperbolic space is the only regular divisible Hilbert geometry. Among homogeneous Hilbert geometries, only the symmetric ones and their products are divisible [53]. Remark that among symmetric Hilbert geometries, the hyperbolic space is the only one to be strictly convex or with C 1 boundary. Apart from the homogeneous cases, the existence of divisible Hilbert geometries has been a long-standing question. From a general point of view, the following result of Y. Benoist is essential: Theorem 2.4 (Y. Benoist [11]). Let .; d / be a divisible Hilbert geometry, divided by a discrete subgroup < Aut./. The following propositions are equivalent: • the convex set is strictly convex; • the boundary @ is C 1 ; • the metric space .; d / is Gromov-hyperbolic; • the group is Gromov-hyperbolic.

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Recall that a geodesic metric space .X; d / is said to be Gromov-hyperbolic if there is some ı > 0 such that any geodesic triangle xyz X of vertices x; y; z 2 X is ı-thin, that is, for any point p on the side Œxz, minfd.p; Œxy/; d.x; Œyz/g 6 ı: When the metric space is Gromov-hyperbolic for the constant ı, we say it is ı-hyperbolic. A finitely-generated group is Gromov-hyperbolic if its Cayley graph with respect to some finite set of generators is Gromov-hyperbolic for the word metric. This property does not depend on the generating set, but the constant ı of hyperbolicity does depend on it. 2.3.5 Strictly convex divisible Hilbert geometries. Examples of strictly convex divisible Hilbert geometries are now available in all dimensions. They are obtained by deformations of compact hyperbolic manifolds following an idea of [32]. In low dimensions, examples can be obtained using Coxeter groups. First examples were provided by these means [33]. Y. Benoist [10] also constructed in this way strictly convex divisible Hilbert geometries which are not quasi-isometric to the hyperbolic space. M. Kapovich [34] found such examples in all dimensions. We refer to Marquis’ contribution [46] for a deeper study of this question. Assume M D = is a compact quotient manifold of a strictly convex Hilbert geometry. Then all elements g 2 are hyperbolic isometries of .; d /, which means the following: • g fixes exactly two points xgC and xg on the boundary @ and acts as a translation (for the Hilbert metric) on the line .xg xgC /, which is called the axis of g; x X fxg g, we have limn!C1 g n y D xgC ; • the point xgC is attractive: for any y 2 C n x y D xg . • the point xgC is repulsive: for any y 2 Xfx g g, we have lim n!C1 g

As elements of SL.n C 1; R/, the elements g 2 are biproximal: their biggest and smallest eigenvalues (in modulus) 0 and n are simple, that is, the corresponding eigenspaces are 1-dimensional; these eigenspaces are the fixed points xgC and xg ; the translation distance on the axis is 12 log n0 . Let us end this paragraph with a result about the group which divides the Hilbert geometry that will be crucial to deduce dynamical rigidity results. It reads as follows: Theorem 2.5 (Y. Benoist [7]). Suppose that .; d / is a strictly convex Hilbert geometry, divided by a discrete subgroup < Aut./. The group is Zariski-dense in SL.n C 1; R/, unless is an ellipsoid. Recall that the Zariski-closure of a subgroup of SL.n C 1; R/ is the smallest algebraic subgroup G of SL.n C 1; R/ which contains . We then say that is Zariski-dense in G.

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The hypothesis of strict convexity in the last theorem is actually unnecessary, but the proof in this case is far more involved [9]. This last theorem will be useful through the following characterization of Zariskidense subgroups of semisimple Lie groups, which is also due to Y. Benoist. Let us explain it in the context of the Lie group SL.n C 1; R/. To each element g in SL.n C 1; R/, we associate the vector log.g/ D .log 0 .g/; : : : ; log n .g// 2 RnC1 ; where 0 .g/ > 1 .g/ > > n .g/ denote the moduli of the eigenvalues of g. For a subgroup of SL.n C 1; R/, let log D flog g; g 2 g. Theorem 2.6 (Y. Benoist, [8]). Let be a subgroup of SL.n C 1; R/. If is Zariskidense P in SL.n C 1; R/, then the subgroup generated by log is dense in the subspace f i xi D 0g.

2.4 The geodesic flow 2.4.1 Definition. For general Hilbert geometries, we consider the geodesic flow ' t W H ! H following projective lines. Let X W H ! TH be the vector field which generates this flow. If we choose an affine chart and a Euclidean metric j j on it, in which is a bounded open convex set, then X can be written as X D mX e , where X e is the generator of the Euclidean geodesic flow, because X and X e have the same orbits. A direct computation gives m.w/ D 2

1 1 C C jxw j jxw j

1

D2

jxw C j jxw j ; jw C w j

w D .x; Œ/ 2 H ;

where w C (resp. w ) denotes the intersection point of @ with the ray leaving x in the direction (resp. ). This link between X and X e , that is, the flatness of Hilbert geometries, is crucial in extending some differential objects in Section 3. The vector field X and the geodesic flow ' t have the same regularity as the boundary of . If M D = is a quotient manifold, the geodesic flow is defined on HM by projection. To study its (local) properties, we will often work directly on H where geodesics are lines and computations can be made. For a compact quotient, we can expect from Benoist’s theorem strong differences for the geodesic flow on HM between the strictly convex (with C 1 boundary) and non-strictly convex (with non-C 1 boundary) cases. We can give easy illustrations of this difference: • at a non-C 1 point of the boundary, there are asymptotic geodesics whose distance does not go to 0; • if I is an open segment in the boundary @, the distance between two geodesic rays ending at some points of I stays bounded.

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We do not know much more than these observations in the non-strictly convex case. On the contrary, more attention has been paid to the strictly convex case, where the flow happens to exhibit strong hyperbolicity properties. This chapter is dedicated to this case, that is, the study of the geodesic flow of a compact quotient M D = with strictly convex (and by Theorem 2.4 its boundary is of class C 1 ). 2.4.2 Closed orbits. Assume M D = is any quotient manifold of a strictly convex (with C 1 boundary) Hilbert geometry .; d /, with a discrete subgroup of Aut./. We use the same notation on and M , on H and HM . The orbit of a point w 2 H is the set ' w D f' t .w/; t 2 Rg. The orbit ' w of w 2 HM is closed if there exists T > 0 such that ' T .w/ D w. The smallest T > 0 which satisfies this equality is the length of the orbit. We also say that w is a periodic point and that T is its period. If G is a group, a non-identity element g of G will be called primitive if there is no h 2 G and k > 2 such that g D hk . The same will be said of the conjugacy class of g in G. For example, if Œg is a free homotopy class in M , that is, the conjugacy class of an element g in the group 1 .M /, Œg is primitive if “it does not make more than one loop”. A closed orbit on HM lifts to an orbit on H which projects down in onto an oriented line .x x C /, x ; x C 2 @, which is left invariant by some non-identity hyperbolic elements of . Among such elements, which are all hyperbolic of axis .x x C /, only one is primitive and has x C as attractive fixed point and x as repulsive fixed point. In this way, we can associate to each closed orbit a primitive hyperbolic element of . Conversely, to such an element is associated the oriented line .x x C /, which yields a closed orbit of the geodesic flow. Two such elements will define the same closed orbit if they are conjugate. Therefore we have the Proposition 2.7 (see [11], Proposition 5.1). Let .; d / be a strictly convex Hilbert geometry with C 1 boundary and M D = a compact quotient manifold, where is a discrete subgroup of Aut./. Closed orbits of the geodesic flow on HM are in bijection with conjugacy classes of primitive hyperbolic elements of . A closed orbit on HM gives by projection on M a closed geodesic, provided with an orientation. If g is a primitive hyperbolic element of , the orbits defined by g and its inverse g 1 project down on the same closed geodesic in M , the closed orbits differ only by the direction in which they run along the geodesic. A direct computation shows that the length of the associated orbit is 12 log n0 , where 0 and n are the moduli of the biggest and smallest eigenvalues of g.

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3 Differential objects in Finsler and Hilbert geometries Given a Riemannian manifold, its geodesic flow benefits a lot from the rich geometric structure of the manifold, and this can be extended to regular Finsler metrics. For less regular Hilbert geometries, this is much more delicate; only part of the objects can be constructed: these defects can be seen as an explanation for some rigidity results we will present later on.

3.1 The Hilbert 1-form For a regular Finsler metric, the geodesic flow is the Reeb flow of a 1-form A called the Hilbert 1-form. This is Proposition 3.2 below. The Hilbert 1-form is defined via the vertical derivative of the Finsler metric: Definition 3.1. Let F be a C 1 Finsler metric on a manifold M . Denote by p W T M ! M and r W T M Xf0g ! HM the canonical bundle projections. The vertical derivative of F is the 1-form dv F on T M defined by F .x; C "dp.Z// F .x; / ; (3.1) "!0 " where .x; / 2 T M and Z 2 T.x;/ T M . It descends by homogeneity on HM to give the Hilbert 1-form A of F , that is, dv F .x; /.Z/ D lim

A D r dv F: The definition of A is made possible because the 1-form dv F depends only Pon the direction Œ: it is invariant under the flow generated by the vector field D D i @@i . Let W HM ! M denote the bundle projection. Since d .X.x; Œ// 2 Œ and F .d .X.x; Œ/// D 1, we can write A.Z/ D lim

"!0

F .x; d .X "Z// 1 : "

Proposition 3.2. Let F be a regular Finsler metric on a manifold M and A its Hilbert 1-form. The generator X of the geodesic flow of F is the unique solution of A.X/ D 1I

dA.X; / D 0:

Moreover, the geodesic flow preserves the volume form A ^ dAn1 that we call the Liouville volume or Liouville measure. There is another way of seeing A as we explain in the next section. We learned this from T. Barthelmé, and more on this question can be found in his PhD thesis [5]; Arnold’s book [3] is still a good reference for the classical facts on Hamiltonian dynamics.

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3.2 The Legendre transform 3.2.1 Definition for regular metrics. We can see the geodesic flow of a regular Finsler metric as a Hamiltonian flow by using the Legendre transform. Definition 3.3. Let F be a regular Finsler metric on a manifold M . The Legendre transform LF W T M ! T M is defined by the formula LF .v/.u/ D

1 d ˇˇ F 2 .x; v C t u/: 2 dt t D0

(3.2)

For a Riemannian metric, the Legendre transform is linear: for a vector v 2 Tx M , the Legendre transform of v is the dual 1-form defined by v: LF .v/.u/ D hv; ui;

u 2 Tx M:

For a regular Finsler metric, we can see the Legendre transform geometrically in the following way. Let Bx .r/ and Sx .r/ be the metric ball and sphere of radius r > 0 of the Finsler norm F .x; :/ on Tx M . The Legendre transform LF .v/ of a vector v 2 Sx .r/ is then the 1-form such that LF .v/.v/ D F .v/2 I

ker LF .v/ D Tv Sx .r/:

3.2.2 Finsler cometrics Definition 3.4. A Finsler cometric is a function F W T M ! Œ0; C1/ such that: • F .x; ˛/ D F .x; ˛/, .x; ˛/ 2 T M , > 0; • F .x; ˛ C ˇ/ 6 F .x; ˛/ C F .x; ˇ/, x 2 M , ˛; ˇ 2 Tx M . We will say that a Finsler cometric F is regular if F is C 2 and the boundary of its unit coballs have positive definite Hessian. Its Legendre transform LF W T M ! T M is defined by LF .v/.u/ D

1 d ˇˇ .F /2 .x; v C t u/: 2 dt tD0

(3.3)

To a Finsler metric F is naturally associated a Finsler cometric F ; this is the usual dual norm, defined as F .x; ˛/ D maxf˛.v/; v 2 Sx .1/g: If F is regular, the Finsler cometric F is also regular. The Legendre transform is then a C 1 -diffeomorphism from T M to T M which sends F to F : for all v 2 v, F .LF .v// D F .v/. Its inverse is the Legendre transform LF of F . LF is trivially homogeneous of degree 1, that is, LF .v/ D LF .v/, so it defines a map lF from HM to H M D T M X f0g=RC that we still call the Legendre transform. Its inverse is the map lF .

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3.2.3 The Hilbert 1-form from the Hamiltonian point of view. The cotangent symplectic form ! being given space T M is canonically a symplectic manifold, theP by the exterior derivative of the Liouville form L D i dxi . The geodesic flow on T M is the Hamiltonian flow of the Hamiltonian function H.x; ˛/ D 12 .F .x; ˛//2 : this is the flow generated by the vector field XH such that dH.Z/ D !.XH ; Z/;

Z 2 T T M:

Let S M D fu 2 T M; F .u/ D 1g denote the unit sphere bundle of F . The restriction of the projection T M ! H M to S M , is a diffeomorphism that we denote by pF W S M ! H M . Through this map, we can define on H M the 1-form pF L, the 2-form pF ! and the vector field pF XH , which are the pushforward under pF of the restrictions of L; ! and XH to S M . The 2-form pF ! is invariant by the flow of pF XH , and the Liouville measure of this flow is the invariant volume given by pF L ^ .pF !/n1 . Proposition 3.5. Let F be a regular Finsler metric on a manifold M . The Hilbert form A, the 2-form dA and the Liouville volume A ^ dAn1 are the respective pullbacks by lF of the Liouville form pF L, the symplectic form pF !, and the volume pF L ^ .pF !/n1 . The Legendre transform lF also conjugates the flow of pF XH on H M and the geodesic flow of F on HM . 3.2.4 General Finsler metrics. For a general Finsler metric, the previous constructions have in general no meaning. Only the Finsler cometric F is well defined. Using the geometrical construction, the Legendre transform could also be seen as a multiplevalued function: for example, to a unit vector v 2 Tx M where the unit sphere Sx .1/ is not C 1 , the Legendre transform would associate the set of linear forms ˛ such that ˛.v/ D 1 and whose kernel is one of the supporting hyperplanes of Sx .1/ at v. If the Finsler metric is C 1 with strictly convex unit balls, the unit balls of the Finsler cometric F are also strictly convex with C 1 boundary. The Legendre transforms LF and lF are then well defined homeomorphisms through the geometrical construction, but they are no more of class C 1 . Thus, even if the differential forms L, ! and the vector field XH are well defined on T M , as well as their projections by (the now C 1 -map) pF on H M , we cannot pull these objects back to HM by the Legendre transform. However, I do not know if we can define a geodesic flow in this context.

3.3 Connection and parallel transport Given a manifold M , there are many ways to identify two distinct tangent spaces Tx M and Ty M . If M is the affine space Rn , then the most natural way is to identify the tangent spaces to the space itself. For a general manifold, such an identification is called a linear connection: any path c from x to y gives an identification, that is, a linear isomorphism, between Tx M and Ty M via parallel transport.

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When M is a Riemannian manifold, we would like the linear isomorphisms to be actual isometries between the tangent spaces. In other words, we want the Riemannian metric to be parallel. The Riemannian or Levi-Civita connection is the unique linear connection without torsion for which the metric is parallel. This construction cannot extend to the regular Finsler context. As we already said, the good geometrical space for a Finsler metric is the tangent bundle T M and not the manifold M . A “Finsler” connection should then be an object defined on T M , which would extend the Levi-Civita connection, in the sense that the latter should be recovered “by projection” in the case of a Riemannian metric. However, depending on which properties we want the connection to have (such as no torsion, parallelism of the Sasaki metric), we will obtain different connections. A way of avoiding these problems was discovered by P. Foulon: if we restrict ourselves to transport along geodesics then we get a well-defined linear transport. The good space to look at is then HM , and the good object is the geodesic flow. In this context, we can define in an intrinsic and canonical way linear objects generalizing the Riemannian ones. Foulon’s contribution appeared in [25], but one can find a shortest version in English in [27]. Here, we just describe the conclusions, which apply in general for C 3 regular Finsler metrics. The vertical bundle of HM is the bundle V HM D ker d , where is the canonical bundle projection W HM ! M . The tangent bundle to HM admits then a horizontal-vertical decomposition as T HM D R X ˚ hF HM ˚ V HM:

(3.4)

(The vector field X is the generator of the geodesic flow.) Furthermore, there is a pseudo-complex structure on hF HM ˚ VHM which exchanges vertical and horizontal subspaces; that is, a bundle isomorphism J F W hF HM ˚ VHM ! hF HM ˚ VHM such that J F .VHM / D hF HM; J F .hF HM / D VHM

and

J F B J F D Id:

Associated to this decomposition is a “partial” covariant derivative D X which defines the differential of a vector field Z defined along an orbit of the geodesic flow in the direction of the flow. This is a differential operator of order 1, D X W THM ! THM , which commutes with J F , and satisfies D X .X / D 0. A vector field Z defined along an orbit of the flow is parallel if D X .Z/ D 0. This allows us to define the parallel transport of a vector Z.w/ 2 Tw HM along the orbit ' w of w: it is the unique parallel vector field Z defined on ' w whose value at w is Z.w/. If we fix a t 2 R, we will denote by T t .Z.w// WD Z.' t .w// the parallel transport of Z.w/ at time t . This yields a bundle isomorphism T t W THM ! THM which sends TwHM on T' t .w/ HM .

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Since D X commutes with J F , the parallel transport T t also commutes with J F and, since and X is parallel, it preserves the decomposition THM D RX ˚hF HM ˚ VHM . The projection d W THM ! T M induces an isomorphism between the space R X.w/ ˚ hF HM.w/ and T.w/ M for each w 2 HM , so we can define a parallel transport along geodesics on M . Let c W R ! M be a geodesic and x D c.0/, w D Œc.0/ P 2 HM . If u 2 Tx M , we consider its lift U.w/ D d 1 .u/ to the space R X.w/ ˚ hF HM.w/, and define the parallel transport of u along the geodesic c by Tct .u/ D d .T t .U.w///;

t 2 R:

This gives a linear isomorphism T t between the tangent spaces Tx M and Tc.t / M .

3.4 Curvature and Jacobi fields The Jacobi operator RF is defined by RF .X/ D 0;

RF .Y / D pvF .ŒX; J F .Y //;

RF commutes with J F ;

where pvF .Z/ denotes the projection on the vertical subspace VHM of the vector Z 2 THM , with respect to the horizontal-vertical decomposition (3.4). In the case X is the geodesic flow of a Riemannian metric g on M , the Jacobi operator allows to recover the curvature tensor Rg of g: for u; v 2 Tx M X f0g, we have Rg .u; v/u D

d .RF V .x; Œu// ; g.u; u/

where V .x; Œu/ is the unique vector in R X.x; Œu/ ˚ hF HM.x; Œu/ such that d .V .x; Œu// D v. Definition 3.6. We will say that a regular Finsler manifold .M; F / is negatively curved if the Jacobi operator RF is negative definite. A Jacobi field is a vector field J on HM which satisfies D X D X J C RF J D 0: Jacobi fields form a 2.2n1/-dimensional vector space, which is J F -invariant. In the Riemannian case, one recovers the usual Jacobi fields by projection on the base. Jacobi fields on HM do not contain more information than the ones on M , the projection only rubs out the J F -symmetry between horizontal and vertical subspaces, as for the tensor curvature.

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3.5 Metrics on H M For a Riemannian metric g on M , there is a canonical associated Riemannian metric on HM called the Sasaki metric. To define it, we use the Levi-Civita connection THM D R X ˚ VHM ˚ hg HM and do the following construction. • The decomposition THM D R X ˚ VHM ˚ hg HM is orthogonal for the Sasaki metric. • Each subspace R X.u/ ˚ hgu HM is isomorphic to the tangent space T.u/ M of M via the projection ; the quadratic form on R X.u/ ˚ hg HM.u/ is defined as the pullback of g.u/ by . • Vertical and horizontal subspaces are identified by the complex structure J g ; the quadratic form on Vu HM is the push-forward by J g of the quadratic form on hgu T M . By construction, the Sasaki metric is invariant by the pseudo-complex structure, and vertical and horizontal subspaces are isometric. There are different ways of extending this definition to the Finsler context. Usually, Finsler geometers consider the following generalization for regular Finsler metrics. The Sasaki metric on HM associated to a regular Finsler metric F on M is the Riemannian metric g F defined by F .Y; Y 0 / D Hess.x;u/ .F 2 /.Y; Y 0 /; g.x;Œu/

Y; Y 0 2 VHM;

where u is the unit vector in Œu, and g F .X; X/ D 1;

g F .h; h0 / D g F .J F .h/; J F .h0 //;

h; h0 2 hF HM:

Equivalently, the formula on VHM can be replaced by g F .Y; Y 0 / D dA.ŒX; Y ; Y 0 /;

Y; Y 0 2 VHM;

where A is the Hilbert 1-form introduced above. Another way would be the following, but this would yield a Finsler metric on HM : we let, for Z D aX C Y C h, 1=2 : kZk D jaj2 C .F .d h//2 C .F .d J F Y //2 When the metric is Riemannian, we recover the Sasaki metric. When the metric is Finsler, this metric is only a Finsler metric (except for surfaces), whose regularity is one less than the original one. The complex structure is still an isometry between vertical and horizontal subspaces. In fact, I do not think the metric k k was already considered before. However, the first construction does not make sense for non-regular Finsler metrics and that is the reason why we consider the second one. In particular, we will use this second metric to study non-regular Hilbert geometries.

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3.6 The case of Hilbert geometries The above constructions work in general for at least C 3 regular Finsler metrics. It is however possible to exploit the flatness of Hilbert geometries to extend this formalism to the case where the convex set is strictly convex with C 1 boundary. Choose an affine chart adapted to the properly convex set RP n and fix a Euclidean metric on it. Denote by X e W H ! H the generator of the Euclidean geodesic flow. The crucial thing is the link X D mX e which exists between the geodesic flow of the Hilbert and the Euclidean metrics and the fact that m and its first derivatives are smooth in the direction of the flow(s). We can find details in [20]. It allows us to use freely Foulon’s dynamical formalism, that is, the objects defined in Sections 3.3 and 3.4. For example, we can prove the following proposition, which is immediate for regular metrics by using the differential dA of the Hilbert form: Proposition 3.7 ([20]). Let be a strictly convex subset of RP n with C 1 boundary and A the Hilbert form of the Hilbert metric F on . Then (i) ker A D VH ˚ hF H ; (ii) A is invariant under the geodesic flow. A general result linking RX and RX and a direct computation prove that such Hilbert geometries have constant curvature 1: e

Proposition 3.8. Let be a strictly convex subset of RP n with C 1 boundary. The Jacobi endomorphism of the Hilbert geometry .; d / is RF D Id on hF H C VH . However, the interpretation of such a result is not the same as in Riemannian geometry. We should not forget that the good geometrical space in Finsler geometry is the tangent bundle and not the manifold. Perhaps, the main consequence of this fact (at least in the context of this chapter) is an explicit computation of Jacobi fields: Lemma 3.9. Let .; d / be a Hilbert geometry defined by a strictly convex set with C 1 boundary. Let w 2 H , and Jw 2 VH C hF H . Then the Jacobi field J along the orbit of w such that J.w/ D Jw and D X J.w/ D ˙Jw is given by J.' t w/ D e ˙t T t Jw : 0 This describes the behaviour of all Jacobi fields since for any Zw ; Zw 2 TH , we can write C C Zw ; Zw D X.w/ C Zw

0 C Zw D 0 X.w/ C Zw Zw ;

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with C D Zw

0 Zw C Zw ; 2

Zw D

0 Zw Zw : 2

Hence the Jacobi field J along the orbit of w such that J.w/ D Zw and D X J.w/ D 0 will be given by Zw C J.' t w/ D .0 t C /X.' t w/ C e t T t Zw C e t T t Zw :

4 Stable and unstable bundles and manifolds We assume in this section that the convex set which defines the Hilbert geometry is strictly convex with C 1 boundary. The geodesic flow ' t W H ! H is then a C 1 flow. We want to study the (spatially infinitesimal) behaviour of the geodesic flow on the manifold H equipped with the Finsler metric k k introduced in Section 3.5. We denote by dH the metric induced by k k on H . Even if we will only consider compact quotients in the other sections, we do not make here any assumption on the existence of a quotient manifold, aiming to provide objects and results that could be useful for more general quotients.

4.1 Busemann functions and horospheres The Busemann function based at 2 @ is defined by b .x; y/ D lim d .x; p/ d .y; p/: p!

In some sense, it measures the (signed) distance from x to y in as seen from the point 2 @. A particular expression for b is given by b .x; y/ D lim d .x; .t // t; t!C1

where is the geodesic leaving y at t D 0 to . When is fixed, then b is a surjective map from onto R. When x and y are fixed, then b: .x; y/ W @ ! R is bounded by a constant C D C.x; y/. The horosphere passing through x 2 and based at 2 @ is the set H .x/ D fy 2 ; b .x; y/ D 0g: H .x/ is also the limit when p tends to of the metric spheres S.p; d .p; x// about p passing through x. In some sense, the points on H .x/ are those which are as far from as x is. The (open) horoball H .x/ defined by x 2 and based at 2 @ is the “interior” of the horosphere H .x/, that is, the set H .x/ D fy 2 ; b .x; y/ > 0g:

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It is easy to see that horospheres have the same kind of regularity as the boundary of . A consequence of (the proof of) Proposition 3.7 is the following Corollary 4.1. Let w D .x; Œ/ 2 H , w ˙ D ' ˙1 .w/ 2 @ and the unit vector in Œ. The projection d .VH C hF H / is the tangent space at x to both Hw C .x/ and Hw .x/ and the tangent space at to the unit sphere of F .x; / in Tx .

4.2 Stable and unstable bundles and manifolds Recall the following notions: Definition 4.2. The stable set of w 2 H is the set of points v 2 H such that lim t !C1 d.' t w; ' t v/ D 0. The unstable set is the set of points v 2 H such that lim t !1 d.' t w; ' t v/ D 0. Knowing a bit of hyperbolic geometry, it is not difficult to “find” what should be the stable and unstable sets of a point w D .x; Œ/ 2 H . Because d > d B , we see that the extreme point of the orbit of a point v 2 W s .w/ must coincide with the one for w, that is, v C D w C . Moreover, the C 1 regularity of the boundary @ implies that the horosphere Hw C ..w// about w C through .w/ satisfies Hw C ..w// D fv 2 H ; d ..' t .w//; .' t .v// D 0g : Thus, the stable set of w has to be a subset of the set W .w/ defined as the set of points v such that v C D w C and whose projection .v/ is on the horosphere through w about w C (see Figure 3): ˚ W .w/ D v 2 H j v C D w C ; .v/ 2 Hw C ..w// : Similarly, the unstable set of w has to be a subset of the set ˚ W C .w/ D v 2 H j v D w ; .v/ 2 Hw ..w// : Both sets W .w/ and W C .w/ are C 1 submanifolds of H . The sets W .w/, w 2 H , foliate H , and similarly the sets W C .w/; in general, these foliations are only C 0 . We will refer to them as the and C foliations. We immediately see that both foliations are invariant under the geodesic flow: W .' t .w// D ' t .W .w//

and

W C .' t .w// D ' t .W C .w//:

The rest of this section is dedicated to the following Theorem 4.3. Let w 2 H . The sets W .w/ and W C .w/ are the stable and unstable sets of w. The unstable and stable tangent bundles are given by E u D fY C J F .Y /; Y 2 VH g

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and E s D fY J F .Y /; Y 2 VH g D J F .E u /:

W − (w)

w−

x

ξ

w+

W + (w)

Figure 3. The submanifolds W C .w/ and W .w/, w D .x; Œ/.

4.2.1 A temporary Finsler metric. Let E and E C be the tangent bundles to the and C foliations. These bundles define a ' t -invariant decomposition of the tangent bundle T HM D R X ˚ E ˚ E C : Corollary 4.1 implies that E ˚ E C D VH C hF H . We define a temporary Finsler metric k k˙ by 1=2 : kZk˙ D jaj2 C 4.F .d Z C /2 C F .d Z /2 / It is not difficult to see that W C .w/ and W .w/ are the stable and unstable sets of w for this metric: this is what Corollary 4.5 below asserts. We first need to make a little computation. To simplify this computation, we will use the projective nature of our objects and choose a good affine chart and a good Euclidean metric on it. Let w D .x; Œ/ 2 H . A good chart at w is an affine chart where the intersection Tw C @ \ Tw @ is contained in the hyperplane at infinity, and a Euclidean structure on it so that the line .xw C / is orthogonal to Tw C @ and Tw @ (see Figure 4). Lemma 4.4. Let w 2 H , Z 2 E .w/ and fix a good chart at w. Set x D .w/, x t D ' t .w/, z D .Z /, z t D d d' t .Z /. We have jx t w C j jzj jx t w C j t ˙ C kd' .Z /k D 2F .z t / D : jxw C j jx t z tC j jx t z t j

Chapter 7. The geodesic flow of Finsler and Hilbert geometries

Tw− ∂Ω

181

Tw+ ∂Ω

x

w−

ξ

w+

Figure 4. A good chart at w D .x; Œ/.

Proof. We have kd' t .Z /k˙ D 2F .z t / by definition of the metric k k˙ . Now, by definition of F , we get jz t j 1 1 F .z t / D ; C jx t z t j 2 jx t z tC j where z tC and z t are the intersection points of the line fx t C z t ; 2 Rg and @ (see Figure 5). Consider the map h t W y 2 Hw C .x/ 7! y t D ' t .y; Œyw C / D .yw C / \ Hw C .x t /: We see that z t is given by z t D dh t .z/ D

jx t w C j z: jxw C j

This gives the result. We have a similar result for Z C 2 E C .w/. If Z C 2 E C .w/, with the same notation, we have jx t w j jzj jx t w j t C ˙ kd' .Z /k D F .z t / D : (4.1) C 2jxw j jx t z tC j jx t z t j The strict convexity of the convex set and the C 1 regularity of its boundary now yield the following Corollary 4.5. Let Z 2 E , Z C 2 E C . The map t 7! kd' t Z k˙ is a strictly decreasing bijection from R onto .0; C1/, and the map t 7! kd' t Z C k˙ is a strictly increasing bijection from R onto .0; C1/.

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zt+

z w

−

ξ x

zt xt

w+

zt− Figure 5. Contraction of stable vectors.

In particular, W .w/ and W C .w/ are the stable and unstable sets of w for the metric k k˙ . We will prove in the sequel that this metric actually coincides with the metric k k. 4.2.2 Identification of stable and unstable bundles for the metric k k. For a Riemannian or regular Finsler manifold M of variable negative curvature, the construction of stable and unstable manifolds is achieved through the understanding of Jacobi fields, via the following fact: for any Zw 2 Tw HM , the vector field Z defined along the orbit of w by Z.' t w/ D d' t Zw is a Jacobi field. This observation stays valid in Hilbert geometry and, as we have seen, the behaviour of Jacobi fields is easy to understand via parallel transport. In this way, we can identify the stable and unstable bundles in terms of the differential objects of Section 3. They are given by E u D fY C J F .Y /; Y 2 VHM g;

E s D fY J F .Y /; Y 2 VHM g D J F .E u /;

as asserts the next Proposition 4.6. Let Z s 2 E s , Z u 2 E u . The map t 7! kd' t Z s k is a strictly decreasing bijection from R onto .0; C1/, and the map t 7! kd' t Z u k is a strictly increasing bijection from R onto .0; C1/. To prove this proposition, we need some preparatory observations. We have THM D RX ˚E u ˚E s and this decomposition is ' t -invariant. As said above, for any Zw 2 Tw HM , the vector field Z defined along ' w by Z.' t w/ D

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u u d' t Zw is a Jacobi field. Furthermore, if Zw 2 E u .w/, we have D X .Z u /.w/ D Zw , and Proposition 3.9 implies that

d' t .Z u / D e t T t Z u :

(4.2)

s s If Zw 2 E s .w/, we have D X .Z s /.w/ D Zw and

d' t .Z s / D e t T t Z s :

(4.3)

To understand the behaviour of kd' t .Z u /k and kd' t .Z s /k, Z u 2 E u , Z s 2 E s , we have to understand the behaviour of kT t Z s k and kT t Z u k. Here is the big difference with the regular Finsler or Riemannian case. In the latter cases, the parallel transport preserves the Sasaki metric. For the Hilbert geometries under consideration, the parallel transport does not preserve the metric k k. In fact, we could prove the Proposition 4.7. Let .; d / be a Hilbert geometry defined by a strictly convex set with C 1 boundary. The following propositions are equivalent: • the boundary of the convex set is C 2 with positive definite Hessian; • for any w 2 H , there exists C > 0 such that C 1 kZk 6 kT t Zk 6 C kZk;

Z 2 Tw H :

To understand the parallel transport, we compare it with the parallel transport of the Euclidean structure. Fix an affine chart and a Euclidean metric j j on it, in which appears as a bounded open convex set. Denote by X e W H ! TH the generator of the Euclidean geodesic flow. Recall that X D mX e . Let w 2 HM and pick a vertical vector Y .w/ 2 Vw HM . Denote by Y and Y e its parallel transports with respect to X and X e along ' w. Let h D J F .Y / and e he D J X .Y e / be the corresponding parallel transports of h.w/ D J F .Y .w// and e he .w/ D J X .Y e .w// along ' w. The main result is the following Lemma 4.8. Along the orbit ' w, we have

m.w/ Y D m

1=2

Ye

and h D LY m X e C .m.w/m/1=2 he

m.w/ LX e m Y e : m

Now, we can complete a Proof of Proposition 4.6. Consider the vector Z.w/ D Y .w/ C h.w/ D Y .w/ J F .Y .w// 2 E s .w/. We use the notation of Proposition 4.4 and its proof. In a good chart at w, LY m D 0 along the orbit ' w, hence kT t Z.w/k D F .d .T t h.w/// D .m.w/m.' t .w///1=2 F .d .he .' t .w///;

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where he is as above. From Corollary 4.1 and the fact that hF H CVH D E s CE u , the vector d .T t .h.w/// is in Tx t H' t .w/ . The Euclidean parallel transport preserves the Euclidean metric so one has jd .he .' t .w//j D jd .he .w//j D jd .h.w//j. Keeping the same notation (see Figure 6), these two observations give t

t

kT Z.w/k D .m.w/m.' .w///

1=2 jd .h.w//j

C 1=2

jx t w j D C.w/ jx t z tC j

2

1 1 C C jx t z t j jx t z t j

jx t w C j1=2 ; C jx t z t j

for some constant C.w/ > 0. This equality is similar to the one in Lemma 4.4. The strict convexity of the convex set and the C 1 regularity of its boundary conclude the proof. zt+

dπ(h(w)) ω−

x

dπ(T t h(w)) xt

ω+

zt− Figure 6. Action of the parallel transport.

4.2.3 Both constructions coincide. We will now conclude the proof of Theorem 4.3, through the Proposition 4.9. Let .; d / a Hilbert geometry defined by a strictly convex set with C 1 boundary. We have E s D E , E u D E C and k k D k k˙ . I do not know a simple proof of this fact. If the geometry is divisible, then the following lemma concludes:

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Lemma 4.10. If k k and k k˙ are bi-Lipschitz equivalent on H , then E s D E , E u D E C and k k D k k˙ . Proof. Pick a vector Z 2 E , decompose it with respect to E s ˚ E u , and use Corollary 4.5 and Proposition 4.6 to conclude. Otherwise, we can do as follows. For related material concerning Benzécri’s theorem and Gromov-hyperbolic Hilbert geometries, Marquis’ contribution [46]. Proof of Proposition 4.9. Let X D f.; x/; x 2 g and X 0 D f.; x/ 2 X; is strictly convex with C 1 boundaryg: For ı > 0, let X ı D f.; x/ 2 X; the Hilbert geometry .; d / is ı hyperbolicg: Finally, let Xh D

[

Xı

ı>0

be the set of Gromov-hyperbolic Hilbert geometries (see Section 2.3.4 for the definition of Gromov-hyperbolicity). The space X is equipped with the topology induced by the Gromov-Hausdorff distance on subsets of RP n on the -coordinate, and the topology of RP n for the x-coordinate. The subsets X 0 , X h , X ı inherit the induced topology. We have X ı X h X 0 for any ı > 0. The space X h contains all .; x/ such that .; d / is regular ([18]), and so X h is dense in X 0 and X . J.-P. Benzécri [12] proved that the action of the projective group PGL.n C 1; R/ on X is proper and cocompact. Y. Benoist [10] proved that X ı is closed in X, so the action of PGL.n C 1; R/ on X ı is also proper and cocompact. Consider the map f W .; x/ 2 X 0 ! R defined by

²

³

kZk˙ ; Z 2 Tw H ; .w/ D x : f .; x/ D max kZk This map is continuous, positive and PGL.n C 1; R/-invariant on X 0 . So, by the Benoist–Benzécri result, there exists Cı such that Cı1 6 f 6 Cı on X ı . Hence, for any Gromov-hyperbolic Hilbert geometry .; d /, k k and k k˙ are bi-Lipschitz equivalent and Lemma 4.10 implies that E s D E , E u D E C and k k D k k˙ . In other words, f 1 on the subset X h of X 0 . Since X h is dense in X 0 and f is continuous, we have f 1 on X 0 , which concludes the proof.

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5 Hyperbolicity and Lyapunov exponents In this section, we want to understand the asymptotic behaviour of the geodesic flow locally around a given orbit; in particular, we want to see when it is locally hyperbolic. Since these are local considerations, we work in the universal cover and on H . Throughout this section, the Hilbert geometry .; d / is assumed to be defined by a strictly convex set with C 1 boundary.

5.1 Hyperbolicity of an orbit The orbit ' w of the point w 2 H of the geodesic flow ends at the point w C on the boundary. A fundamental observation is that most of the local properties of the geodesic flow around the orbit ' w are given by the local shape of the boundary @ at the point w C . Recall the following notions. Definition 5.1. Let k 2 N; 0 < " 6 1 and ˇ > 2. Let f W U Rn ! Rk be defined on an open subset U . Denote by T k f its Taylor expansion up to order k, if defined. The function f is said to be • of class C kC" on U if f is C k on U and, for some constant C > 0, jf .x/ f .y/ T k f .x/.x y/j 6 C jx yjkC" ;

x; y 2 U I

• of class D kC" at x 2 U if f is k-times differentiable at x and, for some constant C > 0 and a neighborhood V of x, jf .x/ f .y/ T k f .x/.x y/j 6 C jx yjkC" ;

y 2VI

• ˇ-convex on U if f is C 1 on U and, for some constant C > 0, jf .x/ f .y/ T 1 f .x/.x y/j > C jx yjˇ ;

x; y 2 U:

• ˇ-convex at x 2 U if f is differentiable at x and, for some constant C > 0 and a neighborhood V of x, jf .x/ f .y/ T 1 f .x/.x y/j > C jx yjˇ ;

y 2 V:

To study the behaviour around the orbit ' w of the point w 2 H , we look at the action of the differential d' t : we pick a vector Z 2 Tw H and look at the asymptotic behaviour of the function t 7! kd' t Zk. This function is constant for Z 2 R X. Corollary 4.5 states that, for Z 2 E s .w/ (resp. Z 2 E u .w/), kd' t Zk decreases to 0 (resp. increases to C1). To go further, we want to see when the contraction/expansion on stable and unstable subspaces are exponential.

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Definition 5.2. A point w 2 H or its orbit ' w is said to be hyperbolic if sup

lim sup

1 log kd' t Z s k < 0; t

inf

lim inf

1 log kd' t Z u k > 0: t

Z s 2E s .w/ t!C1

and Z u 2E u .w/

t!C1

From Equalities (4.2) and (4.3), we see that lim sup

1 1 log kd' t Z s k D 1 C lim sup log kT t Z s k; t t!C1 t

lim inf

1 1 log kd' t Z u k D 1 C lim inf log kT t Z u k; t!C1 t t

t !C1

Z s 2 E s .w/;

and t !C1

Z u 2 E u .w/:

Moreover, from the fact that E u D J F .E s / and from the J F -invariance of the norm, we see that w is hyperbolic if and only if 1 < lim inf

t!C1

1 1 log kT t Z s k 6 lim sup log kT t Z s k < 1; t t!C1 t

Z s 2 E s .w/:

In terms of parallel transport on , w is hyperbolic if and only if 1 < lim inf

t !C1

1 1 log F .Twt v/ 6 lim sup log F .Twt v/ < 1; t t!C1 t

v 2 T.w/ Hw C ..w//:

Define .w/ N D

sup

lim sup

1 log kd' t Z u k t

inf

lim inf

1 log kd' t Z u k: t

Z u 2E u .w/ t!C1

and .w/ D

Z u 2E u .w/ t!C1

These two numbers control the exponential rate of expansion on E u along the orbit ' w. Using formula (4.1), we can prove the Proposition 5.3. An orbit ' w is hyperbolic, with exponents 0 < N < 2 if and only if the boundary @ is of class C 1C" and ˇ-convex at the point w C for all 1 < 1 C " < 2N and ˇ > 2 .

5.2 Regular orbits and Lyapunov exponents The exponential rate of contraction/expansion can depend on the vector that we consider.

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Definition 5.4. A point w 2 H or its orbit ' w is said to be forward weakly regular if for any Z 2 Tw H , the quantity 1t log kd' t Zk admits a limit .Z/ when t goes to C1. It is said to be weakly regular if this quantity has the same limit when t goes to 1. From previous considerations, to see if a point w is forward weakly regular, we only have to check that the limit exists for all Z 2 E s .w/ or that .v/ D lim t !C1 1t log F .Twt v/ exists for all v 2 T.w/ Hw C ..w//. Let w 2 H be a forward weakly regular point. We call the number .v/ the parallel Lyapunov exponent of the vector v. These numbers f .v/; v 2 T.w/ Hw C ..w//g can take only a finite number 1 .w/ < < p .w/ of values, which are called the parallel Lyapunov exponents of w (or its orbit). There is then a decomposition T.w/ Hw C ..w// D E1 ˚ ˚ Ep called Lyapunov decomposition, such that, for any vector vi 2 Ei X f0g, 1 log kTwt .vi /k D i .w/: t!C1 t I made a detailed study of (parallel) Lyapunov exponents in [21]. I specify Proposition 5.3 by making a link between the Lyapunov exponents of w and some regularity properties of the boundary @ at w C . Finally, if we want to look at how the flow transforms volumes along a given orbit ' w, we can look at the quantity det Twt . lim

Definition 5.5. A point w 2 H is said to be regular if w is weakly regular, with parallel Lyapunov exponents 1 .w/ < < p .w/ and Lyapunov decomposition T.w/ Hw C ..w// D E1 ˚ ˚ Ep , and if p

X 1 log det Twt D dim Ei i .w/ DW .w/: t!˙1 t lim

iD1

Remark that in this definition, the determinant is not computed with respect to a Riemannian metric but with respect to a Finsler metric. This depends on the volume we associate to the Finsler metric. Since all natural volumes are equivalent, let us say that the determinant is computed with respect to the Busemann–Hausdorff volume Vol . An important fact is that the Lyapunov exponents of a periodic orbit of the geodesic flow of a quotient manifold M D = can be explicitly computed. As we have seen (Proposition 2.7), such an orbit corresponds to the conjugacy class of a hyperbolic element g 2 , and we get the following result, which involves the eigenvalues of g: Proposition 5.6. Let g be a periodic orbit of the geodesic flow of the manifold M D = , corresponding to a hyperbolic element g 2 . Denote by 0 > 1 > > p > pC1 the moduli of the eigenvalues of g. Then

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• g is regular and has no zero Lyapunov exponent; • the Lyapunov exponents . i .g// of the parallel transport along g are given by

i .g/ D 1 C 2

log 0 log i ; log 0 log pC1

i D 1 : : : pI

• the sum of the parallel Lyapunov exponents is given by

.g/ D .n C 1/

log 0 C log pC1 : log 0 log pC1

6 Compact quotients We now want to consider global properties of the geodesic flow of a compact quotient manifold M D =, in the case the convex set is strictly convex (and then with C 1 boundary).

6.1 The Anosov property We begin with the following definition. Definition 6.1. Let ' t W M ! M be a C 1 flow on a compact Finsler manifold .M; kk/. The flow ' t is an Anosov flow it there exists a ' t -invariant decomposition T HM D R X ˚ F s ˚ F u ; called the Anosov decomposition, and constants C; ˛ > 0 such that for any t > 0, kd' t .Z s /k 6 C e ˛t kZ s k; Z s 2 F s ; kd' t .Z u //k 6 C e ˛t kZ u k; Z u 2 F u : This property is named after D. V. Anosov. In [2], he proved that the geodesic flow of a negatively curved compact Riemannian manifold satisfies this property and used it to prove its ergodicity relative to its invariant volume (see Section 7.1). Theorem 6.2 (D. V. Anosov [2], P. Eberlein [23], P. Foulon [26]). Let F be a regular Finsler metric of negative curvature on a compact manifold M . The geodesic flow of F on HM is an Anosov flow. The proof of this result is based on a study of Jacobi fields, whose behaviour can be understood under the curvature assumption. As we have seen, the same can be done in Hilbert geometry, and it is even easier since the curvature is constant: we get actual equalities relating parallel transport and the action of the flow.

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Theorem 6.3 (Y. Benoist, [11]). Let M D = be a compact manifold. The geodesic flow of the Hilbert metric is an Anosov flow on HM , with invariant decomposition THM D R X ˚ E s ˚ E u : Moreover, the geodesic flow is topologically mixing: for any open subsets A and B of HM , there is a time T 2 R such that for all t > T , ' t .A/ \ B 6D ;. The Anosov property is a direct consequence of Corollary 4.5 together with compactness. We give a short proof of this fact below. For the topological mixing property, we refer to Section 5.3 of Benoist’s paper, or to [19] where the proof is extended to some noncompact quotients. The action of the group on the boundary @ is minimal, and this allows us to see that the set f.xgC ; xg /; g 2 g is dense in @@, that is, periodic orbits of the flow are dense in HM . Topological transitivity2 follows from this last result. Topological mixing then comes from the non-arithmeticity of the length spectrum, which is a consequence of Theorem 2.6 through the fact that the length of closed orbits is given by the eigenvalues of the elements of the group (see Section 2.4.2). Proof of the Anosov property. Choose T > 0. The set E1 D fZ 2 E s j kZk D 1g of unit stable vectors is compact, and the function Z 2 E1 7! kd' T .Z/k is continuous, hence attains its maximum for some vector ZM . Lemma 4.5 tells us that a WD kd' T .ZM /k < 1, so that, for all Z 2 E s , kd' T .Z/k 6 akZk: Finally, for t > 0, and setting n D Œt=T , we get kd' t .Z/k 6 an kd' tn Zk 6 DT an kZk 6

DT log at e kZk; at n

˚ t .Z/k ; Z 2 E s; 0 6 t 6 T . with DT D max kd'kZk This gives the upper bound for stable vectors; the same works for unstable ones.

Notice that there is an important difference with the negatively curved Riemannian case. Here, we do not really know what occurs for small t : we know that the norm of a stable vector decreases but we have no control on the rate of decreasing, unlike in the Riemannian case where even infinitesimal exponential rates are controlled by the bounds on the curvature. 2A flow ' t is topologically transitive if for any open subsets A and B of HM , there is a time T 2 R such that ' T .A/ \ B 6D ;

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6.2 Regularity of the boundary As a corollary of Proposition 5.3, we get Corollary 6.4. Let M D = be a compact quotient manifold. Then the boundary @ of is C 1C" and ˇ-convex for all 1 < 1 C " < ˛./ WD

2 2 ; DW ˇ./ ˇ .HM N / .HM /

where N .HM / D .HM N / D sup .w/; w2HM

inf .w/:

w2HM

In particular, the geodesic flow is C 1C" for some " > 0. From symmetry arguments, we can see that 1 1 C D 1: ˛./ ˇ./ Define ˛./ as the biggest 1 < ˛ D 1 C " 6 2 such that @ is C 1C" at all fixed points of the elements of . Obviously, we have ˛./ ˛./ and O. Guichard gave a geometrical proof of the following: Theorem 6.5 (O. Guichard [30]). Let M D = be a compact quotient manifold. We have ˛./ D ˛./. A dynamical proof had been given by U. Hamenstädt in a more general context. She proved the following which implies the previous result. Theorem 6.6 (U. Hamenstädt [31]). Let M D = be a compact quotient manifold. Let .Per/ N D supf .w/; N w 2 HM periodicg: We have .Per/ N D .HM N /.

6.3 Geometrical rigidity The algebraic nature of locally symmetric spaces confer them very specific properties that often turn to be characteristic. Some of these properties directly involve the dynamics of the geodesic flow. For example, let us recall this beautiful result. Theorem 6.7 (Y. Benoist–P. Foulon–F. Labourie [6]). Let .M; g/ be a compact Riemannian (or reversible regular Finsler) manifold of negative curvature. If the Anosov decomposition is C 1 then .M; g/ is locally symmetric.

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Actually, the authors proved a more general result about contact Anosov flows. In particular, they covered the case of a non-reversible Finsler metric: If the Anosov decomposition is C 1 , then there exists a closed 1-form ˛ on M such that F 0 D F C ˛ is a locally symmetric Riemannian metric on M . A similar result, which is easier to prove, allows to characterize the hyperbolic space among all strictly convex divisible Hilbert geometries. Theorem 6.8 (Y. Benoist [11]). Let .; d / be a strictly convex divisible Hilbert geometry. The following are equivalent: (1) the convex set is an ellipsoid; (2) the boundary @ is C 1C" for all 0 6 " < 1; (3) the Anosov decomposition is C " for all 0 6 " < 1. Proof. The implication .1/ ) .2/ is obvious. .2/ , .3/ comes from the description of stable and unstable bundles as E u D fY C J F .Y /; Y 2 VH g;

E s D fY J F .Y /; Y 2 VH g D J F .E u /

and the fact that the subbundle hF H and the map J F are C " if @ is C 1C" . The implication .2/ ) .1/ uses Corollary 6.4. If the boundary @ is C 1C" for all 1 6 ˛ < 2 this means that .HM N / D 1. In particular, the largest Lyapunov exponent of any periodic orbit is 1. By Proposition 5.6 this implies that 2

log 0 .g/ log i .g/ D1 log 0 .g/ log pC1 .g/

for all g 2 . By Theorem 2.6, this implies is not Zariski-dense in SL.n C 1; R/. Theorem 2.5 concludes that is an ellipsoid.

7 Invariant measures There are various ways of looking at a dynamical system. In this section, we want to look at geodesic flows from a measure-theoretic point of view. Let us first recall some basic definitions. We are given a flow ' t on a topological space X. • A Borel measure on X is invariant by the geodesic flow if ' t D , that is, for any Borel subset of X, we have .' t .A// D .A/. • An invariant measure is ergodic if any invariant Borel subset has zero or full measure. • An invariant probability measure is mixing if for any Borel sets A; B, we have lim t !C1 .' t .A/ \ B/ D .A/.B/; mixing implies ergodicity.

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Geodesic flows of compact negatively curved Riemannian or regular Finsler manifolds preserve lots of probability measures. The simplest are probably those defined by periodic orbits: if w is a periodic point of period Tw > 0, we can define an invariant probability measure by pushing forward by the application t 2 Œ0; Tw 7! ' t .w/ the Lebesgue measure of Œ0; Tw , and normalizing it. If we look at the geodesic flow from this measure-theoretic point of view, the system seems trivial: almost every point is periodic with the same period.

7.1 Absolutely continuous measures Let M be a manifold and the Riemannian measure of an arbitrary Riemannian metric on M . We say that a measure on M is smooth or is a volume if d D f d with f W M ! R everywhere positive and continuous; in the case f is only measurable and nonnegative, we say that is an absolutely continuous measure. Since a regular Finsler geodesic flow is in particular a Hamiltonian flow, it preserves the Liouville measure, which is given by A ^ dAn1 , where A denotes the Hilbert form of the metric (see Section 3.2.3). For a compact negatively curved manifold M , this measure is ergodic and so is the only invariant measure in its Lebesgue class.

7.2 Entropies 7.2.1 Topological entropy. A geodesic flow is a continuous action of the group R by diffeomorphisms. Its topological entropy measures the topological complexity of this action. Let us recall the definition for a flow ' t W X ! X on a compact metric space .X; d /. For t > 0, we define the distance d t on X by: d t .x; y/ D max d.' s .x/; ' s .y//; 06s6t

x; y 2 X:

For any " > 0 and t > 0, we consider coverings of X by open sets of diameter less than " for the metric d t . Let N.'; t; "/ be the minimal cardinality of such a covering. The topological entropy of the flow is then the (well-defined) quantity htop .' t / D lim lim sup 1t log N.'; t; "/ : "!0

t!1

It is important to remark that since the space is compact, the topological entropy does not depend on the metric d , but only on the topology defined by d on the space X : if we replace the metric by a topologically equivalent one, then we get the same number. 7.2.2 Measure-theoretical entropy. To every invariant probability measure m of the flow ' t is attached its measure-theoretical entropy hm .' t /, which measures the complexity of the flow from this measure point of view. We refer to Walters’ monograph for a complete description [54]. As could be expected, measures defined by periodic

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orbits have zero entropy whereas the Liouville volume has always positive entropy (in the negatively curved case). We will give an expression of the entropy of the Liouville measure in a forthcoming section. 7.2.3 Variational principle and measure of maximal entropy. A variational principle makes a link between measure-theoretical and topological entropies. This principle asserts that htop .' t / D sup h .' t /;

where the supremum is taken over all invariant probability measures. A measure which achieves the supremum is called a measure of maximal entropy. In some sense, such a measure is well adapted to describe the topological action. The following theorem gives existence and uniqueness of a measure of maximal entropy for hyperbolic dynamics. Theorem 7.1 (see for instance [35]). Let W be a compact manifold. A topologically transitive Anosov flow ' t W W ! W admits a unique measure of maximal entropy. As a consequence, the geodesic flow of a compact negatively curved manifold admits a unique measure of maximal entropy. It is usual to call it the Bowen–Margulis measure because R. Bowen and G. Margulis gave two independent constructions of it; we will denote it by BM . Both constructions are of particular interest. In his PhD thesis, Margulis [44], [45] constructed this measure as a product measure. Stable, unstable and orbit foliations provide local coordinate systems W s W u ."; "/: each point w has a neighborhood U in which any point is at the intersection of exactly one local stable leaf W s \U , one local unstable leaf W u \U and one local orbit ' w \ U . The measure is then described locally as a product BM D s u dt , where the measures s and u are measures on stable and unstable leaves, uniquely defined by the following transition property, involving topological entropy: ' t sw D e htop t s' t .w/ ;

' t uw D e htop t u' t .w/ ;

w 2 W:

In [16], [17], Bowen proved that for the geodesic flow of a compact hyperbolic manifold, closed orbits are uniformly distributed with respect to the Liouville measure, which in this case is also the measure of maximal entropy. Bowen’s construction extends to the case of a topologically transitive Anosov flow, and finally, we find that closed orbits are uniformly distributed with respect to a specific measure, which actually coincides with the one Margulis introduced.

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7.3 Harmonic measure and measure of maximal entropy We saw (Theorem 6.7) that locally symmetric spaces can be characterized among other Riemannian negatively curved compact manifolds by the regularity of their Anosov decomposition. We expect them to be also characterized by some property of their invariant measures. Of particular interest are the Liouville measure, the Bowen–Margulis measure and the harmonic measure. These three measures are naturally related to various aspects of the manifold: • the Liouville measure is directly defined through the local Euclidean structure defined by the Riemannian metric; • the Bowen–Margulis measure describes the distribution of closed orbits of the geodesic flow, hence it is related to the distribution of closed geodesics on the manifold; • the harmonic measure is related to the Brownian motion on the universal covering of the manifold, hence to the Laplace–Beltrami operator. For locally symmetric manifolds, such as hyperbolic manifolds, these three measures coincide and there are deep relations between the Riemannian structure, the distribution of closed geodesics and the Laplace–Beltrami operator. For non-locally symmetric manifolds, we expect the three measures to be distinct, and the previous links to fail. This has been confirmed for the harmonic measure: Theorem 7.2. Let M be a compact negatively curved Riemannian manifold. • The Bowen–Margulis measure coincides with the Liouville measure if and only if z is asymptotically harmonic, that is, horospheres have the universal covering M constant mean curvature. • The harmonic and Liouville measures coincide if and only if M is locally symmetric. The first point is due to Ledrappier [39]; he also proved in the same work that z is Liouville and harmonic measures coincide if and only if the universal covering M asymptotically harmonic. This last result is used by Besson, Courtois and Gallot [13] together with their minimal entropy rigidity theorem to deduce the second point. Concerning the Bowen–Margulis measure, the situation was clarified by A. Katok [36] for surfaces; Foulon [28] extended it to the more general case of contact Anosov flows. We give the following version: Theorem 7.3 (A. Katok [36], P. Foulon [28]). Let .M; F / be a compact regular Finsler surface of negative curvature. The Liouville measure coincide with the Bowen– Margulis measure if and only if M is a Riemannian manifold of constant curvature.

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7.4 Regular Finsler metrics For negatively curved regular Finsler metrics, comparing the Liouville measure A ^ dAn1 with the Bowen–Margulis one is still a relevant question that has not been considered except for Foulon’s contribution [28]. The extension to negatively curved regular Finsler metrics should not change the conjecture: if both measures coincide, the manifold should be Riemannian and locally symmetric. The case of the harmonic measure is more delicate, because there are various ways to extend the Laplace–Beltrami operator from the Riemannian to the Finsler world, namely because it lives on the manifold and not on its tangent bundle. There have been various attempts to define such an extension, but they have not been studied further. A more recent notion is T. Barthelmé’s definition [4] which is directly linked to the approach to Finsler geometry we have here. Barthelmé proved in his Ph.D. thesis [5] that the harmonic measure is well defined in the negatively curved case, but its properties have not been studied yet.

7.5 Hilbert geometries To my knowledge, there is no natural Brownian motion defined on a general Hilbert geometry. All proposed extensions of the Riemannian definition require the regularity of the metric; in particular, there is no natural way of defining a harmonic measure associated to the Hilbert metric of a convex projective compact manifold M D = (apart from the case of hyperbolic manifolds). Let M D = be a compact manifold, with .; d / a strictly convex Hilbert geometry, and a discrete subgroup of Aut./. We have seen that the geodesic flow on HM is a topologically mixing Anosov flow. In particular, according to Theorem 7.1, it admits a unique measure of maximal entropy BM , which is ergodic and mixing. The following general rigidity result implies in particular that this measure is absolutely continuous if and only if M is Riemannian hyperbolic, that is, is an ellipsoid. Theorem 7.4 (Y. Benoist [11]). Let M D = be a compact quotient manifold, with .; d / a strictly convex Hilbert geometry, and a discrete subgroup of Aut./. If is not an ellipsoid, the geodesic flow on HM admits no absolutely continuous invariant measure. Proof. A. Livšic [41], [42] proved that any absolutely continuous measure had to be smooth. In particular, for any periodic point w 2 HM of period T , the change of variable formulas implies that det dw ' T D 1: This implies in our context (see Section 5) that, for any periodic point w 2 HM of period T , det TwT D 1:

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In the notation of Section 5.2, this implies that, for any g 2 , we have .g/ D 0. According to Proposition 5.6, this is equivalent to a condition on the eigenvalues of the element of , which cannot be satisfied if is Zariski-dense in SL.n C 1; R/ via Theorem 2.6. Theorem 2.5 tells us that is an ellipsoid. As we already noticed in Section 3.2.4, the last result can be seen as a consequence of the lack of regularity of the Legendre transform. In particular, this raises the question of characterizing the pullback of the Liouville volume on H M by the Legendre transform. Finally, it seems that there is no counterpart neither for the harmonic measure nor for the Liouville measure in Hilbert geometry. Nevertheless, we will see in the next section that the Liouville measure is also characterized by another property than its absolute continuity, and this allows us to define an extension (in reality two) of the Liouville measure.

8 Entropies 8.1 Volume entropy The volume entropy of a Hilbert geometry measures the exponential growth of volume of balls. It is defined as 1 hvol D lim sup Vol .B.o; R// R R!C1 where o is an arbitrary point in . The Hilbert geometry defined by the simplex has zero volume entropy while the volume entropy of the .n 1/-dimensional hyperbolic space is n 1. We conjecture that those are the extremal cases: Conjecture. Let .; d / be a Hilbert geometry. Then its volume entropy is less than n 1. The result has been proved in dimension 2 by G. Berck, A. Bernig and C. Vernicos. Recently, Vernicos gave a proof in dimension 3. In higher dimensions, this is known to be true for polytopes, which have zero entropy, and for convex sets whose boundary is C 1C1 , for which hvol D n 1. For divisible convex sets, we can use a dynamical approach using the following Theorem 8.1 (A. Manning [43], see also [20]). Let .; d / be a Hilbert geometry and M D = a compact quotient manifold with G < Isom.; d /. The volume entropy of .; d / equals the topological entropy of the geodesic flow of M .

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8.2 Topological entropy Theorem 8.2. Assume M D = is compact, with strictly convex. The topological entropy of the geodesic flow satisfies htop 6 n 1, with equality if and only if is an ellipsoid. Sketch of the proof. Ruelle’s inequality [49] implies that Z

dBM : htop D hBM 6 n 1 C HM

Oseledec’s Theorem [48] says that the set ƒ HM of regular points has full BM measure. Furthermore, this set is invariant under the flip map W HM ! HM defined by .x; Œ/ D .x; Œ/; and we have B D on ƒ. To get the inequality, we just have to remark that, since F is reversible, conjugates t ' and ' t , hence BM is the measure of maximal entropy of both flows and R BM D BM . We conclude that HM dBM D 0. For the equality case, recall that F. Ledrappier and L.-S.Young proved that equality in Ruelle’s inequality Z h 6

C d

occurs if and only if the measure has absolutely continuous conditional measures on unstable manifolds. But the diffeomorphism sends stable manifolds to unstable ones. Since preserves BM , the Lebesgue class of its conditional measures on stable and unstable manifolds coincide. In particular, there is equality in Ruelle’s inequality for BM if and only if BM is itself absolutely continuous. By Theorem 7.4, this implies is an ellipsoid. The last two theorems imply that Conjecture 8.1 is true for divisible strictly convex Hilbert geometries. The numerous examples of such geometries also provide numerous examples whose volume entropy is positive but strictly smaller than n 1.

8.3 Variations of entropy Given a compact manifold M of dimension d , consider the moduli space ˇ.M / of marked convex projective structures on M . Such a structure can be described by a pair .dev; / consisting of z ! RP n which is a diffeomorphism onto a convex set • a developing map dev W M ; • a faithful morphism W 1 .M / ! PSL.n C 1; R/ called the holonomy map with respect to which dev is 1 .M /-equivariant; its image D .1 .M // divides with quotient = diffeomorphic to M .

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The space ˇ.M / is endowed with the topology given by uniform convergence on compact subsets on the first coordinate and the compact-open topology on the second. Proposition 8.3. Let M be a compact manifold of dimension n. The entropy function hvol W ˇ.M / ! .0; n 1 is continuous. Proof. Consider a deformation . ; dev /; 2 Œ1; 1 of a given structure . 0 ; dev0 /. These structures provide Finsler metrics F on the abstract manifold M . These metrics vary continuously with in the following sense: lim

sup

!0 TM Xf0g

F D 1: F0

For let T 1 M be the unit tangent bundle for F0 . Since T 1 M is compact and 7! dev is continuous, lim sup jF F0 j D 0: !0 T 1 M

Moreover minT 1 M F0 > 0, hence lim sup j

!0 T 1 M

F 1j D 0: F0

Homogeneity gives the result, that is, there exist reals C > 1 such that lim!0 C D 1 and F 6 C : C1 6 sup TM Xf0g F0 z z Denote by dQ the associated distances on R M . Let x; y 2 M , and c be the geodesic from x to y for the metric dQ , such that Fz .c0 .t // dt D dQ .x; y/. Then R R Fz .c0 .t// dt Fz .c00 .t // dt dQ .x; y/ 1 6 C : 6 6 C 6 R R Fz0 .c 0 .t// dt Fz0 .c 0 .t // dt dQ0 .x; y/ 0

z, Thus for any x; y 2 M C1 6

dQ .x; y/ 6 C : dQ0 .x; y/

From that we clearly get Bz .x; R/ Bz0 .x; C R/. Hence 1 cardfg 2 1 .M /; gx 2 Bz .x; R/g R R!1 1 6 lim sup cardfg 2 1 .M /; gx 2 Bz0 .x; C R/g R!1 R

hvol . ; dev / D lim sup

D C hvol . 0 ; dev0 /: Similarly,

C1 hvol . 0 ; dev0 /

6 hvol . ; dev /. This gives the continuity.

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For a 2-dimensional manifold M of genus g D 1, the convex structure is necessarily given by a triangle, whose entropy is zero. If g > 2, the space ˇ.M / has been well described by W. Goldman [29] as a manifold diffeomorphic to R16g16 . X. Nie studied in [47] a special kind of projective structures given by simplicial Coxeter groups, and among other things he showed the following Theorem 8.4 (X. Nie [47]). Let M be a compact 2-dimensional manifold. The entropy function hvol W ˇ.M / ! .0; 1 is a surjective continuous map.

8.4 Sinai–Ruelle–Bowen measures Consider a flow ' t on a compact manifold W preserving a smooth probability measure . Birkhoff’s ergodic theorem asserts that there is set A M of full -measure on which the quantity Z 1 t g.' t .x// dt t 0 converges to some g.x/ N as t goes to C1 for any integrable Borel function g. This limit function x ! g.x/ N is integrable and invariant under the flow.R In particular, in the case is ergodic, gN is (-almost everywhere) constant equal to g d . In general, a flow ' t on a compact manifold W does not preserve a smooth measure but it seems legitimate to ask about the behaviour of a random orbit, chosen randomly with respect to some Lebesgue probability measure . In particular, does there exist a subset of full -measure on which the Birkhoff averages Z 1 t g.' t .x// dt t 0 converge for all continuous functions? If this is the case and the limit is constant (-almost everywhere), we can then define a measure C by Z Z 1 t C g d D lim g.' t .x// dt: t!C1 t 0 Such a measure is called a physical measure for the flow ' t since it describes the asymptotic distribution of Lebesgue-almost every orbit. Theorem 8.5 (R. Bowen–D. Ruelle [15]). Any C 1C" Anosov flow on a compact manifold M admits a unique physical measure. Bowen and Ruelle constructed this measure as the equilibrium measure of the potential d ˇ t f C D ˇ tD0 log det d'E u: dt

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That is, C is the unique measure which maximizes the quantity Z P .f C ; / D h C f C d among all invariant probability measures. In particular, from general results, C is ergodic. Furthermore, the pressure P .f C / D sup P .f C ; / is zero and we have Z h D f C d: This implies that C achieves the equality in Ruelle’s inequality and is the unique measure to do so. In particular, from the Ledrappier–Young theorem [38], C is the only invariant probably measure to have absolutely continuous conditional measures on unstable manifolds. Recall that the flip map W HM ! HM is defined by .x; Œ/ D .x; Œ/. The measure WD C is also invariant by ' t and is in fact the physical measure of ' t . In other words, the Birkhoff averages Z 1 0 g.' t .x// dt jt j t R converge to g d when t goes to 1. Similarly, we find that is the equilibrium measure of the potential d ˇ t f D ˇ tD0 log det d'E s; dt and that it has absolutely continuous conditional measures on stable manifolds. Finally, both measures C and coincide if and only if one of them is absolutely continuous. All of this applies to the geodesic flow of a compact quotient of a strictly convex Hilbert geometry, which is C 1C" from Corollary 6.4. Corollary 8.6. Let M D = be a compact quotient manifold of a strictly convex Hilbert geometry. Unless is an ellipsoid, the three measures BM , C and are mutually singular. As a direct application of the fact that the measure C achieves the equality in Ruelle’s inequality, we can give a lower bound on its entropy, hence on topological entropy: Proposition 8.7 ([20]). Let .; d / be a strictly convex divisible Hilbert geometry where is not an ellipsoid. Assume @ is ˇ-convex for some ˇ 2 .2; C1/. Then htop >

2 .n 1/: ˇ

Another application is given in the next section.

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8.5 Curvature of the boundary of a divisible convex set Let us begin with an old theorem of A. D. Alexandrov [1] about convex functions: Theorem 8.8. Let f W U Rn 7! R be a convex function defined on a convex 2 open set U of Rn . The Hessian matrix Hess.f / D @@i @fj ij exists Lebesgue almost everywhere in U . Let be a bounded convex set of the Euclidean space Rn . It is then possible to compute the Hessian of its boundary at Lebesgue almost every point x 2 @. We will call a D 2 point a point x where this is possible. The Hessian is a positive symmetric bilinear form on the tangent space Tx @. It represents the curvature of the boundary at x. To be degenerate means that the curvature of the boundary is zero in some tangent direction. The Hessian is a Euclidean notion, but its degeneracy is not. Namely, if is a properly convex open set of RP n and x a point of @, we can choose an affine chart centered at x and a metric on it and compute the Hessian of @ at x; its degeneracy does not depend on the choice of the affine chart and the metric. We can measure the vanishing of the curvature of @ in the following way. Fix a smooth measure on the boundary of the dual convex set , and call its pull-back to @. Then can be seen as a measure of the curvature of @. It can be decomposed as D ac C sing ; where ac is an absolutely continuous measure and sing is singular with respect to any Lebesgue measure on @. For example, if @ is not differentiable at some point x then will have an atom at x. The support of ac is the closure of the set of D 2 points with nondegenerate Hessian. Though is convex, it may happen that ac D 0, that is, is singular with respect to some (hence any) smooth measure on @. This is equivalent to the fact that the Hessian is degenerate at almost all D 2 point of @. We then say that the curvature of the boundary is supported on a set of zero Lebesgue measure. The following lemma gives a criterion for this to happen, due to Benzécri [12]: Lemma 8.9. Let Xn denote the set of properly convex open sets of RP n , and pick 2 Xn . If there exists a D 2 point x 2 @ with nondegenerate Hessian, then the closure of the orbit PGL.n C 1; R/ in Xn contains an ellipsoid. Proof. Choose an affine chart and a Euclidean metric on it such that appears as a bounded convex open set of Rn . Let x be a point of @ with nondegenerate Hessian. Let E be the osculating ball of @ at x. It defines a hyperbolic geometry .E; dE /. Pick a point y 2 @E distinct from x, and choose a hyperbolic isometry g of E whose attracting fixed point is y and the repulsive one is x. Now, since @E and @ are tangent up to order 2, it is not difficult to see that g n converges to E when n goes to C1. This proves the statement.

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As a consequence, we get the following Theorem 8.10. Let .; d / be a divisible Hilbert geometry, and assume is not an ellipsoid. Then any D 2 point has degenerate Hessian. In particular, the curvature of @ is supported on a subset of zero Lebesgue measure. Proof. This is a consequence of the last proposition and of the following fact: if is divisible then the orbit of under PGL.n C 1; R/ is closed in Xn (for a proof of this fact, see Proposition 9.18 in Marquis’ contribution [46]). The boundary of a divisible convex set has thus a quite mysterious geometry. In the case of a strictly convex divisible set, we can be more precise using the fact that the unstable conditional measures of C are absolutely continuous: Proposition 8.11 ([21]). Let .; d / be a strictly convex divisible Hilbert geometry. There exists " > 0 such that, for Lebesgue almost every point x 2 @, there exists a 2-dimensional subspace Px intersecting and containing x such that the boundary of the 2-dimensional convex set \ Px is D 2C" at x. In particular, if RP 2 , there exists " > 0 for which the boundary @ is D 2C" at Lebesgue almost every point. Acknowledgement. This work is partially supported by CONICYT (Chile), FONDECYT project NB 3120071.

References [1]

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[10] Y. Benoist, Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. Inst. Hautes Études Sci. 97 (2003), 181–237. [11] Y. Benoist, Convexes divisibles 1. In Algebraic groups and arithmetic, Tata Inst. Fund. Res. Stud. 17, Narosa Publishing House, New Delhi 2004, 339–374. [12] J.-P. Benzécri, Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88 (1960), 229–332. [13] G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5 (1995), no. 5, 731–799. [14] A. Bosché, Symmetric cones, the Hilbert and Thompson metrics. Preprint, arXiv:1207.3214. [15] R. Bowen and D. Ruelle, The ergodic theory of axiom A flows. Invent. Math. 29 (1975), 181–202. [16] R. Bowen, The equidistribution of closed geodesics. Amer. J. Math 94 (1972), 413–423. [17] R. Bowen, Periodic orbits for hyperbolic flows. Amer. J. Math 94 (1972), 1–30. [18] B. Colbois and P. Verovic, Hilbert geometry for strictly convex domains. Geom. Dedicata 105 (2004), 29–42. [19] M. Crampon and L. Marquis, Le flot géodésique des quotients geometriquement finis des géométries de Hilbert. Preprint, arXiv:1105.6275v1. [20] M. Crampon, Entropies of strictly convex projective manifolds. J. Mod. Dyn. 3 (2009), no. 4, 511–547. [21] M. Crampon, Lyapunov exponents in Hilbert geometry. Preprint, arXiv:1204.5005v1. [22] P. de la Harpe, On Hilbert’s metric for simplices. In Geometric group theory (Graham A. Niblo et al., eds.,) Vol. 1, London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge 1993, 97–119. [23] P. Eberlein, When is a geodesic flow of Anosov type? I, II. J. Differential Geom. 8 (1973), 437–463; ibid. 565–577, 1973. [24] J. Faraut and A. Korányi, Analysis on symmetric cones. Oxford Math. Monogr., Oxford Science Publications, New York 1994. [25] P. Foulon, Géométrie des équations différentielles du second ordre. Ann. Inst. Henri Poincaré 45 (1986), 1–28. [26] P. Foulon, Estimation de l’entropie des systèmes lagrangiens sans points conjugués. With an appendix “About Finsler geometry”. Ann. Inst. H. Poincaré Phys. Théor. 57 (1992), no. 2, 117–146. [27] P. Foulon, Locally symmetric Finsler spaces in negative curvature. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 10, 1127–1132. [28] P. Foulon, Entropy rigidity of Anosov flows in dimension three. Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1101–1112. [29] W. M. Goldman, Convex real projective structures on compact surfaces. J. Differential Geom. 31 (1990), 791–845.

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[30] O. Guichard, Régularité des convexes divisibles. Ergodic Theory Dynam. Systems 25 (2005), no. 6, 1857–1880. [31] U. Hamenstädt, Anosov flows which are uniformly expanding at periodic points. Ergodic Theory Dynam. Systems 14 (1994), 299–304. [32] D. Johnson and J. J. Millson, Deformation spaces associated to compact hyperbolic manifolds. In Discrete groups in geometry and analysis, 1984), Progr. Math. 67, Birkhäuser Boston 1987, 48–106. [33] V. G. Kac and E. B. Vinberg, Quasi-homogeneous cones. Mat. Zametki 1 (1967), 347–354. [34] M. Kapovich, Convex projective structures on Gromov-Thurston manifolds. Geom. Topol. 11 (2007), 1777–1830. [35] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Cambridge University Press, Cambridge 1995. [36] A. Katok, Four applications of conformal equivalence to geometry and dynamics. Ergodic Theory Dynam. Systems 8 (Charles Conley Memorial Issue) (1988), 139–152. [37] M. Koecher, The Minnesota notes on Jordan algebras and their applications. Lecture Notes in Math. 1710, Springer-Verlag, Berlin 1999. [38] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ann. of Math. 122 (1985), 509–574. [39] F. Ledrappier, Harmonic measures and Bowen-Margulis measures. Israel J. Math. 71 (1990), no. 3, 275–287. [40] B. Lemmens and C. Walsh, Isometries of polyhedral Hilbert geometries. J. Topol. Anal. 3 (2011), , no. 2, 213–241. [41] A. Livšic, Certain properties of the homology of Y-systems. Mat. Zametki 10 (1971), 555–564. [42] A. Livšic, Cohomology of dynamical systems. Math. USSR Izvestija 6 (1972), 1278–1301. [43] A. Manning, Topological entropy for geodesic flows. Ann. of Math. 110 (1979), 567–573. [44] G. Margulis, Certain measures associated with Y-flows on compact manifolds. Functional Anal. Appl. 4 (1969), 55–67. [45] G. Margulis, On some aspects of the theory of Anosov systems. Springer Monogr. Math., Springer-Verlag, Berlin, 2004. [46] L. Marquis, Around groups in Hilbert geometry. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 207–261. [47] X. Nie, On the Hilbert geometry of simplicial Tits sets. Preprint, arXiv:1111.1288v1. [48] V. I. Osedelec, A multiplicative ergodic theorem. Trans. Moscow Math. Soc. 19 (1968), 197–231. [49] D. Ruelle, An inequality for the entropy of differentiable maps. Bol. Soc. Bras. de Mat. 9 (1978), 83–87. [50] É. Socié-Méthou, Comportements asymptotiques et rigidité en géométrie de Hilbert. PhD thesis, Université de Strasbourg, 2000; available at http://www-irma.u-strasbg.fr/annexes/publications/pdf/00044.pdf.

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[51] É. Socié-Méthou, Caractérisation des ellipsoïdes par leurs groupes d’automorphismes. Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 4, 537–548. [52] E. B. Vinberg, The theory of homogeneous convex cones. Trudy Moskov. Mat. Ob˘sc˘ . (12) (1963), 303–358. [53] E. B. Vinberg, The structure of the group of automorphisms of a homogeneous convex cone. Trans. Math. Mosc. Soc. (13) (1965), 63–93. [54] P. Walters, An introduction to ergodic theory. Grad. Texts in Math. 79, Springer-Verlag, New York 1982.

Chapter 8

Around groups in Hilbert geometry Ludovic Marquis

Contents 1 2

3

4

5

6

7

Generalities on the group Aut./ . . . . . . . . . . . . . . . . . . The automorphisms of a properly convex open set . . . . . . . . . 2.1 The first lemma . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The matrix point of view . . . . . . . . . . . . . . . . . . . . 2.3 The dynamical point of view . . . . . . . . . . . . . . . . . . 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of “large” groups acting on properly convex open subsets 3.1 Homogeneous properly convex open set . . . . . . . . . . . . 3.2 Symmetric properly convex open set . . . . . . . . . . . . . 3.3 Spherical representations of semi-simple Lie groups . . . . . 3.4 Schottky groups . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Strictly convex divisible and quasi-divisible convex sets . . . 3.6 Non strictly convex divisible and quasi-divisible convex sets . Convex hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 An alternative construction . . . . . . . . . . . . . . . . . . . Zariski closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Definitions of proximality . . . . . . . . . . . . . . . . . . . 5.2 Positive proximality . . . . . . . . . . . . . . . . . . . . . . 5.3 Cocompact and finite-covolume case . . . . . . . . . . . . . 5.4 A sketch of proof when is strictly convex . . . . . . . . . . Gromov-hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The first step . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The duality step . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The key lemma . . . . . . . . . . . . . . . . . . . . . . . . . Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 A naturality statement . . . . . . . . . . . . . . . . . . . . . 7.2 A sketch of the proof of closedness . . . . . . . . . . . . . . 7.3 The openness . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 A description of the topology for the surface . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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211 212 212 213 214 220 222 223 224 224 225 226 229 231 231 231 233 233 233 234 235 235 237 237 239 239 240 240 241 242 243

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7.5 Description of the topology for 3-orbifolds . . . . . . 8 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 For strongly convex bodies . . . . . . . . . . . . . . . 8.2 For round convex bodies . . . . . . . . . . . . . . . . 8.3 For any convex bodies . . . . . . . . . . . . . . . . . 9 Benzécri’s theorem . . . . . . . . . . . . . . . . . . . . . . 9.1 The proof . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Natural things are equivalent . . . . . . . . . . . . . . 9.3 Two not-two-lines applications of Benzécri’s theorem 10 The isometry group of a properly convex open set . . . . . . 10.1 The questions . . . . . . . . . . . . . . . . . . . . . . 10.2 The knowledge . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hilbert geometries were introduced by Hilbert as examples of geodesic metric spaces where straight lines are geodesics. We will completely forget this story. We will take Hilbert geometry as a very simple recipe for metric spaces with several different flavours. Hilbert geometries are Finsler manifold, so it is not easy to say that they are non-positively curved, but we hope that at the end of this text the reader will have noticed some flavours of non-positively curved manifolds and will start to see them as “damaged non-positively curved manifold”. The most interesting examples of Hilbert geometries in the context of geometric group theory are called, following Vey in [88], divisible convex sets. These are those properly convex open subsets of the real projective space P d D P d .R/ such that there exists a discrete subgroup of the group PGLd C1 .R/ of projective transformations which preserves and such that the quotient = is compact. In 2006, Benoist wrote a survey [12] of divisible convex sets and in 2010, Quint wrote a survey [75] of the work of Benoist on divisible convex sets. Thus, we will not concentrate on divisible convex sets since the survey of Benoist does this job very well, even if we consider divisible and quasi-divisible convex sets1 as the most important class of convex sets. We want to describe the groups that appear in Hilbert geometry without restriction and also how groups can be used for the purpose of Hilbert geometry. The first two sections of this survey are very elementary. Their goal is to make the reader familiar with the possible automorphisms of a convex set from a matrix point of view and a dynamical point of view. The third section presents existence results on convex sets with a “large” group of symmetries. This section presents the examples that motivated this study. The reader in quest of motivation should skip Section 2 to go straight to the third section. 1 Those

for which the quotient = is of finite volume rather than compact.

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The remaining seven sections are roughly independent. They support the claim that geometric group theory mixed with Hilbert geometry meet (and need) at least: differential geometry, convex affine geometry, real algebraic group theory, metric geometry, moduli spaces, hyperbolic geometry, symmetric spaces, Hadamard manifolds and geometry on manifolds. We have tried to give sketchy proofs in these sections. We do not report on the links with Coxeter group theory and partial differential equations.

Context We will work with the real projective space P d .R/ D P d of dimension d (i.e. the space of lines of the real vector space Rd C1 ). An affine chart of P d is the complement of a projective hyperplane. Every affine chart carries a natural structure of an affine space. A convex subset of the projective space P d is a subset of P d which is either P d or is included in an affine chart A and is convex in this affine chart in the usual sense. A convex subset C of P d is properly convex when there exists an affine chart such that C is bounded in it or, equivalently, when C does not contain any affine line. Our playground will be a properly convex open set, also called a convex body. We will always denote a properly convex open subset of P d by the letter with a subscript if necessary. On a properly convex open set, one can define the Hilbert distance. Given x ¤ y 2 , let p, q be the intersection points of the line .xy/ with the boundary @ of in such a way that x is between p and y and y between x and q (see Figure 1). pC

q

v x

p

y

p

Figure 1. Hilbert distance.

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We set d .x; y/ D

1 jpyj jqxj 1 log Œp W x W y W q D log and d .x; x/ D 0; 2 2 jpxj jqyj

where the quantity Œp W x W y W q is the cross-ratio of the four points p, x, y, q, and x of . j j is any Euclidean norm on any affine chart A containing the closure 2 The cross-ratio is a projective notion. Thus it is clear that d does not depend on A or on the choice of the Euclidean norm on A. We also note that the distance d is invariant by the group of projective automorphisms preserving . The metric space .; d / is geodesic 3 , proper 4 and the topology induced by the Hilbert distance and the projective space coincide. But this space is not uniquely geodesic 5 in general (see Proposition 10.3). This distance d is called the Hilbert distance and has the good taste of coming from a Finsler metric on defined by a very simple formula. Letˇ x be a point in d ˇ d .x; x C t v/ and v a vector in the tangent space Tx of at x; the quantity dt t D0 is homogeneous of degree one in v, therefore it defines a Finsler metric F .x; v/ on . Moreover, if we choose an affine chart A containing and any Euclidean norm j j on A, we get (see Figure 1): 1 d ˇˇ jvj 1 C F .x; v/ D ˇ d .x; x C t v/ D dt tD0 2 jxp j jxp C j The Finsler structure gives rise to a measure on which is absolutely continuous with respect to the Lebesgue measure, called the Busemann volume. To define it, choose an affine chart A containing , a Euclidean norm j j on A and let Leb be the Lebesgue measure on A normalised by the fact that the volume of the unit cube is 1. The Busemann volume 6 of a Borel set A is then defined by the following formula: Z !d .A/ D d Leb.x/ Tang .1// A Leb.Bx Tang

where !d is the Lebesgue volume of the unit ball of A for the metric j j and Bx is the unit ball of the tangent space Tx of at x for the metric F .x; /.

.1/

2 In fact, one should notice that if we allow the symbol 1 in our computation, then we can define d on any affine chart containing . 3A metric space is geodesic when for any two points, there exists a geodesic joining them. 4A metric space is proper when the closed balls are compact, and so a proper space is complete. 5A geodesic metric space is uniquely geodesic when the geodesic between any two points is unique. 6 There is another notion of volume which is often used in Finsler geometry, the so-called Holmes–Thompson volume HT , defined by the following formula:

Z HT .A/ D CoTang

A

CoTang

Leb.Bx !d

.1//

d Leb.x/

where Bx .1// is the unit ball of the cotangent space of at x for the dual norm induced by .Tx ; F .x; //.

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A charming fact about Hilbert geometries is that they are comparable between themselves. Indeed, if 1 2 , then we can compare the distance, the balls, etc. of those two properly convex open sets. The moral statement is the following: The Hilbert distance is decreasing with . The balls (in and in the tangent space) are increasing with . The Busemann volume is decreasing with . The precise statement is the following: Comparision Theorem. Let 1 , 2 be two properly open convex subset such that 1 2 . Then the following holds. • For every x; y 2 1 , d1 .x; y/ 6 d2 .x; y/. • For every x 2 1 , for every v 2 Tx 1 D Tx 2 , F1 .x; v/ 6 F2 .x; v/. • For any Borel set A of 1 , 2 .A/ 6 1 .A/. We will see that regularity conditions of @ have a crucial impact on the geometry of .; d /. A properly convex open set is strictly convex if there does not exist any nontrivial segment in its boundary @. A point p 2 @ is of class C 1 if the hypersurface @ is differentiable at p; since is convex this is equivalent to the uniqueness of a supporting hyperplane7 at p of . A properly convex open set has C 1 -boundary if every point of @ is of class C 1 . We will see that these two properties are dual to each other (see Subsection 2.3). A properly convex set which verifies both properties is called round. Round convex sets have a hyperbolic behaviour. Non-round convex sets have common properties with “the geometry of a normed plane”. An analogy should be made with symmetric spaces. We recall that rank-one symmetric spaces are Gromov-hyperbolic, and that higher rank symmetric spaces contain Euclidean planes. Acknowledgements. The author thanks the anonymous referee for his useful comments and Athanase Papadopoulos for his careful and patient review. The author also wants to thanks Constantin Vernicos and Mickaël Crampon for all the discussions they had around Hilbert geometry. The author is supported by the French program ANR Finsler.

1 Generalities on the group Aut./ The main goal of this chapter is to study the group ˚ Coll˙ ./ D 2 PGLd C1 .R/ j ./ D where is a properly convex open subset of P d . The following fact is very basic and also very useful. 7A

hyperplane H is a supporting hyperplane at p 2 @ when H \ D ¿ and p 2 H .

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Proposition 1.1. The action of the group Coll˙ ./ on is by isometries with respect to the Hilbert distance. Consequently, the action of Coll˙ ./ on is proper, so Coll˙ ./ is a closed subgroup of PGLd C1 .R/ and therefore is a Lie group. One may want to introduce the group Isom./ of isometries of .; d / for the Hilbert metric. This group is rather mysterious. We postpone some remarks on the general knowledge on this group to the end of this chapter. Before going to a “matrix study” of the elements of Coll˙ ./, let us make a remark which allows us to see Coll˙ ./ as a subgroup of the group SL˙ d C1 .R/ of d C1 linear transformations of R with determinant 1 or 1. We denote by P W Rd C1 X f0g ! P d the canonical projection. Let be a properly convex open subset of P d . The cone above is one of the two connected components C of P 1 ./.8 We introduce the group ˚ Aut ˙ ./ D 2 SL˙ d C1 .R/ j .C / D C rather than Coll˙ ./. The canonical morphism W SL˙ d C1 .R/ ! PGLd C1 .R/ is onto with kernel equal to f˙1g. But, the restriction of W Aut˙ ./ ! Coll˙ ./ is an isomorphism since elements of Aut ˙ ./ preserve C . With this in mind, we will now concentrate on Aut./ which is the group of linear automorphisms of Rd C1 preserving C and having determinant one: ˚ Aut./ D 2 SLd C1 .R/ j .C / D C : Sometimes it is very useful to look at the two-fold cover Sd of P d . A projective way to define Sd is to consider it as the space of half-lines of Rd C1 . The group of d projective automorphisms of Sd is the group SL˙ d C1 .R/. One advantage of S is that the definition of convexity is neater in Sd than in P d .

2 The automorphisms of a properly convex open set 2.1 The first lemma Lemma 2.1. Every compact subgroup of Aut./ fixes a point of . Proof. The proof of the lemma relies on the construction of a “center of mass” for every bounded subset of . This construction relies on the Vinberg convex hypersurface, see Lemma 4.2.

8 This

is a little abusive since there are two such cones, but nothing depends on this choice.

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2.2 The matrix point of view Let be an element of SLd C1 .R/. We denote by 1 . / > 2 . / > > d C1 . / the moduli of the eigenvalues of listed in decreasing order with multiplicity. The 1 spectral radius C is by definition 1 ./. We also define D d C1 . / D C1 . C C An element is semi-proximal if or is an eigenvalue of . An element is positively semi-proximal if C is an eigenvalue of . An element is proximal if 1 ./ > 2 . /. This implies that is semi-proximal. An element is positively proximal if is positively semi-proximal and proximal. Since, det./ D 1, we remark that if C D 1 then D 1 also, and i . / D 1 for all i D 1; : : : ; d C 1. An element is bi-“something” if and 1 are “something”. The main use will be for biproximal elements and positively biproximal elements. Let k D R or C. An element is k-semi-simple if in a suitable k-basis is diagonal. An element is S1 -semi-simple 9 if it is C-semi-simple and all its eigenvalue are on the unit circle. An element is unipotent if . 1/d C1 D 0. The power of a unipotent element is the smallest integer k such that . 1/k D 0. For every element 2 SLd C1 .R/, there exists a unique triple consisting of an R-semi-simple element h , an S1 -semi-simple element e and a unipotent element u such that D h e u and these three elements commute with each other (see for example §4.3 of the book [71] of Witte Morris or [72], Chapter 3, Part 2). Let 2 SLd C1 .R/ and be an eigenvalue of . The power of is the size of the maximal Jordan block of with eigenvalue . In other terms, the power of is the multiplicity of in the minimal polynomial of . Of course, when is unipotent the power of 1 is exactly what we previously called the power of . Proposition 2.2. Suppose the element 2 SLd C1 .R/ preserves a properly convex open set . Then is positively bi-semi-proximal. Moreover, if C D 1, then its power is odd. Finally, the projective traces of the eigenspaces ker. C / and ker. / x meet the convex set . Proof. First, we have to prove that is positively semi-proximal. This is exactly the content of Lemma 3.2 of [9]. We give a rough proof. Consider the subspace V of Rd C1 whose complexification VC is the sum of the eigenspaces corresponding to the eigenvalues of modulus D C . If x 2 , then a computation using the Jordan form of shows that any accumulation point of the x \ P .V /. sequence n .x/ belongs to x \ P .V / which is a properly convex open Consider the relative interior 0 of subset of a vector space V 0 and consider the restriction 0 of to V 0 . The element 0 is semi-simple and all its eigenvalues have the same modulus . Therefore, the element 9 Usually, we say that is hyperbolic when is R-semi-simple, and we say that is elliptic when is S1 -semi-simple, but we shall not use this terminology because it will create a conflict with the next subsection.

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00 D 0 = is S1 -semi-simple and preserves 0 . Since 00 is S1 -semi-simple, the group generated by 00 is relatively compact. Lemma 2.1 shows that 1 is an eigenvalue of 00 and that P .ker. 00 1// meet 0 . The “bi” statement is obvious. Finally, we have to show that the Jordan multiplicity of in is odd, when D 1. This is almost contained in Lemma 2.3 of [10]. One has to study the action of Aut./ on Sd , the 2-fold covering of P d . Denote by k the integer minfl j . /l D 0g and take any point x 2 which is not in ker. /k1 . A computation using a Jordan form of shows that lim n .x/ D lim n .x/

n!C1

if k is even, and

n!1

lim n .x/ D lim n .x/

n!C1

n!1

if k is odd. Therefore preserves a properly convex open set if and only if k is odd.

2.3 The dynamical point of view Definition 2.3. Let 2 Aut./. The translation length of on is the quantity

. / D inf d .x; .x//. We will say that is x2

(1) elliptic if ./ D 0 and the infimum is achieved; (2) parabolic if ./ D 0 and the infimum is not achieved; (3) hyperbolic if ./ > 0 and the infimum is achieved; (4) quasi-hyperbolic if ./ > 0 and the infimum is not achieved. A complete classification of automorphisms of properly convex open sets was given in dimension 2 in [19] and [65]. There is a classification under the hypothesis that the convex set is round in [32]. Finally, one can find a classification in the general context in [29]. Our exposition is inspired by the last reference. The faces of a properly convex open set. We present the notion of faces of a convex body. For more details and a different point of view, the reader can consult the book [17], Chapter 1, §1.5. Let be a properly convex open subset of P d . We introduce the following equivx x y when the segment Œx; y can be extended beyond x and alence relation on : x the closure of an open y. The equivalence classes of are called open faces of ; x face is a face of . The support of a face or of an open face is the smallest projective space containing it. The dimension of a face is the dimension of its support. The behaviour of open faces and faces with respect to closure and relative interior is very nice, thanks to convexity. More precisely, the interior of a face F in its support (i.e. its relative interior) is equal to the unique open face f such that fN D F . Finally,

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x then f is a properly convex open set one should remark that if f is an open face of in its support. The notion of face is important to describe the dynamics, more precisely to describe the set of attractive and repulsive fixed points of an automorphism. We give a family of examples. Consider the convex subset of Rd formed by the points with strictly positive coordinates. This is a properly convex open subset of P d . The diagonal matrix 2 SLd C1 .R/ given by .1 ; : : : ; d C1 / with i > 0, …i D 1 and 1 6 2 6 6 d C1 preserves . If 1 > 2 and d > d C1 , then has one attractive fixed point and one repulsive fixed point corresponding to the eigenlines of 1 and d C1 . Now, if 1 D 2 > 3 and d > d C1 , then has a set of attractive fixed points and one repulsive fixed point. More precisely, the set of attractive fixed points of is a segment included in @; it is even a face of . We can build a lot of examples with various sizes of attractive or repulsive fixed points with this convex set . Each time the attractive set and the repulsive set will be faces of . This is a general fact. Horospheres in Hilbert geometry The round case. Horospheres are very important metric objects in the study of the geometry of metric spaces. But horospheres are not easy to define in a general Hilbert geometry. We begin by a definition in the context of round Hilbert geometry. Let be a round convex subset of P d . First, we define the Busemann function at a point p 2 @. Let x; y 2 . We set ˇp .x; y/ D lim d .y; z/ d .x; z/: z!p

x and To make the computation, compute first in any affine chart containing then send p to infinity. Take the supporting hyperplane H of at p and do your computation in the affine chart A D P d X H . You get: jx qx j 1 ˇp .x; y/ D log 2 jy qy j where qx (resp. qy ) is the point p; x/ \ @ (resp. p; y/ \ @). We recall that the horosphere passing trough x is the set of points y of such that ˇp .x; y/ D 0. Therefore, in the affine chart A, the horospheres are just the translates of the set @ X fpg in the direction given by the line p 2 P d which are in (see Figure 2). Hence, we showed that in a round convex set the horospheres are round convex sets and have the same regularity as @.

The general case: algebraic horospheres. Walsh gave a complete description of the horofunction boundary of a Hilbert geometry in [92]. We will not go that far in the study of horospheres, we will content ourselves with simpler objects.

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p X fpg x

qx

Figure 2. Algebraic horosphere for the ellipsoid.

The convergence of Busemann functions is no more true if the convex set is not round. Cooper, Long and Tillmann introduced an alternative definition in [29]. This time a horosphere will not be defined at a point but at a point p together with a supporting hyperplane H of at p. The algebraic horospheres based at .p; H / of a properly convex open set are the translates of @ X H in the affine chart P d X H in the direction10 p 2 P d which are in (see Figure 3).

p x qx

Figure 3. Algebraic horosphere for the triangle.

10 We

insist on the fact that p is a point of P d which is not in A; therefore p is also a direction in A.

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Hence, we get two transverse foliations of , the one given by the straight line ending at p and the one given by the algebraic horosphere based at .p; H /. We need one more notion before stating and proving the classification. Duality. We present the notion of duality for convex bodies. For more details and a different point of view, the reader can consult the books [17], [81]. A convex cone of a real vector space is said to be sharp when it does not contain any affine line. Hence, there is a correspondence between open sharp convex cones of Rd C1 and properly convex open subsets of P d via the natural projection. If C is a convex cone in a vector space E, then we define C D ff 2 E j for all x 2 Cx X f0g; f .x/ > 0g: We first remark that C is a convex open cone of E . Secondly, C is non-empty if and only if C is sharp and C is sharp if and only if C has non-empty interior. This operation defines an involution between the set of all sharp convex cones of E and the set of all sharp convex cones of E . If is a properly convex open subset of P .E/ then we define the dual of /. This defines an involution between the set of all properly convex by D P .C open subsets of P .E/ and the set of all properly convex open subsets of P .E /. One can remark that is a space of hyperplanes of P d , hence is also a space of affine charts since there is a correspondence between hyperplanes and affine charts. Namely, can be identified with the space of affine charts A containing the closure x of or with the space of hyperplanes H such that H \ x D ¿. An important proposition about duality for convex subsets is the following: Proposition 2.4. Let be a properly convex open set. Then is strictly convex if and only if @ is C 1 . We can explain this proposition in dimension 2. If @ is not C 1 at a point p 2 @, then the space of lines containing p and not intersecting is a segment in the boundary of , showing that is not strictly convex. We consider the following set: X D f.; x/ j is a properly convex open subset of P d and x 2 g endowed with the Hausdorff topology. The group PGLd C1 .R/ acts naturally on X . We denote by P d the projective space P ..Rd C1 / /. We also consider its dual, X D f.; x/ j is a properly convex open subset of P d and x 2 g: The following theorem needs some tools, so we postpone its proof to Subsection 9.1.2 (Lemma 9.3) of this chapter. We think that this statement is natural and we hope the reader will feel the same.

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Lemma 2.5. There is a continuous bijection ? W X ! X which associates to a pair .; x/ 2 X a pair . ; x ? / 2 X where is the dual of , and this map is PGLd C1 .R/-equivariant. We are now able to give a dynamical description of the automorphisms. The classification in the general context Proposition 2.6. Let 2 Aut./. The following are equivalent: (1) is elliptic; (2) fixes a point of ; (3) is S1 -semi-simple. Proof. 1/ () 2/ This is the definition. 2/ ) 3/ Since fixes a point x, also fixes the hyperplane x ? dual to x with respect to (cf. Lemma 2.5). Therefore, is a linear transformation of the affine chart A D P d X x ? centered at x which preserves the convex , hence preserves the John ellipsoid11 of (centered at x in A), so is conjugate to an element in SOd . The conclusion follows. 3/ ) 1/ The closure G of the group generated by is compact, therefore every orbit of is bounded, hence is elliptic by Lemma 2.1. Given a point p 2 @ and a supporting hyperplane H at p of @, one can define the group Aut.; H; p/ of automorphisms of which preserve p and H . This group acts on the set of algebraic horosphere based at .p; H / of . Since different algebraic horospheres correspond by translation in the direction p in the chart P d X H , we get a morphism h W Aut.; H; p/ ! R that measures horosphere displacement. The proofs of two following lemmas are left to the reader. Lemma 2.7. Let 2 Aut.; H; p/. If C D 1, then h. / D 0 and preserves all the algebraic horospheres based at .H; p/ of . If C > 1, then h. / ¤ 0. Lemma 2.8 (McMullen, Theorem 2.1 of [68]). Let 2 Aut./. The translation length satisfies 1 log max. C ; 1= ; C = / 6 . / 6 log max. C ; 1= /: 2 In particular, C D 1 if and only if is elliptic or parabolic. x of a properly convex open set When a subset A of P d is included in the closure , we can define its convex hull in , i.e. the smallest convex set of containing A 11 We recall that the John ellipsoid of a bounded convex of a real vector space is the unique ellipsoid with maximal volume included in and with the same center of mass.

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in its closure. We will denote this set by Conv.A/. An element 2 Aut./ is planar when is R-semi-simple and has exactly two eigenvalues. C In that case, the function 1 d .x; .x// is constant on and equal to 2 log = . Proposition 2.9. Let 2 Aut./. The following are equivalent: (1) is hyperbolic or quasi-hyperbolic. (2) C > 1. x which are fixed and the action (3) There exist two disjoint faces F C and F of of on the properly convex set axe D Conv.F [ F C / is planar. In the last case ./ D 12 log. C = /. Moreover, is quasi-hyperbolic if and only if axe @; otherwise is hyperbolic. Proof. Lemma 2.8 shows that 1/ ) 2/. For 2/ ) 3/ and 1/, since C > 1 we have < 1. So consider the projective C x spaces E C D P .ker. C // and E D P .ker. //, and let us set F C D \E x \ E . Since the action of on is proper, the convex sets F C and F and F D are included in @. Moreover, a computation using the Jordan form of shows that for any point x 2 the limit n x as n tend to C1 (resp. 1) belongs to F C (resp. x Finally the restriction of to the F ), therefore F ; F C are non-empty faces of . C convex hull axe D Conv.F [ F / is clearly planar, and so for every x 2 axe we get d .x; .x// D daxe .x; .x// D 12 log. C = /. Hence, . / 6 12 log. C = /, but . / > 12 log. C = / thanks to Lemma 2.8. 3/ ) 2/ Since the action of on axe is planar we get C > 1. Proposition 2.10. Let 2 Aut./. The following are equivalent: (1) is parabolic. (2) C D 1 and is not elliptic. (3) h D 1 and the power of u is odd and > 3. x and there is only one maximal face (4) fixes every point of a face F @ of fixed by . (5) preserves an algebraic horosphere and is not elliptic. Proof. 1/ ) 2/ is a consequence of Lemma 2.8. 2/ ) 3/ is a consequence of Proposition 2.2. For 3/ ) 4/, denote by k the power of u and consider the subspace E D P .Im.u 1/k1 / of P d . From the Jordan form of , we get that every point of X P .ker.u 1/k1 / accumulates on a point of F D @ \ E; hence F is a face x Moreover, every point of F must be fixed by since it is fixed by h , and of . and h commutes. Finally, such an F is unique since otherwise the action of on Conv.F [ F 0 / would be planar and C > 1.

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4/ ) 5/ Take any point p in the relative interior of F . One can find a supporting hyperplane H fixed by that contains F . Since F is unique we have C D 1, and one concludes with Lemma 2.8. 5/ ) 1/ Lemma 2.7 shows that C D 1. Since is not elliptic, Lemma 2.8 shows that is parabolic. The classification in the strictly convex case or C 1 boundary case Proposition 2.11. Suppose the element 2 SLd C1 .R/ preserves a properly convex open set . If is strictly convex or has C 1 boundary then is not quasi-hyperbolic. If is hyperbolic then is positively proximal and if is parabolic then its Jordan block of maximal size is unique 12 . Proof. First, assume that is strictly convex. If is quasi-hyperbolic, then by Proposition 2.9, one gets that axe @, hence is not strictly-convex. Now, if is hyperbolic and not proximal then it is an exercise to check that @ must contain a non-trivial segment. Analogously, if is parabolic and its Jordan block of maximal size is not unique, then @ must contain a non-trivial segment. Finally, if has a C 1 boundary then the dual is strictly convex (Proposition 2.4) and is preserved by t 1 .

2.4 Examples 2.4.1 Ellipsoid. Consider the quadratic form q.x/ D x12 C Cxd2 xd2 C1 on Rd C1 . The projective trace Ed of the cone of timelike vectors Cd C1 D fx 2 Rd C1 j q.x/ < 0g is a properly convex open set. In a well chosen chart, Ed is a Euclidean ball. We call any image of Ed by an element of SLd C1 .R/ an ellipsoid. The ellipsoid is the leading example of a round Hilbert geometry. The group Aut.Ed / is the group SOd;1 .R/13 and .Ed ; dEd / is isometric to the real hyperbolic space of dimension d . In fact, Ed is the Beltrami–Cayley–Klein projective model of the hyperbolic space. The elements of SOd;1 .R/ can be of three types: elliptic, hyperbolic or parabolic. The stabilizer of any point of Ed is a maximal compact subgroup of SOd;1 .R/, and it acts transitively on @Ed . The stabilizer P of any point p of @Ed is an amenable subgroup of SOd;1 .R/ that acts transitively on Ed . Moreover, P splits as a semi-direct product of a compact group isomorphic to SOd 1 and a solvable subgroup S which acts simply transitively on Ed . And S also splits as a semi-direct product S D U ÌRC where RC is the stabilizer of p and any point q ¤ p, q 2 @Ed . It is a group composed uniquely of hyperbolic elements, and U is a unipotent subgroup composed only of parabolic elements. It is isomorphic to Rd 1 and it acts simply transitively on each 12 We

mean by this that has only one Jordan block of size its power. precisely it is the identity component of SOd;1 .R/.

13 More

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horosphere of Ed . Finally, one can show that the Lie algebra u of U is conjugate to the following Lie algebra of sld C1 .R/: 8 0 19 ˇ 0 0 u1 ud 1 > ˆ ˇ > ˆ > ˆ ˇ B C > ˆ 0 0 0 u ˆ 1 ˇ B C> = < C : d 1 ˇ B : :: : .u1 ; : : : ; ud 1 / 2 R ˇ B C 0 : C> : ˇ B ˆ > ˆ ˇ @ ˆ 0 ud 1 A> > ˆ ˇ > ˆ ; : 0 The case of the ellipsoid is not only the illuminating example of the world of round convex sets, its properties are also the main tool for studying groups acting on round convex sets. 2.4.2 Cone over an ellipsoid. The cone of timelike vectors Cd C1 D fx 2 Rd C1 j q.x/ < 0g is a sharp convex cone of Rd C1 , so it is also a properly convex open subset of P d C1 . It is not strictly convex nor with C 1 boundary and its group of automorphisms is isomorphic to SOd;1 .R/ RC . One should remark that the group Aut.Cd C1 / is “affine” since every automorphism preserves the support of the ellipsoidal face of Cd C1 , hence the affine chart Rd C1 . This properly convex open set gives several counter-examples to statements about strictly convex open sets. 2.4.3 Simplex. Consider an open simplex Sd in Rd . This is a polyhedron (and so a properly convex open subset of P ) whose group of automorphisms is the semi-direct product of the group Dd of diagonal matrices with positive entries and determinant one and the alternate group on the d C 1 vertices of Sd . The group Dd is isomorphic to Rd and it acts simply transitively on Sd . One should remark that Aut.Sd / does not preserve any affine chart of P d (i.e. it is not affine). This very simple properly convex open set is more interesting than it looks. One can remark that since every collineation of P d fixing d C 2 points in generic position is trivial and every polyhedron of P d which is not a simplex has d C2 vertices in generic position, we get that the automorphism group of every polyhedron which is not a simplex is finite. Therefore the only polyhedron (as a properly convex open set) which can be of interest for the geometric group theorist is the simplex. 2.4.4 The symmetric space of SLm .R/. Consider the convex cone of symmetric positive definite matrices of Mm .R/. It is a sharp convex cone in the vector space of CC symmetric matrices. We consider its trace Sm on P d where d D .m1/.mC2/ . This 2 is a properly convex open set which is not strictly convex nor with C 1 boundary, if m > 3.

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CC The automorphism group of Sm is SLm .R/ via the representation m .g/ M D gM tg. This representation offers plenty of examples of various conjugacy classes of matrices acting on a convex set. The interested reader can find a very nice description of S3CC and its relation to the triangle S2 in [35].

3 Examples of “large” groups acting on properly convex open subsets Before giving the examples, we need a nice context and for that a notion of indecomposability. This is the goal of the next lines. Let us begin by a remark. Given a properly convex open subset of the projective space P .E/, the cone C above is a properly convex open subset of the projective space P .E ˚ R/. We will say that a properly convex open set is a convex cone if there exists an affine chart such that is a cone in this affine chart. The following definition is very natural: a sharp convex cone C of a vector space E is decomposable if we can find a decomposition E D E1 ˚ E2 of E such that this decomposition induces a decomposition of C (i.e. Ci D Ei \ C and C D C1 C2 ). A sharp convex cone is indecomposable if it is not decomposable. We apply this definition to properly convex open sets. A properly convex open set is indecomposable if the cone C above is indecomposable. This definition suggests a definition of a product of two properly convex open sets which is not the Cartesian product. Namely, given two properly convex open sets 1 and 2 of the projective space P .E1 / and P .E2 /, we define a new properly convex open set 1 ˝ 2 of the projective space P .E1 E2 / by the following formula: if Ci is the cone above i then 1 ˝ 2 D P .C1 C2 /. It is important to note that if i is of dimension di then 1 ˝ 2 is of dimension d1 C d2 C 1. For example if the i are two segments then 1 ˝ 2 is a tetrahedron. Here is a more pragmatic way to see this product. Take two properly convex sets !i of a projective space P .E/ of disjoint support. The !i are not open but we assume that they are open in their supports. Then there exists an affine chart containing both !i and the convex hull, and such an affine chart14 of !1 [ !2 is isomorphic to !1 ˝ !2 . Let us finish by an explanation of why the Cartesian product is not a good product for convex sets from our point of view. The Cartesian product is an affine notion and not a projective notion, namely the resulting convex set depends on the affine chart containing the convex sets. For example if the i are segments (which is a projective notion) then in an affine chart i can be a half-line or a segment. If they are both half-lines then the Cartesian product is a triangle. But if they are both segments then the Cartesian product is a square which is not projectively equivalent to a triangle. 14 Of

course, this does not depend of the choice of the affine chart containing the convex sets.

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We get that a properly convex open set is either indecomposable or decomposable. We stress that a cone C can be written C D ˝ where is just one point, which is a properly convex open set of S0 . One can easily see that 1 ˝ 2 is never strictly convex nor with C 1 boundary, except if 1 ˝ 2 is a segment, and that the automorphism group of 1 ˝ 2 is by a canonical isomorphism the group Aut.1 / Aut.2 / R and that “the group R” can be written in a good basis as a diagonal matrix group with diagonal entries: .; : : : ; ; ; : : : ; / „ ƒ‚ … „ ƒ‚ … d1 C1

with d1 C1 d2 C1 D 1:

d2 C1

3.1 Homogeneous properly convex open set A properly convex open set is said to be homogeneous when its automorphism group acts transitively on it. Homogeneous convex sets have been classified in the sixties in [90], [89] by Vinberg. Rothaus gave an alternative construction of all homogeneous convex sets in [78]. If 1 and 2 are two homogeneous properly convex open sets, then the product 1 ˝ 2 is also homogeneous. In fact, one should also remark that the cone above 1 is a homogeneous properly convex open set with automorphism group G1 R. We will not spend time on homogeneous properly convex open sets because they do not give new examples of divisible or quasi-divisible convex sets. This is due to the following corollary of the classification: Proposition 3.1. A homogeneous properly convex open set is quasi-divisible if and only if it is symmetric. Moreover, it admits a compact and a finite-volume non-compact quotient. A properly convex open set is symmetric15 when it is homogeneous and self-dual16 . Proof. If is symmetric, then G D Aut./ is semi-simple and acts transitively and properly on . Theorem 3.2 shows that there exists a uniform lattice17 and a non-uniform lattice 0 of G, therefore admits a compact and a finite-volume non-compact quotient. If is quasi-divisible, then G D Aut./ admits a lattice, but a locally compact group which admits a lattice has to be unimodular (i.e., its Haar measure is rightinvariant and left-invariant). Therefore, the group G is unimodular. The classification of Vinberg shows that the only case where is homogeneous with Aut./ unimodular is when is symmetric. 15 This is equivalent to the fact that for every point x 2 there exists an isometry of .; d / which fixes x and whose differential at x is Id . 16 That is, and are isomorphic as properly convex open set. 17A lattice in a locally compact group G is a discrete subgroup such that the quotient G= is of finite volume for the induced Haar measure. A lattice is uniform when the quotient is compact.

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Theorem 3.2 (Borel–Harish-Chandra). A semi-simple Lie group admits a uniform lattice and a non-uniform lattice.

3.2 Symmetric properly convex open set Symmetric properly convex open sets have been classified by Koecher in the sixties using the classification of Jordan algebras ([53], [36], [89]). The classification of indecomposable symmetric properly convex open sets is given by the two following theorems: Theorem 3.3. Let be a symmetric properly convex open subset of P d which is strictly convex. Then is an ellipsoid. In other words, is the symmetric space associated to SOd;1 .R/. For d D 1, we warn the reader that the usual name for an ellipsoid of dimension 1 is a segment. Theorem 3.4. Let be an indecomposable symmetric properly convex open subset of P d which is not strictly convex. Then is the symmetric space associated to SLm .K/ where K D R; C; H and m > 3 or to the exceptional Lie group E6.26/ . We can give an explicit description of these symmetric properly convex open sets. For example, the symmetric properly convex open set associated to SLm .R/ is the projective trace of the cone of positive definite symmetric matrices of size m m. For the other non-exceptional ones, just take the Hermitian or quaternionic Hermitian matrices. We remark that such convex sets do not exist in all dimensions and that the smallC2/ D est one is of dimension 5. The real ones are of dimension DR .d / D .d 1/.d 2 5; 9; 14; 20; 27; 35; : : : , the complex ones are of real dimension DC .d / D d 2 1 D 8; 15; 24; 35; 48; 63; : : : , the quaternionic ones DH .d / D .2d C 1/.d 1/ D 14; 27; 44; 65; 90; 119; : : : , the exceptional one of dimension 26.

3.3 Spherical representations of semi-simple Lie groups In [91], Vinberg characterised the representation of a semi-simple real Lie group that preserves a properly convex open set. We use the classical notation. Let K be a maximal compact subgroup of G and let P be a minimal parabolic subgroup of G. The characterisation is the following: Theorem 3.5. Let W G ! SL.V / be an irreducible representation of a semi-simple group G. Then the following are equivalent: (1) The group .G/ preserves a properly convex open subset of P .V /.

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(2) The vector space V K of vectors fixed by K is not zero. (3) The group P preserves a half-line. Furthermore, if one of the previous assertions is true, then dim.V K / D 1 and there exist two properly convex open sets min and max such that for any properly convex open set preserved by we have min max . Such a representation is called a spherical representation. The properly convex open set min is the convex hull of the limit set of G and max is the dual convex to min .

3.4 Schottky groups The following subsection relies on the machinery of representation of real Lie groups. It can be skipped without consequence for the understanding of the rest of the chapter. Let G be a semi-simple real linear connected Lie group and an irreducible representation of G on a real vector space V of finite dimension. One can ask the following question: When does there exist a Zariski-dense subgroup of G such that ./ preserves a properly convex open subset of P .V /? Benoist gives an answer to this question in [5]. To present this theorem we need to introduce some vocabulary. Let G be a linear algebraic semi-simple Lie group and W G ! V an irreducible representation. We suppose that G; and V are defined over the field R. Let B be a Borel subgroup of G containing a maximal torus T of G. Using T , one can decompose the representation into weight spaces V , 2 X .T / D Hom.T; C /. Let …. / be the set of weights of the representation , i.e. …. / D f 2 X .T / j V ¤ 0g. Take any order on X .T / such that the roots of G are positive on B. There is a unique maximal weight 0 (called the highest weight of ) for this order and the corresponding weight space V 0 is a line. This line is the unique line of V stabilized by B. We also need some “real” tools. The maximal torus T contains a maximal R-split torus S and the Borel subgroup B is contained in a minimal parabolic R-subgroup P of G. Using S, one can decompose the representation into weight spaces V , 2 X .S / D Hom.S; R /. Let …R . / be the set of restricted weights of the representation , i.e. …R . / D f 2 X .S / j V ¤ 0g. The order on X .T / induces an order on X .S/ such that the roots of G are positive on P . There is a unique maximal restricted weight 0 (called the highest restricted weight of ) for this order and the corresponding weight space V 0 has no reason to be a line. The representation is proximal when dim.V 0 / D 1. We denote by PG the lattice of restricted weights associated to P . We have the following equivalences due to Abels, Margulis and So˘ıfer in [1]: (1) The representation of G is proximal. (2) The group .GR / contains a proximal element.

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(3) Every element in the interior of a Weyl chamber of S is mapped to a proximal element. (4) V 0 is stabilized by P . Furthermore, if one of the previous assertions is true, then the subset of GR of all elements g 2 G such that .g/ is proximal is Zariski-dense in G. A representation is orthogonal (resp. symplectic) when G preserves a nondegenerate symmetric (resp. antisymmetric) bilinear form on V . We say that two irreducible representations , 0 are equal mod 2 when the difference of their restricted highest weight is in 2PG . Theorem 3.6 (Benoist [5]). Let G be a semi-simple real linear connected Lie group with finite center and an irreducible representation of G on a real vector space V of finite dimension. (1) G contains a Zariski-dense subgroup which preserves a properly convex open subset of P .V / if and only if is proximal and not equal mod 2 to an irreducible proximal symplectic representation. (2) Every Zariski-dense subgroup 0 of G contains a Zariski-dense subgroup which preserves a properly convex open subset of P .V / if and only if is proximal and equal mod 2 to an irreducible proximal orthogonal representation. Remark 3.7. All subgroups constructed by Benoist for this theorem are Schottky groups, so in particular they are discrete free groups. We give three examples. The interested reader can find more examples in the article by Benoist. The canonical representation of SLd C1 .R/ on Rd C1 satisfies (1) if and only if d > 2 and it never satisfies (2). The canonical representation of SOp;q .R/ on RpCq satisfies (1) and (2). The canonical representation of SLm .C/ on C m never satisfies (1) nor (2).

3.5 Strictly convex divisible and quasi-divisible convex sets A properly convex open set is divisible (resp. quasi-divisible) if the automorphism group of contains a discrete subgroup such that the quotient = is compact (resp. of finite volume). Figures 4 and 5 were made by Xin Nie, and they show tilings of 2-dimensional open convex bodies obtained by triangular Coxeter groups. In the first example, each tile is compact. In the second example, each tile is of finite volume but not compact. 3.5.1 Existence of non-trivial examples Theorem 3.8 (Folklore). In every dimension d > 2, there exists a divisible (resp. quasi-divisible but not divisible) convex set which is strictly convex and is not an ellipsoid.

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Figure 4. Three divisible convex sets divided by the triangular Coxeter group .3; 3; 7/.

Let us say a few words about the history of divisible and quasi-divisible convex sets: The fact that the ellipsoid is divisible can be found at least in small dimensions in the work of Poincaré. The general case is due to Borel and Harish-Chandra (Theorem 3.2). The fact that the only strictly convex open set which is homogeneous and divisible is the ellipsoid is a consequence of the work of Vinberg and Koecher as explained in Sections 3.2 and 3.1. Kac and Vinberg showed using triangular Coxeter groups in [48] that in dimension 2 there exist divisible convex sets which are not homogeneous, but they did not show that their examples are strictly convex (this is a consequence of a theorem of Benzécri [14] (Theorem 3.14) or the article [55] of Kuiper). Johnson and Millson described in [47] a deformation18 of the projective structure of “classical arithmetic” hyperbolic manifolds. A previous theorem of Koszul in [54] (Theorem 7.7) showed that the deformed manifold is actually a convex projective manifold if the deformation is small enough. Finally, Benoist showed that the convex sets given by this deformation are strictly convex [8] (Theorem 6.1) and that in fact the deformed structure is always convex even in the case of a big deformation [9] (Theorem 7.1). In [67], the author shows the theorem in dimension 2 by describing explicitly the moduli space of convex projective structures of finite volume on a surface. This explicit description is a small extension of Goldman’s parametrisation in the compact 18 This

deformation, introduced by Thurston in [84] for quasi-fuchsian groups, is called bending.

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Figure 5. Three quasi-divisible convex sets quasi-divided by the triangular Coxeter group .3; 3; 1/.

case [40]. In [64], the author shows the theorem in any dimension, using a “bending construction”, but in that case, the convexity of the projective structure is obtained “by hand”, i.e. without a theorem like Koszul’s theorem. Theorem 3.8 needs a security statement: Proposition 3.9. If is an indecomposable properly convex open set which has a compact and a finite-volume non-compact quotient, then is a symmetric properly convex open set. Proof. This proposition is a consequence of Theorem 5.3 which states that either is symmetric or Aut./ is Zariski-dense in SLd C1 .R/. Therefore, either is symmetric or Aut./ is discrete, since a Zariski-dense subgroup of a quasi-simple Lie group is either discrete or dense. Now, the case where Aut./ is discrete is excluded since the quotient of by Aut./ would have to be compact and non-compact at the same time. 3.5.2 A non-existence result on exotic examples in small dimensions Proposition 3.10. Every group dividing (resp. quasi-dividing) a strictly convex properly convex open set of dimension 2 or 3 is a uniform lattice (resp. a lattice) of SO2;1 .R/ or SO3;1 .R/.

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Proof. The articles [32] and [29] show that such a quotient = is the interior of a compact manifold. Moreover, such a manifold is aspherical since its universal cover is homeomorphic to Rd . In dimension 2, = is homeomorphic to a compact surface with a finite number of punctures and this surface has negative Euler characteristic since is strictly convex; therefore is a lattice of SO2;1 .R/. In dimension 3, we distinguish the case of a compact quotient from the case of a non-compact quotient. Suppose = is of finite volume and non-compact, then = is the interior of an aspherical atoroidal compact 3-manifold with non-empty boundary, hence = is a Haken manifold and Thurston’s hyperbolization Theorem of Haken manifolds implies that is a non-uniform lattice of SO3;1 .R/. Finally, if = is compact then = is an aspherical atoroidal compact 3-manifold and does not contain Z2 since is Gromov-hyperbolic (Theorem 6.1). Hence, Perelman’s theorem on Thurston’s geometrization conjecture shows that is a uniform lattice of SO3;1 .R/. 3.5.3 An existence result of exotic examples in higher dimensions Theorem 3.11 (Benoist (d D 4) [11], Kapovich (d > 4) [49]). In every dimension d > 4, there exists a strictly convex divisible convex set which is not quasi-isometric to the hyperbolic space. We just point out that the examples of Benoist are obtained using a Coxeter group, and that Kapovich constructed a convex projective structure on some “Gromov– Thurston manifold”. We recall that Gromov–Thurston manifolds are sequences of manifolds Mn of dimension d > 4 such that none of them carries a Riemannian metric of constant curvature 1 but all of them carry a Riemannian metric of variable negative curvature such that the pinching constant converges to zero as n goes to infinity ([44]). The following question is then natural and open: Open question 1. Does there exist in every dimension d > 4 a strictly convex quasidivisible convex set which is not quasi-isometric to the hyperbolic space and which is not divisible?

3.6 Non strictly convex divisible and quasi-divisible convex sets 3.6.1 Intermission: Vey’s theorem. The following theorem describes the splitting of a divisible convex set into indecomposable pieces. Theorem 3.12 (Vey [88]). Let be a discrete subgroup of SLd C1 .R/ that divides a properly convex open set . There exists a subgroup 0 of of finite index such that the following holds.

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(1) There exists a decomposition V1 ˚ ˚ Vr of Rd C1 such that 0 preserves this decomposition and for each i D 1; : : : ; r, the group i0 D Stab 0 .Vi / acts strongly irreducibly on Vi . x are faces of . We denote by !1 ; : : : ; !r the corre(2) The convex sets P .Vi / \ sponding open faces. We have D !1 ˝ ˝ !r . (3) Each !i is an indecomposable divisible convex set divided by i0 . Corollary 3.13. Let be a discrete subgroup of SLd C1 .R/ that divides a properly convex open set . Then is indecomposable if and only if is strongly irreducible. This theorem leads to the following open question: Open question 2. Does Vey’s theorem hold if quasi-divides ? 3.6.2 A non-existence theorem of non-trivial examples in small dimensions Theorem 3.14 (Kuiper [55] or Benzécri [14] (divisible), Marquis [65] (quasi-divisible)). In dimension 2, the only non-strictly convex divisible properly convex open set is the triangle, and there does not exist any quasi-divisible non-divisible non-strictly convex properly convex open subset of dimension 2. Corollary 3.15. In dimension 2, every indecomposable quasi-divisible convex open set is strictly convex with C 1 boundary. 3.6.3 An existence result Theorem 3.16 (Benoist (3 6 d 6 7, divisible) [10], Marquis (d D 3, quasi-divisible) [66]). In every dimension 3 6 d 6 7 (resp. d D 3), there exists an indecomposable divisible (resp. quasi-divisible non-divisible) convex set which is not strictly-convex nor with C 1 boundary. We stress the fact that Benoist describes very precisely the structure of every divisible convex set in dimension 3 in [10]. In particular, he shows that every segment in the boundary of is in fact in a unique triangle T such that @T @ and such that every triangle in the boundary is stabilized by a virtually Z2 -subgroup of . Moreover, every Z2 -subgroup of stabilizes such a triangle. Finally the projection of T on the quotient = is a Klein bottle or a torus, giving a geometric version of the Jaco–Shalen–Johannson decomposition of the quotient. 3.6.4 A question in higher dimensions Open question 3. Does there exist in every dimension d > 4 an indecomposable divisible (resp. a quasi-divisible not divisible) convex set which is not-strictly convex?

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4 Convex hypersurfaces A very important tool in the study of groups acting on Hilbert geometries is the notion of a convex hypersurface. Informally, it consists in building the analogue of the hyperboloid associated to an ellipsoid for any properly convex open set.

4.1 The theorem Let X D f.; x/ j is a properly convex open subset of P d and x 2 g endowed with the Hausdorff topology. The group PGLd C1 .R/ acts naturally on X . Theorem 4.1 (Vinberg [89]). Let be a properly convex open subset of P d , and C the cone above . There exists a map D W ! C which defines a strictly convex analytic embedded hypersurface of Rd C1 which is asymptotic to C and this map is Aut./-equivariant. In fact, one can define this map as X ! Rd C1 so that it becomes PGLd C1 .R/-equivariant. A convex hypersurface of Rd C1 is an open subset of the boundary of a convex set of . A convex hypersurface † is asymptotic to an open convex cone C containing R † if every affine half-line contained in the cone C intersects the hypersurface (see Figure 6). d C1

† †

Figure 6. An asymptotic and a non-asymptotic convex hypersurface.

R Proof. Consider the map ' W C ! RC given by '.x/ D C e f .x/ df . We leave it as an exercise that the level set of this map gives the intended hypersurface.

4.2 Consequences Existence of centers of mass Lemma 4.2. Let B be a bounded part of a properly convex open set. There exists a point xB 2 Conv.B/ such that for every 2 Aut./, if .B/ D B then .xB / D xB .

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Proof. Denote by D W ! † the convex hypersurface given by Theorem 4.1. Consider C D Conv.D.B//. It is a bounded convex part of Rd C1 . The action of Aut./ on Rd C1 is linear, so the projection on of the center of mass of C is the point xB we are looking for. Existence of an invariant Riemannian metric Lemma 4.3. Let be a properly convex open set. There exists on a Riemannian metric which is Aut./-invariant. In fact, one can define this metric from X so that it becomes PGLd C1 .R/-equivariant. Proof. Since the hypersurface of Theorem 4.1 is strictly convex, its Hessian is definite positive at every point, therefore it defines an invariant analytic Riemannian metric.

Existence of a convex locally finite fundamental domain Lemma 4.4. Let be a properly convex open set. There is a map H from to the space of hyperplanes intersecting such that for every x; y 2 the hyperplane H.x; y/ separates x from y and H is Aut./-equivariant, and such that if yn ! p 2 @ then H.x; yn / tends to a supporting hyperplane at p. Proof. Consider the affine tangent hyperplanes Hx and Hy at the point D.x/ and D.y/ of the convex hypersurface † D D./. Set H.x; y/ D P .Vect.Hx \ Hy //. It is an exercise to check that H.x; y/ does the job (see Figure 7).

†

H.x; y/

y

x

O Figure 7. Bisector.

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The existence of convex locally finite fundamental domains for actions of discrete groups is very useful. For the hyperbolic space, the existence of fundamental domains is due to Dirichlet. A direct application of the Dirichlet techniques does not work in Hilbert geometry since the bisector of two points is no longer a hyperplane. Indeed, in the case of Hilbert geometry the two connected components given by a bisector have no reason to be convex. Nevertheless, thanks to Lemma 4.4, Lee shows: Theorem 4.5 (Lee [57] or [58]). Let be a discrete subgroup of SLd C1 .R/ acting on a properly convex open set . There exists a locally finite 19 convex fundamental domain for the action of on . The method of Lee relies on the existence and geometry of affine spheres (cf. next paragraph). The reader can also find a proof which relies only on Vinberg’s hypersurface in [65].

4.3 An alternative construction There is an another construction of similar convex hypersurfaces. Theorem 4.6 (Cheng–Yau, Calabi–Nirenberg). Let be a properly convex open 0 W ! C which defines a strictly convex subset of P d . There exists a map D 20 embedded hyperbolic affine sphere of Rd C1 which is asymptotic to C and this map is Aut./-equivariant. In fact, one can define this map from X to Rd C1 so that it becomes PGLd C1 .R/-equivariant. The main application of this theorem is Theorem 7.9.

5 Zariski closure 5.1 Definitions of proximality A subgroup of SLd C1 .R/ is proximal if it contains a proximal element. The action of a group on P d is proximal if for every x; y 2 P d , there exists a 2 such that .x/ and .y/ are arbitrarily close. We give the definition of a proximal representation of a semi-simple real Lie group in Subsection 3.4. 19A fundamental domain is locally finite when each compact subset of intersects only a finite number of translates of the fundamental domain. 20 We refer to the survey [62] of Loftin for a definition of an affine sphere and for references, see also Chapter 11 of this volume [52].

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These three definitions of proximality have the following link. When is strongly irreducible, the action of on P d is proximal if and only if is proximal. Moreover, if G is the identity component of the Zariski closure of , then the representation W G ! SLd C1 .R/ is proximal if and only if G is proximal if and only if is proximal.

5.2 Positive proximality The following theorem characterizes irreducible subgroups of SLd C1 .R/ which preserve a properly convex open set. A proximal group is positively proximal when every proximal element is positively proximal. Theorem 5.1 (Benoist [5]). A strongly irreducible subgroup of SLd C1 .R/ preserves a properly convex open set if and only if is positively biproximal. If G is the identity component of the Zariski closure of then G has no reason to be positively biproximal. The group G is positively biproximal if and only if the representation W ! SLd C1 .R/ is spherical (cf. Section 3.3). This phenomenon is not exceptional: just take any divisible convex set which is not an ellipsoid. Any group dividing it is Zariski dense in SLd C1 .R/ by Theorem 5.3 below and SLd C1 .R/ is not positively proximal. Finally, we stress on the following fact about positively proximal groups. Theorem 5.2. Suppose is a strongly irreducible group which preserves a properly convex open set. Then there exists a unique closed -invariant minimal subset ƒ of P d ; hence there exist two properly convex open sets min and max preserved by such that for every properly convex open set preserved by , we have min max . Therefore, if we start with a strongly irreducible group preserving a properly convex open set, by taking its Zariski closure we get a reductive 21 group G. We then get an irreducible representation W G ! SLd C1 .R/ which is proximal. Since we assume that 6 SLd C1 .R/, we get that G is in fact semi-simple 22 . The irreducible representations of a semi-simple group are completely classified. The next question is: what can we say about this representation? Theorem 3.6 gives a complete answer to this question. But we can say more in the case of a finite-covolume action. linear Lie group is reductive when it does not contain any non-trivial normal unipotent subgroup. since G is reductive, we just need to show that the center of G is discrete. Take any element g in the center of G; g has to preserve all the eigenspaces of all the proximal elements of G, hence g is a homothety, so g D ˙1 since G 6 SLd C1 .R/. 21A

22 Indeed,

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5.3 Cocompact and finite-covolume case The following theorem of Benoist completely describes the Zariski closure of a group dividing a properly convex open set. Theorem 5.3 (Benoist [6]). Suppose that divides an indecomposable properly convex open set which is not homogeneous. Then is Zariski-dense. The following question is open: Open question 4. Suppose that quasi-divides an indecomposable convex set which is not homogeneous. Is Zariski-dense? The following theorem answers Question 4 for quasi-divisible convex sets in the strictly convex case. Theorem 5.4 (Crampon–Marquis [32]). Suppose that quasi-divides a strictly convex open set which is not an ellipsoid. Then is Zariski-dense. For results of this kind in the context of geometrically finite actions, the reader is invited to read [32].

5.4 A sketch of proof when is strictly convex We only sketch the proof when the convex set is strictly convex. The actual techniques for the non-strictly convex case are different but the hypothesis of strict convexity already gives a nice feeling of the proof. Lemma 5.5 (Vey [88]). Let be a discrete subgroup of SLd C1 .R/ that divides a properly convex open set . Then, for every x 2 , the convex hull of the orbit of x is , i.e. Conv. x/ D . In particular, if is strictly convex, then ƒ D @. Proof. The action is cocompact so there exists a number R0 > 0 such that given any point x 2 , the projection of the ball Bx .R0 / of radius R0 is = ; in other words, the ball Bx .R0 / meets every orbit. Let p be an extremal point of @. We are going to show that p 2 x, and this will prove the first part of the lemma. Suppose that p … x. Then there exists a neighbourhood U of p such that U \ x D ¿; but since p is extremal, the set U \ contains balls of .; d / of arbitrary size, contradicting the first paragraph. Since @ is closed and -invariant we always have ƒ @. Now if is strictly convex, suppose that ƒ ¨ @. Then since is strictly convex we get Conv.ƒ / ¨ ; in particular the convex hull of the orbit of any point of Conv.ƒ / is not all of , contradicting the first part of this lemma. Hence, ƒ D @ when is strictly convex.

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Proof of 5.3 in the case where is strictly convex. Let G be the identity component of the Zariski closure of . Since is strictly convex, ƒ D @ by Lemma 5.5, is strongly irreducible by Theorem 3.12 and positively proximal by Theorem 5.1; hence the group G is semi-simple and the representation W G ! SLd C1 .R/ is irreducible and proximal. Consider the limit set ƒG . The limit set is an orbit of G because the action of G on P d is by projective transformations, hence every orbit is open in its closure. Hence, the limit set is the unique closed orbit of G acting on P d . It is the orbit of the line of highest restricted weight. Hence, ƒG ƒ D @. There are two possibilities, ƒG D @ or ƒG D P d . In the first case, the maximal compact subgroup K of G acts also transitively on ƒG D @. But K fixes a point x of and a point x ? of since K is compact, hence K preserves the John ellipsoid E of centered at x in the affine chart P d X x ? . As K acts transitively on @, we get D E. In the second case, the lemma below shows that either G D SLd C1 .R/ or G D Sp2d .R/ and preserves a symplectic form. But a group which preserves a properly convex open set cannot preserve a symplectic form (Corollary 3.5 of [5]). Lemma 5.6 (Benoist, Lemma 3.9 of [5]). Let G be a linear semi-simple connected Lie group and . ; V / a faithful irreducible and proximal representation. The action of G on P .V / is transitive if and only if (1) G D SLd C1 .R/ and V D Rd C1 with d > 1, or (2) G D Sp2d .R/ and V D R2d with d > 2. If one reads carefully the proof, one should remark that we only used the hypothesis “ is strictly convex” to get ƒ D @. So, in fact we have shown that: Theorem 5.7. Let be a discrete group of SLd C1 .R/ that preserves a properly convex open set . If acts minimally on @, then is Zariski-dense or is an ellipsoid. This leads to the following question: Open question 5. Let be a discrete group of SLd C1 .R/ that divides (or quasidivides) an indecomposable properly convex open set which is not homogeneous. Does act minimally on @? If the answer to this question is yes, then we get an alternative proof of Theorem 5.3. We note that Benoist answers the last question in dimension 3 in [10]. Namely, he shows that the action of any group dividing an indecomposable properly convex open set is minimal on @.

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6 Gromov-hyperbolicity The notion of Gromov-hyperbolicity is a very powerful tool in geometric group theory and metric geometry. The goal of this section is to catch the link between Gromovhyperbolicity and roundness of convex bodies. A proper geodesic metric space X is Gromov-hyperbolic when there exists a number ı such that given any three points x; y; z 2 X, and given any geodesics Œx; y, Œy; z and Œz; x, the geodesic Œx; y is included in the ı-neighbourhood of Œy; z [ Œz; x. A group of finite type is Gromov-hyperbolic if its Cayley graph with respect to one of its finite generating sets 23 is Gromov-hyperbolic for the word metric. Theorem 6.1 (Benoist [8]). Let be a divisible convex set divided by a group . Then the following are equivalent: (1) The metric space .; d / is Gromov-hyperbolic. (2) The convex set is strictly convex. (3) The boundary of is C 1 . (4) The group is Gromov-hyperbolic. Note that a similar statement is true by [29] in the case of quasi-divisible convex sets. There is also a statement of this kind in [22] for non-compact quotients. And, finally, there is a weaker statement of this kind for geometrically finite actions in [32]. We will not review these results. We shall present a rough proof of this theorem.

6.1 The first step Proposition 6.2 (Benoist [8], Karlsson–Noskov [50]). Let be a properly convex open set. If the metric space .; d / is Gromov-hyperbolic, then the convex set is strictly convex with C 1 boundary. Proof. Suppose is not strictly convex and take a maximal non trivial segment s @. Choose a plane … containing s and intersecting , choose a sequence of points xn and yn in \ … converging to the different endpoints of s, and, finally, take any point z 2 \ …. It is an exercise to show that sup u2Œxn ;yn

d .u; Œxn ; z [ Œz; yn / ! 1;

which shows that is not Gromov-hyperbolic (see Figure 8)

23 This

property does not depend on the choice of the generating set.

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z

u

xn

yn

s

Figure 8. Strict convexity proof.

Suppose that @ is not C 1 but Gromov-hyperbolic. Then there exist a point z 2 @, a plane … 3 z such that \ … ¤ ¿, a triangle T of … containing \ … such that z is a vertex of T and such that the two segments of T issuing from z are tangent to \ … at z (see Figure 9).

T

x

0 n

y

xn y n

n

z

Figure 9. C 1 proof.

Now, choose two points x and y in \ … and two sequences xn and yn on Œx; zŒ and Œy; zŒ converging to z. Consider the triangle n of vertices xn , yn and z. We want to show that supu 2 Œxn ; yn d .u; Œxn ; z [ Œz; yn / ! 1. This needs some attention. One should remark that the Comparison Theorem (p. 211) shows that

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for any u 2 Œxn ; yn , we have dT .u; Œxn ; z [ Œz; yn / 6 d .u; Œxn ; z [ Œz; yn /. But the sequence dT .u; Œxn ; z [ Œz; yn / is eventually constant and strictly positive. In particular, the sequence dT .u; Œxn ; z [ Œz; yn / does go to 1, unfortunately. Nevertheless, one must remember that must be strictly convex thanks to the previous paragraph. Hence, we can find a triangle 0n whose sides tend to the side of T . Such a triangle contradicts Gromov-hyperbolicity.

6.2 The duality step We recall the following proposition. Proposition 6.3. Let be a properly convex open set. Then is strictly convex if and only if @ is C 1 . The following proposition is very simple once you know the notion of virtual cohomological dimension. The cohomological dimension of a torsion-free group of finite type is an integer d such that if acts properly on Rd then d > d and the quotient Rd = is compact if and only if d D d . The virtual cohomological dimension of a virtually torsion-free 24 group is the cohomological dimension of any of its torsion-free finite-index subgroup. A reference is [82]. Proposition 6.4. Let be a discrete group of SLd C1 .R/ acting on a properly convex open set . Then the action of on is cocompact if and only if the action of on is cocompact. We grab the opportunity to mention the following open question: Open question 6. Let be a discrete group of SLd C1 .R/ acting on a properly convex open set . Is it true that the action of on is of finite covolume if and only if the action of on is of finite covolume? It is known that the answer is yes in dimension 2 from [65] and also yes in any dimension if you assume that the convex set is strictly convex or with C 1 boundary by [29] and [32].

6.3 The key lemma Theorem 6.5. Suppose that is divisible and that is strictly convex. Then is Gromov-hyperbolic. 24 We

recall that Selberg’s lemma shows that every finite type subgroup of GLm .C/ is virtually torsion-free.

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This theorem is the heart of the proof of Theorem 6.1. We postpone its proof until Subsection 9.3.2 where we will show a more general theorem (Corollary 9.15). Proof of Theorem 6.1 assuming Theorem 6.5. 1/ ) 2/ and 1/ ) 3/ are the content of Proposition 6.2. 2/ ) 1/ is the content of Theorem 6.5. The equivalence 1/ () 4/ is a consequence of the fact that since divides , with the word metric is quasi-isometric to .; d / and Gromov-hyperbolicity is invariant by quasi-isometry. Let us show that 3/ ) 4/ to finish the proof. The properly convex open set dual to is strictly convex, by Proposition 6.3, and the action of on is cocompact by Proposition 6.4. Therefore, by Theorem 6.5, the metric space . ; d / is Gromovhyperbolic. But the group acts by isometries and cocompactly on . ; d /, hence the group is Gromov-hyperbolic. Theorem 6.1 has a fascinating corollary: Corollary 6.6. Suppose a group divides two properly convex open sets and 0 . Then is strictly convex if and only if 0 is strictly convex. We also want to stress that Theorem 6.1 is the only way known by the author to show that a divisible convex set (which is not an ellipsoid) of dimension at least 3 is strictly convex. Hence, the actual proof of the existence of a strictly convex divisible convex set relies on this theorem. We shall see in Subsection 8.2 that Gromov-hyperbolicity also implies some regularity of the boundary of .

7 Moduli spaces 7.1 A naturality statement Let be a group of finite type and d > 2 an integer. The set of homomorphisms Hom.; SLd C1 .R// can be identified with a Zariski closed subspace of SLd C1 .R/N , where N is the number of generators of . We put on it the topology induced from Hom.; SLd C1 .R//. We denote by ˇ the subspace of representations of in SLd C1 .R/ which divide a non-empty properly convex open set of P d . The following striking theorem shows that convex projective structures are very natural. Theorem 7.1 (Koszul–Benoist). Suppose M is a compact manifold of dimension d and D 1 .M / does not contain an infinite nilpotent normal subgroup. Then the space ˇ is a union of connected components of the space Hom.; SLd C1 .R//.

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We will see in the next paragraph that Koszul showed the “open” part of this theorem in [54]. The “closed” part was proved by Choi and Goldman when d D 2 in [23], by Kim when d D 3 and is a uniform lattice of SO3;1 .R/ in [51] and, finally, Benoist showed the general case in [9] (which is the hard step of the proof). One can also find a version of this theorem in the finite volume context in dimension 2 in [67]. Remark 7.2. In [9] (Corollary 2.13), Benoist showed that if divides a properly convex open set, then the group does not contain any infinite nilpotent normal subgroup if and only if is strongly irreducible.

7.2 A sketch of the proof of closedness The proof of the fact that ˇ is closed is quite hard in the general case. In dimension 2, we can give a simpler proof due to Choi and Goldman [23]. The following proposition is a corollary of a classical theorem of Zassenhaus ([95]): Proposition 7.3. Let be a discrete group that does not contain an infinite nilpotent normal subgroup. Then any limit of a sequence of discrete and faithful representations n W ! SLd C1 .R/ is also discrete and faithful. One can find a proof of this proposition in [42] or [75]. This theorem explains the beginning of the story. Lemma 7.4 (Choi-Goldman [23]). Let be a discrete group such that does not admit any infinite nilpotent normal subgroup. Let .n ; n / be a sequence where n is a properly convex open subset of P d and n a sequence of discrete and faithful representations, n W ! SLd C1 .R/, such that n ./ is a subgroup of Aut.n /. Suppose that n ! 1 2 Hom.; SLd C1 .R//. If 1 is irreducible, then 1 preserves a properly convex open set. Proof. Endow the space of closed subsets of P d with the Hausdorff topology. The subspace of closed convex subsets of P d is closed for this topology, therefore it is compact. Consider an accumulation point K of the sequence .n /n2N in this space. The convex set K is preserved by 1 . We have to show that K has non-empty interior and is properly convex. To do that, consider the 2-fold cover Sd of P d and rather than taking an accumulation point K of .n /n2N in P d , take it in Sd and call it Q. We have three possibilities for Q: (1) Q has empty interior; (2) Q is not properly convex; (3) Q is properly convex and has a non-empty interior. Suppose Q has empty-interior. As Q is a convex subset of Sd , this implies that Q spans a non-trivial subspace of Rd C1 which is preserved by 1 . This is absurd since

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1 is irreducible. Suppose now that Q is not properly convex. Then Q \ Q ¤ ¿ spans a non-trivial subspace of Rd C1 . This is again absurd. Therefore Q is properly convex and has non-empty interior. In dimension 2, using Lemma 7.5, one can show easily that every accumulation point of ˇ is an irreducible representation, hence the preceding lemma shows that ˇ is closed. Unfortunately, this strategy does not work in high dimensions and the machinery to show closedness is highly more involved. Lemma 7.5. Every Zariski-dense subgroup of SL2 .R/ contains an element with a negative trace. Every element of SL3 .R/ preserving a properly convex open set is such that Tr./ > 3. Remark 7.6. The second assertion of this lemma is a trivial consequence of Proposition 2.2. The first part is a lemma of [23]. Proof of the closedness in dimension 2 assuming Lemma 7.5. We choose a sequence n 2 ˇ which converges to a representation . Suppose is not irreducible. Then up to conjugation and transposition, the image of restricted to Œ; is included in the special affine group of the canonical affine charts of RP 3 (this subgroup is of course isomorphic to SL2 .R/ Ì R2 ). Hence, Lemma 7.5 shows that there exists an element 2 such that Tr. 1 .// < 1. But Lemma 7.5 shows that for every element , we have Tr. n .// > 3. Hence, the representation is irreducible. Therefore by Lemma 7.4, the representation preserves a properly convex open set . The action of on is proper using the Hilbert metric and the quotient is compact since the cohomological dimension of is 2.

7.3 The openness Let M be a manifold. A projective manifold 25 is a manifold with a .P d ; PGLd C1 .R//structure 26 . A marked projective structure on M is a homeomorphism h W M ! M where M is a projective manifold. A marked projective structure is convex when the projective manifold M is convex, i.e., the quotient of a properly convex open set by a discrete subgroup of Aut./. Two marked projective structures h; h0 W M ! M; M0 are equivalent if there exists an isomorphism i of .P d ; PGLd C1 .R//-structures between M and M 0 and the homeomorphism h01 B i B h W M ! M is isotopic to the identity. 25All projective structures are assumed to be flat all along this chapter. Here is an alternative definition: two torsion-free connections on a manifold M are projectively equivalent if they have the same geodesics, up to parametrizations. A class of projectively equivalent connections defines a projective structure on M . A projective structure is flat if every point has a neighbourhood on which the projective structure is given by a flat torsion-free connection. 26 See [85], [43] and Chapter 11 of the present volume [52] for the definition of a .G; X /-structure.

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We denote by P .M / the space of marked projective structures on M and by ˇ.M / the subspace of convex projective structures. There is a natural topology on P .M /, see [43] for details. Theorem 7.7 (Kozsul [54]). Let M be a compact manifold. The subspace ˇ.M / is open in P .M /. One can find a proof of this theorem in [54], but also in a lecture of Benoist [4] or in a paper of Labourie [56]. In the last reference, Labourie did not state the theorem but his Theorem 3.2.1 implies Theorem 7.7. The formalism of Labourie is the following: a projective structure on a manifold M is equivalent to the data of a torsion-free connexion r and a symmetric 2-tensor h satisfying a compatibility condition. Namely, let r h be a connexion on the vector bundle TM R ! M given by the formula: rXh .Z; / D rX Z C X; LX ./ C h.Z; X / where X, Z are vector fields on M , is a real-valued function on M and LX is the derivation of along X. We say that the pair .r; h/ is good if r preserves a volume and r h is flat. Labourie shows that every good pair defines a projective structure and that every projective structure defines a good pair. Finally, Labourie shows that a projective structure is convex if and only if the symmetric 2-tensor h of the good pair is definite positive. Hence, he shows that being convex is an open condition.

7.4 A description of the topology for the surface Let † be a compact surface with a finite number of punctures. We denote by ˇ.†/ and Hyp.†/ the moduli spaces of marked convex projective structures and hyperbolic structures on †, and by ˇf .†/ and Hypf .†/ the finite-volume ones (for the Busemann volume). In dimension 2, we can give three descriptions of the moduli space ˇ.†/. The first one comes from Fenchel–Nielsen coordinates, the second one comes from Fock– Goncharov coordinates on higher Teichmüller space and finally the third one was initiated by Labourie and Loftin independently. Theorem 7.8 (Goldman). Suppose that † is a compact surface with negative Euler characteristic . Then the space ˇ.†/ is a ball of dimension 8. Goldman shows this theorem for compact surfaces in [40]27 by extending Fenchel– Nielsen coordinates on Teichmüller space. Choi and Goldman extend this theorem to compact 2-orbifold in [24]. Finally, the author of this chapter extends Goldman’s theorem to the case of finite-volume surfaces in [67]. 27 This

is by the way a very nice door to the world of convex projective geometry.

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At a different time by completely different methods, Fock–Goncharov find a system of coordinates for higher Teichmüller space described in [37]. Since the space ˇ.†/ is the second simplest higher Teichmüller space (after Teichmüller space itself), their results give another system of coordinates on ˇ.†/. The situation of SL3 .R/ is much simpler than the situation of a general semi-simple Lie group, and there is a specific article dealing with Fock–Goncharov coordinates in the context of ˇ.†/, that the reader will be happy to read: [38]. Note that the Fock–Goncharov coordinates are nice enough to describe very simply and efficiently ˇf .†/ in ˇ.†/. This leads us to the last system of coordinates: Theorem 7.9 (Labourie [56] or Loftin [61] (compact case), Benoist–Hulin (finite volume case [13])). There exists a fibration ˇf .†/ ! Hypf .†/ which is equivariant with respect to the mapping class group of †.

7.5 Description of the topology for 3-orbifolds 7.5.1 Description of the topology for the Coxeter polyhedron. There is an another context where we can describe the topology of ˇ . It is the case of certain Coxeter groups in dimension 3. Take a polyhedron G , label each of its edges e by a number e 20; 2 and consider the space ˇG of marked 28 polyhedra P of P 3 with a reflection s across each face s such that for every two faces s, t of P sharing an edge e D s \ t , the product of the two reflections s and t is conjugate to a rotation of angle e . We define the quantity d.G / D e C 3 where e C is the number of edges of G not labelled 2 . We need an assumption on the shape of P to get a theorem. Roughly speaking, the assumption means that P can be obtained by induction from a tetrahedron by a very simple process. This process is called “écimer” in French. The English translation of this word is “to pollard” i.e. cutting the head of a tree. An ecimahedron is a polyhedron obtained from the tetrahedron by the following process: see Figure 10.

Figure 10. Ecimation or truncation.

28 This

means that you keep track of a numbering of the faces of P .

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A bad 3-circuit of G is a sequence of three faces r; s; t that intersect each other but such that the edges e D r \ s, f D s \ t and g D t \ s do not intersect and such that

e C f C g > and one of the angles e ; f ; g is equal to 2 . Theorem 7.10 (Marquis [66]). Suppose G is an ecimahedron without bad 3-circuits, which is not a prism and such that d.G / > 0. Then the space ˇG is a finite union of balls of dimension d.G /. Moreover, if G is a hyperbolic polyhedron, then ˇG is connected. The number of connected components of ˇG can be computed, but we refer the reader to [66] for a statement. The expression “G is a hyperbolic polyhedron” means that there exists a hyperbolic polyhedron which realizes G . Andreev’s theorem perfectly answers this question (see [2] or [77]). The assumption that G is not a prism is here just to simplify the statement. The case of the prism can be worked out very easily. We also mention that Choi shows in [21] that under an “ordonnability” condition, ˇ.G / is a manifold of dimension d.G /. Choi, Hodgson and Lee in [25] and Choi and Lee in [26] study the regularity of the points corresponding to the hyperbolic structure in ˇ.G /. 7.5.2 Description of the topology for 3-manifolds. Heusener and Porti show the “orthogonal” theorem to Theorem 7.10. Namely, in practice, Theorem 7.10 shows that one can find an infinite number of hyperbolic polyhedra such that their moduli space of convex projective structures is of dimension as big as you wish. Heusener and Porti show: Theorem 7.11 (Heusener and Porti [46]). There exist an infinite number of compact hyperbolic 3-manifolds such that ˇ.M / is a singleton. Cooper, Long and Thistlethwaite compute explicitly, using a numerical program, the dimension of ˇ.M / for M a compact hyperbolic manifold, using an explicit presentation and a list of 3-manifolds in [28]. It would be very interesting to find a topological obstruction to the existence of a deformation. And also a topological criterion for the smoothness of the hyperbolic point in ˇ.M /, as started by Choi, Hodgson and Lee. We also want to mention that very recently Ballas obtained in [3] a version of Theorem 7.11 in the context of finite-volume convex manifolds.

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8 Rigidity For us, a rigidity result is a theorem that says: “If a convex set have some regularity plus a big isometry group, then this convex has to belong to this list”. We give three precise statements. The interested reader can also read the chapter [45] of this Handbook.

8.1 For strongly convex bodies We say that a properly convex open set is strongly convex if its boundary is C 2 with positive Hessian. Colbois and Verovic show in [27] that a strongly convex body of P d endowed with its Hilbert metric is bi-lipschitz equivalent to the hyperbolic space of dimension d . Theorem 8.1 (Sasaki [79], Socié-Méthou [83]). Let be a strongly convex open set. If the group Aut./ is not compact, then is an ellipsoid. Sasaki proves this theorem in [79] in the case where the boundary is C 1 and d > 4. Podestà gives in [74] a detailed proof of Sasaki’s theorem with some refinements. Finally Socié-Methou gives in [83] the proof in full generality. In fact, Socié-Methou shows a more precise statement; she shows that if a properly convex open set admits an infinite-order automorphism which fixes a point p 2 @, then if @ admits an osculatory ellipsoid at p then is an ellipsoid. We remark that the technique of Sasaki-Podestà and Socié-Methou are completely different. The first one uses geometry of affine spheres and the second one uses only elementary techniques.

8.2 For round convex bodies The following theorem shows that the boundary of a quasi-divisible convex open set cannot be too regular unless it is an ellipsoid. Theorem 8.2 (Benoist (divisible) [8], Crampon–Marquis (quasi-divisible) [30]). Let be a quasi-divisible strictly convex open set. Then the following are equivalent: (1) The convex set is an ellipsoid. (2) The regularity of the boundary of is C 1C" for every 0 6 " < 1. (3) The boundary of is ˇ-convex for every ˇ > 2. Let ˛ > 0 and consider the map W Rd ! R, x 7! jxj˛ , where j j is the canonical Euclidean norm. The level set of defines a properly convex open subset E˛ of P d which is analytic outside the origin and infinity. A point p 2 @ is of class C 1C" (resp. ˇ-convex) if and only if one can find an image of E" (resp. Eˇ ) by a projective

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transformation inside (resp. outside) and such that the point origin of E" (resp. Eˇ ) is sent to p. We recall that a quasi-divisible strictly convex open set is Gromov-hyperbolic, therefore there exists an " > 0 and a ˇ > 2 such that the boundary of is C 1C" and ˇ-convex ([7]). The reader should find more details about this theorem in Chapter 7 of this volume [33].

8.3 For any convex bodies The following statement is just a reformulation of Theorem 5.3. We stress that this statement can be seen as a rigidity theorem. Theorem 8.3 (Benoist [6]). Let be a divisible convex open set. Then the following are equivalent: (1) The group Aut./ is not Zariski-dense. (2) The convex set is symmetric.

9 Benzécri’s theorem Let X D f.; x/ j is a properly convex open subset of P d and x 2 g. The group PGLd C1 .R/ acts naturally on X . The following theorem is fundamental in the study of Hilbert geometry. Theorem 9.1 (Benzécri’s Theorem). The action of PGLd C1 .R/ on X is proper and cocompact. Roughly speaking, even if the action of PGLd C1 .R/ on X is not homogeneous, the quotient space is compact; therefore it means that we can find some homogeneity in the local geometric properties of Hilbert geometry. Precise examples of this rough statement can be found in Subsection 9.2 of the present text.

9.1 The proof To prove Theorem 9.1, we will need two lemmas. The first one is very classical, the second one is less classical. 9.1.1 John’s ellipsoid Lemma 9.2. Given a bounded convex subset of the affine space Rd with center of mass at the origin, there exists a unique ellipsoid E with center of mass at the origin which is included in and of maximal volume. Moreover, we have E d E.

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9.1.2 Duality. The following lemma may sound strange to the reader not used to jump from the projective world to the affine world and vice versa; nevertheless, once we understand it, the lemma should sound right; but its proof needs some analytic tools. Lemma 9.3. Given a properly convex open set of P d and a point x 2 , there exists a unique hyperplane x ? 2 such that x is the center of mass of viewed in the affine chart P d X x ? . Moreover, the map X ! .P d / defined by .; x/ 7! x ? is continuous, PGLd C1 .R/-equivariant and the map ! given by x 7! x ? is analytic. We first give a Proof of Benzécri’s Theorem assuming Lemmas 9.2 and 9.3. We consider the space S D f.E; x/ j E is an ellipsoid of P d and x 2 Eg. The action of PGLd C1 .R/ on S is transitive with stabilizer PSOd .R/, a compact subgroup of PGLd C1 .R/, so S is a PGLd C1 .R/-homogeneous space on which PGLd C1 .R/ acts properly. We are going to define a fiber bundle of X over S with compact fiber which is PGLd C1 .R/-equivariant. Namely, the map ' W X ! S which associates to the pair .; x/ the pair .E; x/ where E is the John ellipsoid of viewed in the affine chart P d X x ? . The map ' is continuous and well-defined, thanks to Lemma 9.3. Lemma 9.2 shows that the fibers of this map are compact and this map is clearly equivariant. Since the action of PGLd C1 .R/ on S is proper and cocompact, it follows that the action of PGLd C1 .R/ on X is proper and cocompact. 9.1.3 A sketch of the proof of Lemma 9.3. In [89] Vinberg introduces a natural diffeomorphism between a sharp convex cone and its dual. Let C be a sharp convex cone of Rd C1 . The map ' W C ! C ; 7! Center of mass.C .d C 1//; where C .d C 1/ D fu 2 C j .u/ D d C 1g, is an equivariant diffeomorphism. For a proof of this statement, the reader can consult [89] but also [41]. We give a geometric presentation of this diffeomorphism but in practice an analytic presentation is needed to understand it correctly. Proof of Lemma 9.3. Let be a properly convex open set and x a point in . Consider C and C , the cone above and respectively, and the diffeomorphism ' W C ! C. Take a point u in C such that Œu D x. Let 2 C be the linear form defined by D '1 .u/. Then .u/ D d C 1 and u is the center of mass of C .d C 1/. Therefore, x is the center of mass of in the affine chart P d X 1 .0/. In other words, x ? D Œ 2 is the point we are looking for.

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9.2 Natural things are equivalent 9.2.1 Definitions. We denote by X the following space: X D f j is a properly convex open subset of P d g: A projective volume is a map from X which associates to a properly convex open set an absolutely continuous measure on with respect to Lebesgue measure. A projective metric is a map F from X which associates to a pair .; x/ 2 X a norm F .x/ on the tangent space Tx of at x. Each of these notions is said to be natural when it is invariant by PGLd C1 .R/ and continuous. The two basic examples are the Hilbert distance which gives rise to the Busemann volume and the Holmes–Thompson volume. We give a brief definition of both volumes in the introduction. We will denote by Hil any measure, distance, norm, etc. coming from the Hilbert distance. 9.2.2 Volume Proposition 9.4. In every dimension d , given any natural projective volume , there exist two constants 0 < ad < bd such that for any properly convex open set we have ad 6 Hil 6 bd . Proof. Given a properly convex open set , let us denote by f (resp. g ) the density of (resp. Hil ) with respect to Lebesgue measure. We get a map from X to RC f .x/ given by .; x/ 7! g .x/ . This map is continuous since and Hil are absolutely continuous. This map is also PGLd C1 .R/-invariant. So, Theorem 9.1 shows that this map attains its maximum and its minimum which are two strictly positive constants. The two following corollaries are now trivial. We just recall some definitions for the convenience of the reader. Given a projective volume , a properly convex open set is said to be -quasi-divisible if there exists a discrete subgroup of Aut./ such that .=/ < 1. Corollary 9.5. If and 0 are two natural projective volumes then any properly convex open set is -quasi-divisible if and only if it is 0 -quasi-divisible. This corollary justifies the usual definition of quasi-divisible convex set which uses Busemann volume. Given a projective volume , the -sup-volume entropy of is the quantity log..B.x; R/// ; R!1 R lim

Hil where x is any point 29 of and B.x; R/ is the ball of radius R of .; d /. 29 This

quantity does not depend on x.

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Corollary 9.6. If and 0 are two natural projective volumes then for any properly convex open set the -sup-volume entropy and the 0 -sup-volume entropy coincide. Of course, the same result is true if we change the supremum into an infimum. 9.2.3 Metric Proposition 9.7. In every dimension d , given any natural projective metric F there exist two constants 0 < ad < bd such that for any properly convex open set and for any point in x, we have ad F .x/ 6 FHil .x/ 6 bd F .x/. Proof. We denote by F Hil the Hilbert metric and we introduce a slightly bigger space than X . We take X ¾ D f.; x; v/ j .; x/ 2 X ; v 2 Tx and FHil .x/.v/ D 1g: The action of PGLd C1 .R/ on X ¾ is again proper and cocompact since X ¾ is a PGLd C1 .R/-equivariant fiber bundle over X with compact fiber. Now the following map is continuous, PGLd C1 .R/-invariant and takes strictly positive values: .; x; v/ 2 X ¾ 7! F .x/.v/. The conclusion is straightforward. 0 Corollary 9.8. If d and d are two distances coming from natural projective metrics 0 / are then for any properly convex open set , the metric spaces .; d / and .; d bi-Lipschitz equivalent through the identity map.

We remark that there is no reason for which the volume entropy of two bi-Lipschitz spaces is the same. We also remark that if a convex set is Gromov-hyperbolic for the Hilbert distance, it is Gromov-hyperbolic for all natural distances. 9.2.4 Curvature. It is hard to give a meaning to the sentence “Hilbert geometries are non-positively curved” since the Hilbert distance is a Finsler metric and not a Riemannian metric. A Hilbert geometry which is CAT(0) is an ellipsoid, see [34]. Lemma 4.2 allows to construct two natural Riemannian metrics on a Hilbert geometry, gvin and gaff . The first one is constructed using the Vinberg hypersurface (Theorem 4.1) and the second one thanks to the affine hypersurface (Theorem 4.6). Proposition 9.9. In any dimension, there exist two numbers 1 6 2 such that for any properly convex open set, any point x 2 and any plane … containing x, the sectional curvature of gvin and gaff at .x; …/ is between 1 and 2 . The triangle gives an example where the curvature is constant and equal to 0. The ellipsoid is an example where the curvature is constant equal to 1. It would have been good if 2 6 0. Unfortunately, Tsuji shows in [86] that there exists a properly convex open set , a point x and a plane … such that the

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sectional curvature of gaff at .x; …/ is strictly positive. Since the example of Tsuji is a homogeneous properly convex open set, we stress that gvin D gaff for this example. Hence, Hilbert geometry cannot be put in the world of non-positively curved manifolds using a Vinberg hypersurface or an affine sphere. Finally, we note that Calabi shows in [18] that the Ricci curvature of gaff is always non-positive.

9.3 Two not-two-lines applications of Benzécri’s theorem 9.3.1 The Zassenhaus–Kazhdan–Margulis lemma in Hilbert geometry. The use of the Margulis constant has proved to be very useful in the study of manifolds of non-positive curvature. The following lemma says that this tool is also available in Hilbert geometry. Theorem 9.10 (Choi [20], Crampon–Marquis [31], Cooper–Long–Tillmann [29]). In any dimension d , there exists a constant "d such that for every properly convex open set , for every point x 2 , for every real number 0 < " < "d and for every discrete subgroup of Aut./, the subgroup " .x/ generated by the set f 2 j d .x; x/ 6 "g is virtually nilpotent. The following lemma of Zassenhaus is the starting point: Lemma 9.11 (Zassenhaus [95]). Given a Lie group G, there exists a neighbourhood U of e such that for any discrete group of G, the subgroup U generated by the set f 2 j 2 Ug is nilpotent. First let us “virtually” prove the lemma in the case where is an ellipsoid E. Since the action of Aut.E/ on E is transitive and since the Margulis constant is a number depending only on the geometry of the space at the level of points, we just have to prove it for one point. We choose a point O 2 E and we have to show that if " is small enough then every discrete group of Aut.E/ generated by elements moving O at a distance less that " is virtually nilpotent. The Zassenhaus lemma gives us a neighbourhood U of e in Aut.E/ such that for any discrete subgroup of G, the subgroup U generated by f 2 j 2 Ug is nilpotent. The open set O D f 2 Aut.E/ j dE .O; .O// < "g is contained in the open set StabO U if " is small enough. What remain is that U is of finite index N in the group " generated by \ O. For this we have to show that N is less than the number of translates of U by StabO needed to cover entirely StabO . In the case of Hilbert geometry, we have to replace the fact that the action of Aut.E/ on E is transitive by Benzécri’s theorem 9.1. The proof of Benzécri’s theorem gives us an explicit and simple compact space D of X such that PGLd C1 .R/ D D X . Namely, choose any affine chart A, any point O 2 A, any scalar product on A, denote

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by B the unit ball of A and set: D D f.; O/ j O is the center of mass of in A and B dBg; where dB is the image of B by the homothety of ratio d and center at O in A. We need to link the Hilbert distance with the topology on PGLd C1 .R/. This is done by the following two lemmas: Lemma 9.12. For every " > 0, there exists a ı > 0 such that for every .; O/ 2 D and every element 2 Aut./, we have: d .O; .O// 6 " H) dPGLd C1 .R/ .1; / 6 ı: Lemma 9.13. For every " > 0, there exists a ı > 0 such that for every .; O/ 2 D and every element 2 Aut./, we have: d .O; O/ 6 ı H) dPGLd C1 .R/ .Stab .O/; / 6 ": The rest of the proof is like in hyperbolic geometry. The details can be found in [29] or [31]; the strategy is the same and can be traced back to the two-dimensional proof of [20]. 9.3.2 A characterisation of Gromov-hyperbolicity using the closure of orbits under PGLdC1 .R/. Recall that X is the space of properly convex open subsets of P d endowed with the Hausdorff topology. For every ı > 0, we denote by Xı the space of properly convex open subsets which are ı-Gromov-hyperbolic for the Hilbert distance. Theorem 9.14 (Benoist [7]). (1) The space Xı is closed in X . (2) Conversely, for every closed subset F of X which is PGLd C1 .R/-invariant and contains only strictly convex open sets, there exists a constant ı such that F Xı . Proof. We only sketch the proof. We start by showing the first point. Take a sequence n 2 Xı converging to a properly convex open set 1 . We have to show that 1 is ı-hyperbolic; we first show that it is strictly convex. Suppose 1 is not strictly convex. Then there exists a maximal non-trivial segment Œx1 ; y1 1 . Therefore, we can find (see the proof of Lemma 6.2) a sequence of triangles zn ; xn ; yn included in 1 whose “size” in 1 tends to infinity. If n is large enough, these triangles are in fact included in n and so their size in n should be less than ı. Passing to the limit we get a contradiction. To show that 1 is Gromov-hyperbolic, we do almost the same thing. Suppose it is not in Xı . Take a triangle of 1 whose size is more than ı. Since 1 is strictly convex, the side of such a triangle is a segment, and one can conclude like in the previous paragraph. Now for the second affirmation. Suppose such a ı does not exist. Then we can find a sequence of triangles Tn whose size goes to infinity in a sequence of properly

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convex open sets n . Precisely, this means that there exists a sequence xn , yn , zn , un of points of n such that un 2 Œxn ; yn and the quantity dn .un ; Œxn ; zn [ Œzn ; yn / tends to infinity. Using Benzécri’s theorem, we can assume that the sequence .n ; un / converges to .1 ; u1 /. Since F is closed, we get that 1 is strictly convex. One can also assume that the triangle xn , yn , zn converges to a triangle of 1 , which cannot be degenerate since its size is infinite. But the strict convexity of 1 contradicts the fact that the size of the limit triangle is infinite. Corollary 9.15. Let be a properly convex open subset of P d . The following properties are equivalent: (1) The metric space .; d / is Gromov-hyperbolic. (2) The closure of the orbit PGLd C1 .R/ in X contains only strictly convex properly convex open sets. (3) The closure of the orbit PGLd C1 .R/ in X contains only properly convex open sets with C 1 boundary. Proof. We just do 1/ () 2/. If the closure F of the orbit PGLd C1 .R/ in X contains only strictly convex properly convex open sets, since F is closed and PGLd C1 .R/-invariant, Theorem 9.14 shows that all the elements of F are in fact Gromov-hyperbolic, hence is Gromov-hyperbolic. Now, if is Gromov-hyperbolic, there is a ı > 0 such that 2 Xı . This space is a closed and PGLd C1 .R/-invariant subset of X, hence the closure of PGLd C1 .R/ is included in Xı . Hence Proposition 6.2 concludes the proof. This corollary has a number of nice corollaries. The first one is an easy consequence, using Proposition 6.3. Corollary 9.16. The metric space .; d / is Gromov-hyperbolic if and only if the space . ; d / is Gromov-hyperbolic. The second one is the conclusion of Theorem 6.1 that we have left so far. Corollary 9.17. A divisible convex set which is strictly convex is Gromov-hyperbolic. To show this corollary, one just needs to know the following proposition which is a direct consequence of Benzécri’s theorem. Proposition 9.18. Let be a divisible convex set. Then the orbit PGLd C1 .R/ is closed in X. Proof. Benzécri’s theorem shows that the action of PGLd C1 .R/ on X is proper, hence the orbit PGLd C1 .R/ .; x/ is closed for every point x 2 . Now, we remark that the orbit PGLd C1 .R/ is closed in X if and only if the union [ PGLd C1 .R/ .; x/ PGLd C1 .R/ .; / WD x2

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is closed in X . Now, since there is a group which divides , one can find a compact subset K of such that K D . Hence the conclusion follows from this computation: [[ PGLd C1 .R/ .; .K// PGLd C1 .R/ .; / D x2K 2

D

[

PGLd C1 .R/ .; K/:

x2K

10 The isometry group of a properly convex open set 10.1 The questions We basically know almost nothing about the group of isometries Isom./ of the metric space .; d /. Open question 7 (raised by de la Harpe in [35]). Is Isom./ a Lie group? Is Aut./ always a finite-index subgroup of Isom./? If yes, does this index admit a bound Nd depending only of the dimension? Does Nd D 2? By the Arzelá–Ascoli theorem, the group Isom./ is a locally compact group for the uniform convergence on compact subsets. We shall see that the answer to the first question is yes and it is already in the literature. The other questions are open and a positive answer would mean that the study of Aut./ or Isom./ is the same from a group-geometrical point of view. The first question is a corollary of a wide open conjecture: Conjecture (Hilbert–Smith). A locally compact group acting effectively on a connected manifold is a Lie group. This conjecture is known to be true in dimension d D 1; 2 and 3. For the proof in dimension 1 and 2, see [70], or Theorem 4.7 of [39] for a proof in dimension 1. The proof in dimension 3 is very recent and due to Pardon [73] (2011). The first article around this conjecture is due to Bochner and Montgomery [15], ˘ cepin who proved it for smooth actions. Using a theorem of Yang ([94]), Repov˘s and S˘ showed in [76] that the Hilbert–Smith conjecture is true when the action is by biLipschitz homeomorphism with respect to a Riemannian metric. There is also a proof of Maleshich in [63] for Hölder actions. For a quick survey on this conjecture, we recommend the article of Pardon [73]. The author of the present chapter is not an expert in this area and this paragraph does not claim to be an introduction to the Hilbert–Smith conjecture. ˘ cepin works for Finsler metrics since it uses only the The proof of Repov˘s and S˘ fact that the Hausdorff dimension of a manifold with respect to a Riemannian metric

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is the dimension of the manifold. We sketch the argument for the reader. We want to show that: ˘ cepin [76]). A locally compact group acting effectively Theorem 10.1 (Repov˘s and S˘ by Lipschitz homeomorphisms on a Finsler manifold is a Lie group. In particular, the group Isom.; d / is a Lie group. Proof. A classical and very useful reduction (see for example [59]) of the Hilbert– Smith conjecture shows that we only have to show that the group Zp of p-adic integers cannot act on M . Since every neighbourhood of the identity of Zp contains a copy of Zp , we can assume that Zp acts by Lipschitz homeomorphisms with bounded Lipschitz constant. Now, since Zp is compact, we can assume that this action is in fact by isometries by averaging the metric using the Haar measure of Zp . ˘ cepin. We have the following Now comes the key argument of Repov˘s and S˘ inequality where dim, dimH and dimZ mean respectively the topological dimension, the Hausdorff dimension and the homological dimension. dim.M / D dimH .M / > dimH M=Zp > dim M=Zp > dimZ M=Zp D dim.M / C 2 The first equality comes from the fact that the Hausdorff dimension of a Finsler manifold is equal to its topological dimension.30 The first inequality follows from the fact that since the action is by isometries, the quotient map is distance non-increasing. The second and third inequalities follow from the fact that we always have dimH .X / > dim.X / > dimZ .X/ when X is a locally compact Hausdorff metric space.31 The last equality is the main result of Yang in [94].

10.2 The knowledge Since in this context subgroups of finite-index are important, we think that the group Aut˙ ./ of automorphisms of of determinant ˙1 is more adapted to the situation. Proposition 10.2 (de la Harpe [35]). Let be a properly convex open set. Suppose that the metric space .; d / is uniquely geodesic. Then Aut˙ ./ D Isom./. Proof. If the only geodesics are the segments, this implies that the image of a segment by an isometry g is a segment. Therefore the fundamental theorem of projective geometry implies that g is a projective transformation. topological dimension of a compact topological space X is the smallest integer n such that every finite open cover A of X admits a finite open cover B of X which refines A in which no point of X is included in more than n C 1 elements of B. If no such minimal integer n exists, the space is said to be of infinite topological dimension. The topological dimension of a non-compact locally compact Hausdorff metric space is the supremum of the topological dimensions of its compact subspaces. 31 For the definition of the homological dimension we refer to [94]. For the inequalities, we refer to [80] for the first and [94] for the second. 30 The

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The following proposition is a nice criterion for uniqueness of geodesics .; d /. Proposition 10.3 (de la Harpe [35]). Let be a properly convex open set. Then the metric space .; d / is uniquely geodesic if and only if for every plane … intersecting , the boundary of the 2-dimensional convex set \ … contains at most one maximal segment. We stress the following corollary: Corollary 10.4. If is strictly convex, then Isom./ D Aut˙ ./. Theorem 10.5 (de la Harpe [35] for dimension 2, Lemmens–Walsh [60] for the general case). Suppose that is a simplex. Then Aut˙ ./ is of index two in Isom./. Proof. We give a rough proof in the case of a triangle . A nice picture will show you that the balls of the triangle are hexagons. Moreover, the group R2 acts simply transitively on the triangle . The induced map is an isometry from the normed vector space R2 with the norm given by the regular hexagon to with the Hilbert distance. Now, every isometry of a normed vector space is affine. So the group Isom./ has twelve connected components since the group Isom./=Isom0 ./ is isomorphic to the dihedral group of the regular hexagon, where Isom0 ./ is the identity component of Isom./. But the group Aut˙ ./ has six connected components and the group Aut ˙ ./=Aut˙0 ./ is isomorphic to the dihedral group of the triangle. Theorem 10.6 (Molnar [69] for the complex case, Bosché [16] for the general case). Suppose that is a non-strictly convex symmetric properly convex open set. Then Aut˙ ./ is of index two in Isom./. In the case where is the simplex given by the equations D fŒx1 W x2 W W xd C1 j xi > 0g; then an example of a non-linear automorphism is given by Œx1 W x2 W W xd C1 7! Œx11 W x21 W W xd1C1 : In the case where is a non-strictly convex symmetric space, then can be described as the projectivisation of a convex cone of symmetric definite positive matrices (or of positive Hermitian symmetric complex matrices, or the analogue with the quaternions or the octonions), and the non-linear automorphism is given by M 7!t M 1 (or analogous). In both cases, the non-linear automorphisms are polynomial automorphisms of the real projective space, the group Aut˙ ./ is normal in Isom./, and even better, Isom./ is a semi-direct product of Aut ˙ ./ with a cyclic group of order 2. Finally, Lemmens and Walsh showed the following theorem:

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Theorem 10.7 (Lemmens and Walsh [60]). Suppose that is a polyhedron which is not a simplex. Then Isom./ D Aut ˙ ./. Lemmens and Walsh conjecture that Isom./ ¤ Aut˙ ./ if and only if is a simplex or is symmetric but not an ellipsoid. For a survey on the isometry group of Hilbert geometry, see Chapter 5 of this volume [93].

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Chapter 9

Dynamics of Hilbert nonexpansive maps Anders Karlsson

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic numbers associated to semicontractions . . . . . . . . . . . Horofunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . Horofunctions and semicontractions . . . . . . . . . . . . . . . . 4.1 An argument of Beardon . . . . . . . . . . . . . . . . . . . 4.2 A general abstract result . . . . . . . . . . . . . . . . . . . 5 Horofunctions and the asymptotic geometry of Hilbert geometries 6 Some consequences for Hilbert metric semicontractions . . . . . 6.1 Karlsson–Noskov . . . . . . . . . . . . . . . . . . . . . . . 6.2 Monotone orbits . . . . . . . . . . . . . . . . . . . . . . . 6.3 Linear drift . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Two-dimensional geometries . . . . . . . . . . . . . . . . . 7 The theorems of Lins and Nussbaum . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

263 265 266 267 267 268 269 270 270 270 271 271 272 272

1 Introduction In a letter to Klein, published in Mathematische Annalen in 1895, Hilbert noted that a formula which Klein observed giving the projective model of the hyperbolic plane, provides a metric on any bounded convex domain. Hilbert’s concern was in the foundations of geometry, going back to the famous Euclid postulates, and his fourth problem asks about geometries where straight lines are geodesics. Today, with an extensive development of metric geometry, we consider these Hilbert geometries as beautiful concrete examples of metric spaces interpolating between (certain) Banach spaces and the real hyperbolic spaces. To a certain limited extent, the Hilbert metric could also be viewed as a simpler analogue of Kobayashi’s (pseudo-) metric in several variable complex analysis, such as the Teichmüller metric. It is then all the more remarkable that this metric discovered and discussed from a foundational geometric point of view (by Hilbert and Busemann) is a highly useful metric for other mathematical sciences, providing yet another example of research driven by purely theoretical curiosity leading to unexpected applications. This was

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probably first noticed, seemingly independently and simultaneously, by Birkhoff [5] and Samelson [27]. The Hilbert metric is defined via the cross-ratio from projective geometry. By virtue of being invariant, it is therefore not surprising that it is useful in the study of groups of projective automorphisms, see for example [26]. A philosophical explanation for the abundance of applications beyond group theory is that positivity (say of population sizes in biology, concentrations in chemistry or prices in economics) is ubiquitous and that many even nonlinear maps contract Hilbert distances. The contraction property leads to fixed point theorems which in turn are a main source of existence of solutions to equations. Recently, a very good book was published on this subject, nonlinear Perron–Frobenius theory, written by Lemmens and Nussbaum [17]. The following conjecture was made independently by Nussbaum and the author of this chapter about a decade ago: Conjecture. Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Then either there is a fixed point of f in C or the forward orbits f n .x/ accumulate at one unique closed face. In view of Nussbaum [23] we know the first part, namely if the orbits do not accumulate on the boundary of C , then indeed there is a fixed point. That the accumulation set does not depend much on which point x in C is iterated is not difficult to see, but what is not quite settled yet is whether the limit set necessarily belongs to the closure of only one face. In this brief text, I will focus on this issue and explain the known partial results. Taken together, they provide convincing support for the conjecture as they treat in particular two basic and opposite situations, the strictly convex case (Beardon) and the polyhedral case (Lins). A general result of Noskov and the author shows that the conjecture in general is at least almost true, see below. As Volker Metz originally indicated to me, in practice it is often hard to show that orbits are bounded (this is needed for establishing the existence of a fixed point), but arguing by contradiction and assuming that the orbits are unbounded, then if one can show the existence of a limiting face, this sometimes leads to the desired contradiction. There are several studies on the bounded orbit case, which is not considered here; for that we refer to [16], [15], [17], [18] and references therein. There is a wealth of applications of the Hilbert metric, see [23], [25], [17]; let me just add two papers, one on decay of correlation [21] and another on growth of groups [3]. A final remark is that from a certain point of view, the conjecture is somewhat reminiscent of the notorious invariant subspace problem in functional analysis, although surely much less difficult. Acknowledgements. I have at various times benefited from discussions on this general topic with Alan Beardon, Gena Noskov, Volker Metz, Thomas Foertsch, Roger Nussbaum, Brian Lins, Bruno Colbois, Constantin Vernicos, Enrico Le Donne, and Bas Lemmens. I would also like to thank Athanase Papadopoulos and Marc Troyanov for numerous improvements on the text. This work is supported in part by the Swiss NSF grant 200021 132528/1.

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2 Basic numbers associated to semicontractions Let .X; d / be a metric space. A semicontraction f W X ! X is a map such that d.f .x/; f .y// d.x; y/ for every x; y 2 X. Other possible terms for this are 1-Lipschitz, non-expanding or nonexpansive maps . Since f is a self-map we may study it by iterating it. If one orbit ff n .x/gn>0 is bounded, then every orbit is bounded, since d.f n .x/; f n .y// d.x; y/ for every n. Therefore, having bounded or unbounded orbits is a well-defined notion. We use the same terminology for subsets, semigroups or groups of semicontractions. For example, if a semigroup of semicontractions fixes a point, then trivially any orbit is bounded. The translation number of f is 1 d.x; f n .x//; n which exists (and is independent of x) by the following lemma: f D lim

n!1

Lemma 2.1. Let an be a subadditive sequence of real numbers, that is, anCm an C am . Then the following limit exists: 1 1 an D inf am 2 R [ f1g: n!1 n m>0 m lim

Proof. Given " > 0, pick M such that aM =M inf an =n +". Decompose n as kn M C rn , where 0 rn < M . Hence kn =n ! 1=M . Using the subadditivity and considering n big enough (n > N."// 1 1 1 1 am an akn M Crn .kn aM C arn / m>0 m n n n 1 1 aM C " inf am C 2": m>0 m M Since " is at our disposal, the lemma is proved. inf

Furthermore we define ıf .x/ D lim d.f n .x/; f nC1 .x//; n!1

which exists because it is the limit of a bounded monotone sequence. Finally, let Df D inf d.x; f .x// x2X

be the minimal displacement, sometimes also called the translation length. Since d.x; f n .x// d.x; f .x// C C d.f n1 .x/; f n .x// nd.x; f .x//

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we have 0 f Df ıf .x/: If f is an isometry, then ıf is constant along every orbit, which means that typically ıf .x/ > Df . Moreover, it might happen that 0 D f < Df , for example if f has finite order and X is an orbit. A metric space is proper if every closed and bounded subset is compact. A theorem of Calka [6] asserts that provided that X is proper, if an orbit is unbounded, then it actually escapes every bounded set, so that d.f n x; x/ ! 1 as n ! 1 for any x. This applies to the Hilbert metric case that we consider here. In this case this fact was independently discovered by Nussbaum [23]. For non-proper spaces this is not true in general.

3 Horofunctions Horospheres first appeared in noneuclidean geometry in the work of Lobachevsky. Horofunctions appear later in the work of Busemann. Consider the unit disk D in the complex plane. The metric with constant Gaussian curvature 1 is ds D

2 jdzj 1 jzj2

;

and in particular d.0; z/ D log

1 C jzj : 1 jzj

This metric is called the Poincaré metric and for a point 2 @D one has the horofunction j zj2 : h .z/ D log 1 jzj2 These appear explicitly or implicitly for example in the Poisson formula in complex analysis, Eisenstein series, and the Denjoy–Wolff theorem. An abstract definition was later introduced by Busemann who defines the Busemann function associated to a geodesic ray to be the function h .z/ D lim d..t /; z/ t: t!1

Note here that the limit indeed exists in any metric space, since the triangle inequality implies that the expression on the right is monotonically decreasing and bounded from below by d.z; .0//. The convergence is moreover uniform if X is proper as can be seen from a 3"-proof using the compactness of closed balls. Horoballs are sublevel sets of horofunctions h./ C . In Euclidean geometry horoballs are halfspaces and in the Poincaré disk model of the hyperbolic plane horoballs, or horodisks in this case, are Euclidean disks tangent to the boundary circle.

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There is a more general definition, perhaps first considered by Gromov around 1980. Namely, let X be a complete metric space and let C.X / denote the space of continuous real functions on X equipped with the topology of uniform convergence on bounded subsets. Fix a base point x0 2 X . Consider now the map ˆ W X ! C.X / defined by ˆ W z 7! d.z; / d.z; x0 /: A related map was considered by Kuratowski [14] and independently K. Kunugui [13] in the 1930s. We will sometimes denote by hz the function ˆ.z/. Note that every hz is 1-Lipschitz because jhz .x/ hz .y/j D jd.z; x/ d.z; x0 / d.z; y/ C d.z; x0 /j d.x; y/ which applied to y D x0 gives jhz .x/j d.x; x0 /: The map ˆ is a continuous injection. We denote the closure ˆ.X) by Xx H or X [ @H X and call it the horofunction compactification of X. The elements in @H X are called horofunctions. In some sense, horofunctions are to metric spaces what linear functionals (of norm 1) are to vector spaces. More on horofunction boundaries, and in particular on the horofunction boundary of Hilbert geometry, is contained in Chapter 5 of this volume [29], by Cormac Walsh.

4 Horofunctions and semicontractions It has long been observed, starting with the studies of Wolff and Denjoy on holomorphic self-maps, that the notion of horofunction is relevant for the dynamical behaviour of semicontractions. See §4.2 of Chapter 10 of this volume [18].

4.1 An argument of Beardon Here is a sketch of a nice argument due to Beardon [4]. Suppose that the semicontraction f of a proper metric space X can be approximated by uniform contractions fk , so that for every x one has fk .x/ ! f .x/. By the contraction mapping principle, every fk has a unique fixed point yk since X is complete. Now take a limit point y of this sequence in a compactification Xx of X . For simplicity of notation, we assume that yk ! y. If the limit point belongs to X , then it must be fixed by f . Indeed: f .y/ D lim fk .yk / D lim yk D y: k!1

k!1

If the limit point instead belongs to the associated ideal boundary @X D Xx n X , then we define the associated “horoballs”: z 2 Hfyk g .x/ if and only if z is the limit of

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points zk belonging to the closed balls centered at yk with radius d.x; yk /. These are invariant in the sense that f .Hfyk g .x// Hfyk g .x/, as is straightforward to verify ([4]). In the special case that the compactification is the horofunction compactification and the ideal boundary @H X, the horoballs are not dependent on the sequence, just the limit y and one has for any x 2 X , hy .f .x// D lim d.yk ; f .x// d.yk ; x0 / k!1

D lim d.yk ; fk .x// d.yk ; x0 / k!1

D lim d.fk .yk /; fk .x// d.yk ; x0 / k!1

lim d.yk ; x/ d.yk ; x0 / k!1

D hy .x/: This implies in particular: Theorem 4.1 ([4]). Let C be a bounded strictly convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Then either there is a fixed point of f in C or the forward orbits f n .x/ converge to a unique boundary point z 2 @C . In a more recent paper of Gaubert and Vigeral [8], an interesting sharpening (in some sense) of Beardon’s argument is used to establish a result inspired by the Collatz– Wielandt characterisation of the Perron root in linear algebra. Theorem 4.2 ([8]). Let f be a semicontraction of a complete metrically star-shaped hemi-metric space .X; d /. Then, Df D f D max inf h.x/ h.f .x//: x H x2X h2X

This is just the first part of the theorem; it has an interesting second part as well. For precise definitions we refer to the paper in question, but roughly speaking, the metric space is allowed to be asymmetric and should have a canonical choice of geodesics which have a Busemann non-positive curvature property. If .X; d / is not proper, one has to interpret the horofunctions appropriately. To a certain extent this theorem unifies Beardon’s result with the result in the next paragraph in the setting considered. Both Beardon and Gaubert–Vigeral results apply to semicontractions of the Hilbert metric.

4.2 A general abstract result Beardon’s argument depends on finding approximations to f . Here is a quite general version of the relationship between semicontractions and horofunctions, not using any approximating sequence:

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Proposition 4.3 ([9]). Let .X; d / be a proper metric space. If f is a semicontraction with translation number , then for any x 2 X there is a function h 2 Xx H such that h.f k .x// k for all k > 0. Proof. Given a sequence i & 0 we set bi .n/ D d.f n .x/; x/. i /n. Since these numbers are unbounded, we can find a subsequence such that bi .ni / > bi .m/ for any m < ni and by sequential compactness we may moreover assume that f ni .x/ ! h 2 Xx H , for some h. We then have, for any k 1, h.f k .x// D lim d.f k .x/; f ni .x// d.x; f ni .x// i!1

lim inf d.x; f ni k .x// d.x; f ni .x// i!1

lim inf bi .ni k/ C . i /.ni k/ bi .ni / . i /ni i!1

lim k C i k D k; i!1

as was to be shown.

5 Horofunctions and the asymptotic geometry of Hilbert geometries Being a convex domain in Euclidean space, C has a natural extrinsic boundary @C . One faces the challenge to compare it with the intrinsic metric boundary @H C . Two papers that consider this comparison are [12] and [28]. In the former, only the case of simplices is treated and a very precise description is obtained. In the latter, reference to the general situation is considered which is useful here. It is observed that the horoballs are convex sets, and this means in particular that any intersection of horoballs where the base points tend to the boundary of C can at most contain one closed face. Walsh also shows that any sequence convergent in the horofunction boundary also converges to a point in @C . We will make free use of these facts below. Notice that since any two orbits lie at a bounded distance from each other, it follows from simple well-known estimates on the Hilbert metric that the closed face, as predicted by the conjecture, must be independent of the point x being iterated. This is a consequence of the following more general statement: Proposition 5.1 ([10]). Let C be a bounded convex domain with the Hilbert metric d . Fix y 2 C and define the Gromov product: .x; x 0 / D

1 .d.x; y/ C d.x 0 ; y/ d.x; x 0 //: 2

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Assume that we have two sequences of points in C that converge to boundary points: N z, N xn ! xN 2 @C and zn ! zN 2 @C . If @C does not contain the line segment Œx; then there is a constant K D K.x; N zN / such that lim sup.xn ; zn / K: n!1

As remarked in the same paper, this is useful for the study of iteration of semicontractions f , see Theorem 6.1 below. The result has also found other applications, such as in [7]. To explain the above point about different orbits in detail, notice first that d.f n .x/; f n .z// d.x; z/ for all n > 0. Hence for any unbounded f one has 1 .d.f n .x/; y/ C d.f n .z/; y/ d.f n .x/; f n .z/// ! 1: 2 In view of the proposition, any subsequence of the iterates on x and the same for z must accumulate on the same closed faces. .f n .x/; f n .z// D

6 Some consequences for Hilbert metric semicontractions The previous results have a number of applications for 1-Lipschitz maps in Hilbert geometry.

6.1 Karlsson–Noskov In [10], a certain simple asymptotic geometric fact in terms of the Gromov product is established as recalled above. Now following [9] we select a subsequence ani of an WD d.f n .x/; x/ such that ani > ak for all k < ni . It follows that the Gromov product .f ni .x/; f k .x// tends to infinity. By selecting a converging subsequence of the ni ’s, we then conclude from Proposition 5.1 that: Theorem 6.1 ([10]). Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Then either there is a fixed point of f in C or there is a boundary point z 2 @C such that the line segment between z and any limit point of a forward iteration f n .x/ is contained in @C . In [11] the same result but in a more general setting is discussed.

6.2 Monotone orbits If the sequence d.x; f n .x// grows monotonically, then every convergent subsequence can play the role of ni as in the proof of Proposition 4.3. This means that in Theo-

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rem 6.1, the special property of z is held by any limit point, thus verifying the conjecture. Explicitly: Theorem 6.2. Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Suppose that for some x 2 X , d.x; f n .x// % 1 monotonically. Then there is a closed face that contains every accumulation of the iterations of f . The monotonicity can be weakened for this argument to go through with some smaller addition, although this may not suffice for the general case.

6.3 Linear drift If f is strictly positive, then by the general abstract result, Proposition 4.3, the orbit goes deeper and deeper inside the horoballs of a fixed horofunction. Since these are convex sets, their intersection is convex and must therefore be a closed face. The orbit can only accumulate here since the horoballs contain all orbit points: Theorem 6.3. Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Suppose that f > 0; then there is a closed face that contains every accumulation point of the iterations of f . It is interesting to note, as is done in [18], that the conjecture also holds in the quite opposite situation where lim inf k!1 d.f k .x/; f kC1 .x// D 0, as shown by Nussbaum in [25].

6.4 Two-dimensional geometries In this note we have described two methods: Beardon’s method with approximations and another one coming from [9] in terms of orbits. Unfortunately, there is at present no known relation between the associated boundary points these two arguments give. If they would give the same point, then the conjecture would follow. In dimension two, where the extrinsic geometry is limited, one can play out these two boundary points to conclude: Theorem 6.4. The Conjecture is true in dimension 2. Proof. By Karlsson–Noskov, there is a star containing all orbit points. At worst, this is made up of two adjacent line segments; suppose this is the case since otherwise we are done. Beardon’s boundary point must lie in one of these. But no matter in which, since the horoballs are invariant sets, we again get that since the orbit accumulates at all the faces in the star we get a contradiction, and there can only be one closed face

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containing all limit points. In the case Beardon’s point is the corner point, then this coincides with the point in [9]. So Beardon’s point is a limit point, but in this case the closed face is just that same point. This proves the conjecture.

7 The theorems of Lins and Nussbaum Lins showed in his thesis [20] that the conjecture is true for every polyhedral domain. This is particularly interesting since it seems to be the case most opposite to the one that Beardon handled, i.e. the strictly convex domains. The proof goes via an isometric embedding into a finite-dimensional Banach space. Lins and Nussbaum [19] showed that for projective linear maps, a stronger version of the conjecture is true: the orbits converge to a finite number of points. (The paper [7], which proves the convergence of geodesics rays, is somewhat analogous to this). This constitutes a beautiful addendum to the classical Perron–Frobenius theorem. Further established cases of the conjecture for specific maps of practical interest can be found in [1], [19], [25], [17]. We conclude by remarking that, in contrast to the linear case of [19], Lins [20] has shown that in some sense the conjecture is best possible: for any simplex together with a convex subset of the boundary, he constructs a semicontraction whose limit set contains this boundary subset.

References [1]

M. Akian, S. Gaubert, B. Lemmens, and R. Nussbaum, Iteration of order preserving subhomogeneous maps on a cone. Math. Proc. Cambridge Philos. Soc. 140 (2006), 157–176.

[2]

J. C. Alvarez Paiva, Hilbert’s fourth problem in two dimensions. In MASS selecta, Amer. Math. Soc., Providence, RI, 2003, 165–183.

[3]

L. Bartholdi andA. Erschler, Groups of given intermediate word growth. Ann. Inst. Fourier, to appear; preprint, arXiv:1110.3650.

[4]

A. F. Beardon, The dynamics of contractions. Ergodic Theory Dynam. Systems 17 (1997), 1257–1266.

[5]

G. Birkhoff, Extensions of Jentzsch’s theorem. Trans. Amer. Math. Soc. 85 (1957), 219–227.

[6]

A. Całka, On conditions under which isometries have bounded orbits. Colloq. Math. 48 (1984), 219–227.

[7]

T. Foertsch and A. Karlsson, Hilbert metrics and Minkowski norms. J. Geom. 83 (2005), 22–31.

[8]

S. Gaubert and G. Vigeral, A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. Math. Proc. Cambridge Philos. Soc. 152 (2012), 341–363

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A. Karlsson, Non-expanding maps and Busemann functions. Ergodic Theory Dynam. Systems 21 (2001), 1447–1457.

[10] A. Karlsson and G.A. Noskov, The Hilbert metric and Gromov hyperbolicity. Enseign. Math. (2) 48 (2002), 73–89. [11] A. Karlsson, On the dynamics of isometries. Geom. Topol. 9 (2005), 2359–2394. [12] A. Karlsson, A. Metz, and G.A. Noskov, Horoballs in simplices and Minkowski spaces. Internat. J. Math. Math. Sci. 2006, Art. ID 23656, 20 pp. [13] K. Kunugui, Applications des espaces à une infinité de dimensions à la théorie des ensembles. Proc. Imp. Acad. 11 (1935), 351–353. [14] C. Kuratowski, Topologie. I. Espaces métrisables, espaces complets. 2d ed., Monografie Matematyczne 20, Warszawa, Wrocław 1948. [15] B. Lemmens and C. Walsh, Isometries of polyhedral Hilbert geometries. J. Topol. Anal. 3 (2011), 213–241. [16] B. Lemmens, Nonexpansive mappings on Hilbert’s metric spaces. Topol. Methods Nonlinear Anal. 38 (2011), 45–58 [17] B. Lemmens and R. Nussbaum, Nonlinear Perron–Frobenius theory. Cambridge Tracts Math. 189, Cambridge University Press, Cambridge 2012. [18] B. Lemmens and R. Nussbaum, Birkhoff’s version of Hilbert’s metric and its applications in analysis. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 275–303. [19] B. Lins and R. Nussbaum, Iterated linear maps on a cone and Denjoy–Wolff theorems. Linear Algebra Appl. 416 (2006), 615–626. [20] B. Lins, A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains. Math. Proc. Cambridge Philos. Soc. 143 (2007), 157–164. [21] B. Lins, and R. Nussbaum, Denjoy–Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators. J. Funct. Anal. 254 (2008), 2365–2386. [22] C. Liverani, Decay of correlations. Ann. of Math. 142 (1995), 239–301. [23] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 75 (1988), no. 391. [24] R. D. Nussbaum and S.M. Verduyn Lunel, Generalizations of the Perron–Frobenius theorem for nonlinear maps. Mem. Amer. Math. Soc. 138 (1999), no. 659. [25] R. D. Nussbaum, Fixed point theorems and Denjoy–Wolff theorems for Hilbert’s projective metric in infinite dimensions. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 199–249. [26] J.-F. Quint, Convexes divisibles (d’après Yves Benoist). In Séminaire Bourbaki, Volume 2008/2009. Exposés 997—1011, Astérisque 332 (2010), Exp. No. 999, vii, 45–73. [27] H. Samelson, On the Perron–Frobenius theorem. Michigan Math. J. 4 (1957), 57–59. [28] C. Walsh, The horofunction boundary of the Hilbert geometry. Adv. Geom. 8 (2008), 503–529. [29] C. Walsh, The horofunction boundary and isometry group of Hilbert geometry, In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 127–146.

Chapter 10

Birkhoff’s version of Hilbert’s metric and its applications in analysis Bas Lemmens and Roger Nussbaum

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . Birkhoff’s version of Hilbert’s metric . . . . . . . 2.1 Birkhoff’s contraction ratio . . . . . . . . . 2.2 An application to Perron–Frobenius operators 3 Special cones . . . . . . . . . . . . . . . . . . . . 3.1 Simplicial cones . . . . . . . . . . . . . . . 3.2 Polyhedral cones . . . . . . . . . . . . . . . 3.3 Symmetric cones . . . . . . . . . . . . . . . 4 Non-expansive mappings on Hilbert geometries . . 4.1 Periodic orbits . . . . . . . . . . . . . . . . 4.2 Denjoy–Wolff type theorems . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction In the nineteen-fifties Garrett Birkhoff [8] and Hans Samelson [64] independently discovered that one can use Hilbert’s metric and the contraction mapping principle to give an elegant proof of the existence of a positive eigenvector for a variety of linear mappings that leave a cone in a real vector space invariant. Their results can be seen as a direct generalization of Perron’s theorem [59], [60] concerning the existence and uniqueness of a positive eigenvector of square matrices with positive entries. In the past decades the ideas of Birkhoff and Samelson has been further developed by numerous authors. A partial list of contributors include, [12], [13], [19], [20], [23], [33], [35], [36], [37], [38], [42], [52], [62], [68], [73]. It has resulted in a remarkably detailed understanding of the eigenvalues, eigenvectors, and iterative behavior of a variety of linear, and nonlinear, mappings on cones. This body of work belongs to an area in mathematical analysis known as nonlinear Perron–Frobenius theory. A recent introductory account of this field is given in [42].

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Recall that Hilbert’s metric [25] is defined as follows. Let A be a real n-dimensional affine normed space, and denote the norm on the underlying vector space by k k. Consider a bounded, open, convex set A. For x; y 2 , let `xy denote the straight line through x and y in A, and denote the points of intersection of `xy and @ by x 0 and y 0 , where x is between x 0 and y, and y is between x and y 0 , as in Figure 1. ............................ .............. .... .......... ... . . . . . . . . . . ... . . . . . . . ... ..... . . . ... . . . 0 .. ... x .. x y ... .... ... ... . . .... ..... ... .. ....... .. ......... . . .......... ... ............ ....................................

y0

Figure 1. Hilbert’s cross-ratio metric.

On , Hilbert’s metric is defined by

kx 0 yk ky 0 xk ı.x; y/ D log kx 0 xk ky 0 yk

(1.1)

for all x ¤ y in , and ı.x; x/ D 0 for all x 2 . The metric space .; ı/ is called the Hilbert geometry on . In mathematical analysis one uses an alternative version of Hilbert’s metric, which is defined on a cone C in a real vector space V . The version we will use here was popularized by Bushell in [12], and for simplicity we shall refer to it as Birkhoff’s version of Hilbert’s metric. As we shall see, Birkhoff’s version ties together the partial ordering induced by C and the metric. This idea has proved to be very fruitful. The main objective of this survey is to discuss some of the applications of Birkhoff’s version of Hilbert’s metric in the analysis of nonlinear mappings on cones, and to illustrate some of the advantages of Birkhoff’s version by reproving several known geometric results for Hilbert geometries. Among other results, we shall see how Birkhoff’s version can be used to give a simple proof of the well-known fact that the Hilbert geometry on an open n-simplex is isometric to and n-dimensional normed space, whose unit ball is a polytope with n.n C 1/ facets. As many important applications of Hilbert’s metric in analysis are in infinitedimensional Banach spaces, we shall state some of the basic results in an infinitedimensional setting and occasionally point out the difference between the infinitedimensional and finite-dimensional cases. Acknowledgement. This work is partially supported by EPSRC grant EP/J008508/1 (B.L.) and by NSF grant is NSF DMS 1201328 (R.N.).

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2 Birkhoff’s version of Hilbert’s metric To define Birkhoff’s version of Hilbert’s metric we need to recall some elementary concepts from the theory of partially ordered vector spaces. Let V be a (possibly infinite-dimensional) real vector space. A subset C of V is called a cone if (C1) C is convex, (C2) C C for all 0, and (C3) C \ .C / D f0g. If C V satisfies (C1) and (C2) it is called a wedge. A cone C V induces a partial ordering, C , on V by x C y if y x 2 C: If C is merely a wedge, then C is only a pre-order, as it may fail to be anti-symmetric. In analysis one often considers a closed cone C in a Banach space .V; k k/. The cone C is called normal if there exists a constant > 0 such that kxk kyk whenever 0 C x C y. It is known, see Lemma 1.2.5 in [42], that every closed cone in a finite-dimensional normed space is normal, but a closed cone in an infinitedimensional Banach space may fail to be normal. Given a Banach space .V; k k/, we denote the dual space of continuous linear functionals on V by V . If C is a closed cone in a Banach space .V; k k/, the dual wedge is given by, C D f 2 V W .x/ 0 for all x 2 V g: In general C is only a wedge. However, if C is a total cone, i.e., the closure of C C D fx y W x; y 2 C g is V , then C is a cone. In particular, if C is a closed cone in a finite-dimensional vector space V and C has nonempty interior, then C is a closed cone in V with nonempty interior. For 2 C we let † D fx 2 C W .x/ D 1g. It is not hard to show, see Lemma 1.2.4 in [42], that if C is a closed cone with nonempty interior in a finitedimensional normed space, and in the interior of C , then † is a nonempty, compact, convex subset of V . However, in infinite dimensions it may happen that there does not exist 2 C such that † is bounded. A simple example is the Hilbert space, `2 , with the standard positive cone C D f.x1 ; x2 ; : : :/ 2 `2 W xi 0 for all i g. As mentioned earlier, Birkhoff’s version of Hilbert’s metric ties together the partial ordering C and the distance. To define it, let C be a cone in a vector space V . For x 2 V and y 2 C , we say that y dominates x if there exist ˛; ˇ 2 R such that ˛y C x C ˇy. In that case, we write M.x=y/ D inffˇ 2 R W x C ˇyg and m.x=y/ D supf˛ 2 R W ˛y C xg: Obviously, if x; y 2 V are such that y D 0 and y dominates x, then x D 0, as C is a cone. On the other hand, if y 2 C n f0g and y dominates x, then M.x=y/ m.x=y/.

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Using the domination relation one obtains an equivalence relation on C by x C y if y dominates x and x dominates y. The equivalence classes are called the parts of C . Obviously f0g is a part of C . Moreover, if C is a closed cone with nonempty interior, C B , in a Banach space, then C B is a part of C . The parts of a finite-dimensional cone are related to the faces of C . Indeed, if C is a finite-dimensional closed cone, then it can be shown that the parts correspond to the relative interiors of the faces of C , see [42], Lemma 1.2.2. Recall that a face of a convex set S V is a subset F of S with the property that if x; y 2 S and x C .1 /y 2 F for some 0 < < 1, then x; y 2 F . The relative interior of a face F is the interior of F in its affine span. It is easy to verify that if x; y 2 C n f0g, then x C y if, and only if, there exist 0 < ˛ ˇ such that ˛y C x C ˇy. Furthermore, m.x=y/ D supf˛ > 0 W y C ˛ 1 xg D M.y=x/1 :

(2.1)

Birkhoff’s version of Hilbert’s metric on C can now be defined as follows: M.x=y/ d.x; y/ D log m.x=y/ for all x C y with y ¤ 0, d.0; 0/ D 0, and d.x; y/ D 1 otherwise. If C is a closed cone in a Banach space, then d is a genuine metric on the set of rays in each part of the cone as the following lemma shows. Lemma 2.1. If C is a cone in V , then for each x C y C z with y ¤ 0, (i) d.x; y/ 0, (ii) d.x; y/ D d.y; x/, (iii) d.x; z/ d.x; y/ C d.y; z/, (iv) d.x; y/ D d.x; y/ for all ; > 0. Moreover, if C is a closed cone in a Banach space .V; k k/, then d.x; y/ D 0 if, and only if, x D y for some 0. Proof. To prove the first assertion we note that for each 0 < ˛ < m.x=y/ and 0 < M.x=y/ < ˇ we have ˛y C x C ˇy: It follows that y C .ˇ=˛/y, and hence ˇ=˛ 1. Thus M.x=y/=m.x=y/ 1 and hence d is nonnegative. Furthermore, note that by (2.1), d.x; y/ D log M.x=y/M.y=x/ D d.y; x/; which shows that d is symmetric. To show that d satisfies the triangle inequality, we note that for each 0 < ˛ < m.x=y/ and 0 < < m.y=z/ we have ˛y C x and z C y, and hence ˛z C x. This implies that m.x=z/ m.x=y/m.y=z/. In the same way it can be shown that M.x=z/ M.x=y/M.y=z/. Thus, M.x=y/M.y=z/ M.x=z/ ; m.x=z/ m.x=y/m.y=z/

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which proves (iii). For ; > 0, it is easy to verify that M.x=y/ D

M.x=y/

m.x=y/ D

and

m.x=y/;

which gives the fourth assertion. Finally, assume that C is a closed cone in .V; kk/ and x C y with y ¤ 0. As C is closed, m.x=y/y C x C M.x=y/y. If d.x; y/ D 0, then M.x=y/m.x=y/1 D 1, so we get that y C m.x=y/1 x C y from which we deduce that x D m.x=y/y. On the other hand, if x D y for some > 0, then d.x; y/ D 0 by the previous assertion. To understand the relation with Hilbert’s cross-ratio metric, ı, given in (1.1), we consider a cone C in an .n C 1/-dimensional real normed space .V; k k/ and we assume that the interior of C is nonempty. Let H V be an n-dimensional affine hyperplane such that C WD H \ C B is a (relatively) open, bounded, convex set in H . Theorem 2.2. The restriction of d to C coincides with ı. Proof. Consider x ¤ y in C . Let ˛ D m.x=y/ D M.y=x/1 and ˇ D M.x=y/. As C is closed, ˛y C x and x C ˇy. Write u D x ˛y 2 @C and w D y x=ˇ 2 @C . Let `xy denote the straight line through x and y and denote the points of intersection with @C by x 0 and y 0 , as in Figure 2. We know that there exist ; > 1 x0 x

y0

y

u w

Figure 2. Theorem 2.2.

such that

x 0 D y C .x y/

y 0 D x C .y x/:

and

Let be a linear functional on V such that H D fz 2 V W .z/ D 1g. So, u x ˛y y C .x y/ D x 0 D D ; .u/ 1˛ which implies that ˛ D . 1/= . Similarly, x C .y x/ D y 0 D

y x=ˇ w D ; .w/ 1 1=ˇ

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which gives ˇ D =.1 /. Thus, 1 ky x 0 k D D D M.y=x/ kx x 0 k 1 ˛ and

(2.2)

kx y 0 k D D ˇ D M.x=y/; ky y 0 k 1

which shows that d.x; y/ D ı.x; y/ on C . Remark 2.3. From equation (2.2) it follows that log

kx y 0 k D log M.x=y/ ky y 0 k

for all x ¤ y in C . The function FC W C B C B ! R given by FC .x; y/ D log M.x=y/ is called the Funk (weak) metric on C B , see [58], [70]. Remark 2.4. Birkhoff’s version allows one to give a natural definition of Hilbert’s metric on any open, convex, possibly unbounded, subset of an infinite-dimensional Banach space. Indeed, let .V; k k/ be an open, convex set and assume that contains no straight lines, i.e., for each v 2 there exists no u 2 V n f0g such that v C t u 2 for all t 2 R. Let W D R V with norm k.s; v/kW D jsj C kvk for all .s; v/ 2 W . Consider the following set in W : C D f.s; sv/ 2 W W s > 0 and v 2 g: It is easy to verify that C is open, and C C for all > 0. Furthermore its closure, C , is a cone in W . Indeed, suppose that w D .s; u/ 2 C and w 2 C . As s 0, we know that s D 0. Now let wk D .sk ; sk vk / 2 C be such that wk ! w and vk 2 for all k. Then for each t > 0 fixed and k sufficiently large, 0 sk t 1. Thus, for each v 2 and k large, we have .1 sk t /v C sk t vk 2 : Note that k.1 sk t /v C sk t vk .v C t u/k sk t kvk C t ksk vk uk ! 0 as k ! 1, x But for > 0 small we have v C u 2 , as is open. As t was so that v C t u 2 . arbitrary and is convex, we deduce from Proposition 8.5 of [66] that v C t u 2 for all t 0. In the same way we can use the assumption that w 2 C to show that v C t u 2 for all t 0. This, however, contradicts the hypothesis that contains no lines. The restriction of d to f.1; v/ 2 W W v 2 g is a genuine metric, which provides a natural definition of Hilbert’s metric on . Remark 2.5. From an analyst’s point of view Hilbert’s metric on cones has one disadvantage – namely, it is only a metric on the pairs of rays in the cone rather than on the pairs of points in the cone. In applications in analysis one therefore often also

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considers the following variant, which was introduced by Thompson [68]. Given a closed cone C in .V; k k/, Thompson’s metric is defined by ˚ dT .x; y/ D max log M.x=y/; log M.y=x/ for x C y and y ¤ 0, dT .0; 0/ D 0, and dT .x; y/ D 1 otherwise. The reader can verify that dT is a genuine metric on each part of C . It is known, see [68], that if C is a closed normal cone in a Banach space .V; k k/ and P C is a part of C , then .P; dT / is a complete metric space whose topology coincides with the norm topology of the underlying space. Furthermore, if q W C n f0g ! .0; 1/ is a continuous homogeneous (degree 1) function and †q D fx 2 C W q.x/ D 1g, then the metric space .†q \ P; d / is also a complete metric space and its topology coincides with the norm topology. Particularly, interesting examples of homogeneous functions q include q 2 C n f0g and q W x 7! kxk.

2.1 Birkhoff’s contraction ratio The usefulness of Hilbert’s metric lies in the fact that linear, but also certain nonlinear, mappings between cones are non-expansive with respect to this metric. Recall that a mapping f from a metric space .X; dX / into a metric space .Y; dY / is non-expansive if dY .f .x/; f .y// dX .x; y/ for all x; y 2 X: It is said to be a Lipschitz contraction with constant 0 c < 1, if dY .f .x/; f .y// cdX .x; y/ for all x; y 2 X: Note that if C is a cone in V , K is a cone in W , and L W V ! W is a linear mapping, then L.C / K is equivalent to Lx K Ly whenever x C y. A mapping f W C ! K is called order-preserving if x C y implies f .x/ K f .y/. It is said to be homogeneous of degree r if f .x/ D r f .x/ for all x 2 C and > 0. The following result is elementary, but very useful. Proposition 2.6. Let C V and K W be cones. If f W C ! W is orderpreserving and homogeneous of degree r > 0, then M.f .x/=f .y// M.x=y/r and m.x=y/r m.f .x/=f .y// for all x; y 2 C with x C y. Proof. As the proofs of the two inequalities are very similar, we shall only show the first one. If x; y 2 C and x C y, then for each ˇ > 0 with x C ˇy we have that f .x/ K ˇ r f .y/. So, M.f .x/=f .y// ˇ r for all ˇ > M.x=y/, which gives M.f .x/=f .y// M.x=y/r . As an immediate consequence we obtain the following result.

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Corollary 2.7. Let C V and K W be cones. If f W C ! K is order-preserving and homogeneous (of degree 1), then f is non-expansive with respect to d on each part of C . Remark 2.8. It is interesting to note that every order-preserving homogeneous mapping f is also non-expansive with respect to Thompson’s metric. In fact, it can be shown (see [42], Lemma 2.1.7) that an order-preserving mapping f W C ! K is non-expansive under Thompson’s metric if, and only if, f is subhomogeneous, i.e., f .x/ K f .x/ for all x 2 C and 0 1. We should also remark that it is also easy to show that every order-reversing, homogeneous degree 1 mapping is non-expansive under d , see [42], Corollary 2.1.5. From Corollary 2.7 we see that every linear mapping L W V ! W with L.C / K is non-expansive with respect to d on each part of C . Furthermore, if L is invertible and L.C / D K, then L must be an isometry. In fact, in that case we have M.Lx=Ly/ D M.x=y/

and

m.Lx=Ly/ D m.x=y/

for all x C y in C . The non-expansiveness of linear mappings, L W C ! C , on cones provides a way to analyze the eigenvalue problem, Lx D x, on C using contraction mapping arguments. Given cones C V , K W and a linear mapping L W V ! W with L.C / K, the Birkhoff contraction ratio of L is defined by .L/ D inffc 0 W d.Lx; Ly/ cd.x; y/ for all x C y in C g: Moreover, the projective diameter of L is given by .L/ D supfd.Lx; Ly/ W x; y 2 C with Lx K Lyg: Theorem 2.9 (Birkhoff). Let C be a cone in a vector space V and K be a cone in a vector space W . If L W V ! W is a linear mapping with L.C / K, then

1 .L/ ; .L/ D tanh 4 where tanh.1/ D 1. So, if .L/ < 1, then L is a Lipschitz contraction on each part of C , with contraction constant tanh..L/=4/ < 1. In numerous cases one can prove that .L/ < 1. Indeed, Birkhoff showed this for matrices with positive entries and for certain integral operators with positive kernels. In case the linear map L is given by a positive m n matrix A D .aij /, so aij > 0 for all i and j , there exists the following well-known explicit formula, see Appendix A in [42]: api aqj < 1; .A/ D max d.Aei ; Aej / D log max i;j i;j;p;q apj aqi where e1 ; : : : ; en denote the standard basis vectors in Rn .

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Remark 2.10. In [27], [28] Hopf proved a result closely related to Birkhoff’s Theorem 2.9. Hopf was apparently unaware of Birkhoff’s work and did not use Hilbert’s metric. The results of Birkhoff and Hopf have been extended and their connections have been unraveled in numerous papers including, [2], [12], [13], [19], [20]. A detailed exposition of these works can be found in [42], Appendix A. Birkhoff’s Theorem 2.9 can be used to prove the existence and uniqueness of positive eigenvectors of continuous linear mappings that leave a closed, normal cone in a Banach space invariant. In fact, one has the following slightly more general set-up. A mapping L from a cone C V into a cone K W is said to be cone linear, if L.˛x C ˇy/ D ˛Lx C ˇLy for all ˛; ˇ 0 and x; y 2 C . There exist examples, see [10], of cone linear mappings L from a closed cone C in a Banach space V into a closed cone K in a Banach space W such that L is continuous on C , but its linear extension to cl.C C / is not continuous, even if cl.C C / D V . For cone linear mappings there exists the following result, see Theorem A.7.1 in [42]. Theorem 2.11. Let C be a closed normal cone in a Banach space V , and let L W C ! C be a cone linear mapping, which is continuous at 0. If there exists an integer p 1 such that .Lp / < 1 and LpC1 .C / ¤ f0g, then L has a unique eigenvector v 2 C , with kvk D 1, such that Lv D rC .L/v, where rC .f / D lim kLk k1=k C k!1

is the cone spectral radius of L and kLk kC D supfkLk xk W x 2 C with kxk D 1g: Moreover, if we let c D tanh..Lp /=4/ < 1, then d.Lkp x; v/ c k d.x; v/ for all x 2 C n f0g:

2.2 An application to Perron–Frobenius operators In this section we briefly discuss an application of Birkhoff’s contraction ratio, or, more precisely, Theorem 2.11, to so-called Perron–Frobenius operators. These operators play a central role in the study of Hausdorff dimensions of invariant sets given by graph directed iterated function systems, see [55], [57]. For simplicity we restrict attention here to operators arising from iterated function systems rather than the more general case of graph directed iterated function systems. However, the same ideas can be applied in the graph directed case. Let .S; / be a bounded, complete metric space with positive diameter. Let Cb .S / denote the Banach space of bounded, continuous, real-valued functions, f W S ! R, with kf k D sups2S jf .s/j. Write D diam.S /, so 0 < < 1. For 0 < 1 and M > 0 define

K.M; / D ff 2 Cb .S/ W f .s/ f .t /e M.s;t / for all s; t 2 Sg:

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It can be shown, see [57], Lemma 3.2, that K.M; / is a closed normal cone in Cb .S /. By using the assumption that > 0, it is easy to verify that if f 2 K.M; /, then f .s/ 0 for all s 2 S, and f .s/ D 0 for all s 2 S whenever f .t / D 0 for some t 2 S . Furthermore for f 2 K.M; / we have sup f .s/ e M inf f .s/ : s2S

s2S

Now fix M0 > 0 and 0 < 1 and let bi 2 K.M0 ; / for i 2 N. Assume that

i W S ! S are Lipschitz mappings for i 2 N, with ²

³

. i .s/; i .t // Lip. i / D sup W s ¤ t in S c < 1: .s; t /

(2.3)

Assume, in addition, that there exists s 2 S such that X bi .s / < 1: i

Now we define a Perron–Frobenius operator , L W Cb .S / ! Cb .S /, by X .Lf /.t/ D bi .t/f . i .t // for t 2 S:

(2.4)

i

Obviously, kLf k D sup j t2S

X

bi .t/f . i .t//j

i

X

bi .s /e M0 kf k < 1

i

for all f 2 Cb .S/, and hence L is a continuous linear mapping from Cb .S / to Cb .S /. We shall now see how we can use Birkhoff’s Theorem 2.9 to prove that L has a unique eigenvector, v 2 K.M2 ; /, with kvk D 1, for all M2 > M0 =.1 c /. Here c is given in (2.3). Furthermore, for each g 2 K.M2 ; / n f0g,

Lk g v D 0: lim k k!1 kL gk

We will need the following lemma. Lemma 2.12. Let d2 denote the Hilbert metric on K.M2 ; /. If 0 < M1 < M2 , then K.M1 ; / n f0g is a part of K.M2 ; / and supfd2 .f; g/ W f; g 2 K.M1 ; / n f0gg < 1: Proof. Let 2 denote the partial ordering induced by K.M2 ; / on Cb .S /. We need to show that there exists 0 < ˛ ˇ such that ˛f 2 g 2 ˇf

for all f; g 2 K.M1 ; /:

(2.5)

As d2 .f; g/ D d2 .f; g/ for all ; > 0, we may as well assume that kf k D kgk D 1. Furthermore, by interchanging the roles of f and g in (2.5), we see that it suffices

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to prove that there exists ˛ > 0 such that ˛f 2 g for all f; g 2 K.M1 ; / with kf k D kgk D 1. Recall that ˛f 2 g is equivalent to

g.s/ ˛f .s/ .g.t/ ˛f .t//e M2 .s;t /

for all s; t 2 S:

(2.6)

Since f; g 2 K.M1 ; /, we know that

g.s/ g.t /e M1 .s;t/

and

f .s/ f .t /e M1 .s;t /

and

e M1 f .t / 1;

(2.7)

for all s; t 2 S. This implies that

e M1 g.t / 1

for all t 2 S, from which we deduce that

e M1

g.t/ e M1 f .t /

for all t 2 S:

For convenience we shall assume that

˛ < e M1 ; which ensures that g.t/ ˛f .t/ > 0 for all t 2 S . This allows us to rewrite (2.6) as g.s/ ˛f .s/ e M2 .s;t / g.t / ˛f .t/

for all s; t 2 S:

(2.8)

Now using (2.7) we see that for s; t 2 S,

g.t /e M1 .s;t / ˛f .t /e M1 .s;t / g.s/ ˛f .s/ g.t/ ˛f .t/ g.t / ˛f .t /

.g.t/=f .t//e M1 .s;t / ˛e M1 .s;t / : .g.t /=f .t // ˛

This implies for s; t 2 S that

˛.e M1 .s;t / e M1 .s;t / / g.s/ ˛f .s/ e M1 .s;t/ C g.t/ ˛f .t/ e M1 .s;t / ˛

(2.9)

Write r D M2 .s; t / , so r 0 and M1 .s; t / D r, where D M1 =M2 < 1. From (2.9) we see that (2.8) is satisfied if 0 < ˛ < e M1 , can be chosen so that

˛ e M1 ˛

e r e r e r e r

1:

(2.10)

Consider the function W Œ0; 1/ ! R given by

.r/ D

e r e r for r > 0 e r e r

and

.0/ D

2 : 1

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Observe that is continuous on Œ0; 1/ and limr!1 .r/ D 0. A simple calculation gives .1 /e .C1/r 2 C . C 1/e .1/r

0 .r/ D : .e r e r /2 By a power series expansion, for r > 0 we have .1 /e .C1/r 2 C . C 1/e .1/r 1 1 X X . C 1/j 1 r j .1 /j 1 r j 2 C < 0: D .1 / jŠ jŠ j D2

j D2

Thus, achieves its maximum on Œ0; 1/ at 0. Let D .M2 M1 /=.M2 C M1 / and assume that ˛ satisfies 0 < ˛ e M1 . By the previous remarks equation (2.10) will be satisfied if 2 1: 1 1 However, a calculation shows that the left-hand-side of the above expression actually equals 1, so that we can take ˛ D e M1 , and the proof is complete. Note that we have actually shown that for each f; g 2 K.M1 ; / with kf k D kgk D 1, we can choose

M2 M1 M1 e ˛D M2 C M1 This implies that d2 .f; g/ 2 log

and

M2 C M1 M1 ˇD e : M2 M1

M2 C M1 C 2e M1 : M2 M 1

for all f; g 2 K.M1 ; /. We shall also need the following result. Lemma 2.13. Let L W Cb .S/ ! Cb .S/ be given by (2.4), and assume that at least one bi is not identically 0. If M2 > M0 =.1 c / and M1 D M0 C c M2 , then L.K.M2 ; / n f0g/ K.M1 ; / n f0g: Proof. If f 2 K.M2 ; / n f0g, then f B i 2 K.c M2 ; /, because

f . i .s// f . i .t//e M2 .i .s/;i .t // f . i .t //e c

M .s;t / 2

for all s; t 2 S. As bi 2 K.M0 ; /, we have

bi .s/ bi .t/e M0 .s;t /

for all s; t 2 S:

Combining these two inequalities gives bi .s/f . i .s// bi .t/f . i .t //e .M0 Cc

M /.s;t / 2

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for all s; t 2 S , which shows that bi ./f . i .// 2 K.M1 ; / for all i 2 N. Since K.M1 ; / is a closed cone, we find that L.f / 2 K.M1 ; /. It remains to show that L.f / ¤ 0 for f 2 K.M2 ; / n f0g. Note that as f ¤ 0 and f 2 K.M2 ; /, we know that f .s/ > 0 for all s 2 S. As each bi 2 K.M0 ; /, bi .s/ 0 for all s 2 S. Furthermore, there exists j such that bj ¤ 0, and hence bi .t / > 0 for all t 2 S . Thus, L.f /.t/ bj .t /f . j .t // > 0 for all t 2 S , which completes the proof. We can now use Birkhoff’s Theorem 2.9 to prove the following result. Theorem 2.14. Let L W Cb .S/ ! Cb .S/ be given by (2.4), and assume that at least one bi is not identically 0. If M2 > M0 =.1 c /, then there exists a unique v 2 K.M2 ; / with kvk D 1, such that Lv D kLvkv. Furthermore, for each g 2 K.M2 ; / n f0g we have k L g v D 0: (2.11) lim k k!1 kL gk Proof. Let 2 denote the partial ordering from K.M2 ; / and denote the restriction of L to K.M2 ; / by L0 . Note that it follows from Lemmas 2.12 and 2.13 that the projective diameter .L0 / < 1. Now let g 2 K.M2 ; / n f0g. It follows from Lemma 2.13 that L0 g 2 K.M1 ; / and L0 g ¤ 0. As K.M1 ; / K.M2 ; / the same applies to L0 g. So, L20 g 2 K.M1 ; / n f0g, and hence L20 .K.M2 ; // ¤ f0g. It now follows from Theorem 2.11 that there exists a unique v 2 K.M2 ; /, with kvk D 1, such that Lv D kLvkv, and for each g 2 K.M2 ; / n f0g, Equation (2.11) holds. Remark 2.15. As Theorem 2.14 holds for all M2 > M0 =.1 c /, the unique eigenvector v in Theorem 2.14 belongs to K.M0 =.1 c /; /. It can be shown that the elements of K.M; / are Hölder continuous functions with Hölder exponent , and hence v is Hölder with Hölder exponent .

3 Special cones Particularly important examples of finite-dimensional cones arising in analysis are the standard positive cone, RnC D fx 2 Rn W xi 0 for i D 1; : : : ; ng and the Lorentz cone, ƒnC1 D f.s; x/ 2 R Rn W s 2 x12 C C xn2 and s 0g: Other interesting examples are cones of linear operators, such as the cone …n .R/ consisting of positive semi-definite n n matrices in the real vector space Symn of all

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n n symmetric matrices. The cones mentioned above are all examples of so-called symmetric cones. Recall that a cone C , with nonempty interior, in a finite-dimensional inner product space .V; h j i/ is called self-dual if C D C . Furthermore C is called homogeneous if Aut.C / D fA 2 GL.V / W A.C / D C g acts transitively on C B . The interior of a self-dual homogeneous cone is called a symmetric cone. It is well known that the symmetric cones in finite dimensions are in one-toone correspondence with the interiors of the cones of squares of Euclidean Jordan algebras. This fundamental result was proved by Koecher [32] and Vinberg [69]. A detailed exposition of the theory of symmetric cones can be found in [21]. Recall that a Euclidean Jordan algebra is a finite-dimensional real inner-product space .V; h j i/ equipped with a bilinear product .x; y/ 7! xy from V V into V such that for each x; y 2 V , (J1) xy D yx, (J2) x.x 2 y/ D x 2 .xy/, (J3) for each x 2 V , the linear mapping L.x/ W V ! V defined by L.x/y D xy satisfies hL.x/y j zi D hy j L.x/zi for all y; z 2 V: Note that a Euclidean Jordan algebra is commutative, but in general not associative. For example, Symn can be equipped with the usual inner product hA j Bi D tr.BA/ and (Jordan) product A B B D .AB C BA/=2. In this case the interior of the cone of squares in Symn is precisely …n .R/B , defined as …n .R/B D fA 2 Symn W A is positive definiteg D fA2 W A 2 Symn gB : When studying a Hilbert geometry .; ı/, it is often useful to view as the (relative) interior of the intersection of a cone in a vector space and a hyperplane H , and use Birkhoff’s version of Hilbert’s metric. In many interesting cases it gives an alternative formula to compute the distance and provides additional tools to analyze the Hilbert geometry. To illustrate this we need to recall some basic concepts. Let C be a closed cone with nonempty interior in a finite-dimensional vector space V . For u 2 C B define †u D f 2 C W .u/ D 1g, which is a compact convex set in V . Let Eu denote the closure of the set of extreme points of †u . Recall that 2 †u is an extreme point if there exist no ; 2 †u such that D C .1 / for some 0 < < 1. A basic result in convex analysis, due to Minkowski, says that each compact convex set in a finite-dimensional vector space is the closed convex hull of its extreme points, see [63], Corollary 13.5. This result, and its infinite-dimensional extension, is usually called the Krein–Milman theorem. It is easy to show that if C is a closed cone with nonempty interior and u 2 C B , then for x; y 2 V we have x C y if, and only if, .x/ .y/ for all 2 Eu . Indeed, it follows from the Krein–Milman theorem that .x/ .y/ for all 2 C is equivalent to .x/ .y/ for all 2 Eu . Now if x —C y, then y x 62 C . So, by the Hahn–Banach separation theorem, there exist ˛ 2 R and 2 V such that

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.y x/ < ˛ and .z/ > ˛ for all z 2 C . As z 2 C for all 0 and z 2 C , we see that .z/ 0 for all z 2 C , and hence 2 C . Obviously, .0/ D 0, and hence ˛ < 0. Thus, .y/ < .x/. On the other hand, x C y implies .x/ .y/ for all 2 C . So, we have the following result. Lemma 3.1. If C is a closed cone with nonempty interior in a finite-dimensional vector space, and u 2 C B , then for x 2 V and y 2 C B we have M.x=y/ D max

2Eu

.x/ .y/

and m.x=y/ D min

2Eu

.x/ : .y/

(3.1)

Remark 3.2. The identities in (3.1) also hold for closed cones in infinite-dimensional topological vector spaces. Indeed, let C be a closed cone, with nonempty interior in a topological vector space V . Given u 2 C B , one can define the order unit norm, k ku , by kxku D inff˛ > 0 W ˛u C x C ˛ug for all x 2 V . With respect to this norm each linear functional, W V ! R, with .V / Œ0; 1/, is continuous, because j.x/j .u/ for all x 2 V with kxku 1. Now let C D f W V ! R j .V / Œ0; 1/g be the dual of C in .V; k ku / and let †u D f 2 C W .u/ D 1g. Note that †u is a closed subset of the unit ball, B , of .V; k ku / . So, †u is a weak- compact set, as B is weak- compact by the Banach–Alaoglu theorem. If we now let Eu be the weak- closure of the set of extreme points of †u , then Eu is weak- compact and the equalities in (3.1) hold.

3.1 Simplicial cones A cone C in an n-dimensional vector space V is called a simplicial cone if there exist v1 ; : : : ; vn 2 V , linearly independent, such that ˚ Pn C D iD1 i vi W i 0 for all i : A basic example is the standard positive cone RnC . We can apply Lemma 3.1 to derive an explicit formula for Hilbert’s metric on .RnC /B , and use this formula to give a simple proof of the well-known fact that the Hilbert geometry on the open standard Pn x D 1g, is isometric to .n 1/-dimensional simplex, Bn1 D fx 2 .RnC /B W i iD1 an .n 1/-dimensional normed space, see [17], [52], [61]. Indeed, Lemma 3.1, or a direct simple argument, gives M.x=y/ D max xi =yi

and

i

m.x=y/ D min xj =yj j

.RnC /B .

for all x:y 2 Now consider the mapping Log W .RnC /B ! Rn , given by Log.x/ D .log x1 ; : : : ; log xn / for x 2 .RnC /B , and equip Rn with the variation norm, kwkvar D max wi min wj : i

j

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Note that log M.x=y/ D log maxi xi =yi D maxi log xi log yi . Likewise, we see that log m.x=y/ D minj log xj log yj . Thus, the mapping, x 7! Log.x/, is an isometry from ..RnC /B ; d / onto .Rn ; k kvar /. If we let H D fx 2 .RnC /B W xn D 1g, then .H; d / is a genuine metric space, which is isometric to .Bn1 ; d / by Lemma 2.1 (iv). Clearly Log.H / D fx 2 Rn W xn D 0g, which we can identify with Rn1 by projecting out the last coordinate. It follows that .Bn1 ; d / is isometric to the .n 1/-dimensional normed space .Rn1 ; k kH /, where kxkH D maxfx1 ; : : : ; xn1 ; 0g minfx1 ; : : : ; xn1 ; 0g: If we now use Theorem 2.2, we arrive at the following result. Theorem 3.3. The Hilbert geometry .Bn ; ı/ is isometric to .Rn ; k kH /. The unit ball of kkH in Rn is a polytope with n.nC1/ facets. In fact, it is a hexagon, when n D 2, which was already known to Phadke [61], and a rhombic-dodecahedron, when n D 3. If D convfv1 ; : : : ; vnC1 g is a simplex with nonempty interior in an n-dimensional vector space V , then C D f.v; / 2 V R W v 2 and 0g is a simplicial cone 0 D .vnC1 ; 1/ is a basis for V 0 and in V 0 DPV R. Note that v10 D .v1 ; 1/; : : : ; vnC1 0 C D f i i vi W i 0 for all ig. Moreover, the linear mapping A W V 0 ! RnC1 given by X X i vi0 D i ei ; A i

i

where 1 ; : : : ; nC1 2 R and e1 ; : : : ; enC1 are the standard basis vectors in RnC1 , B is invertible and maps C onto RnC1 C . So, A is an isometry from .C ; d / onto nC1 B ..RC / ; d /. By combining this observation with Theorem 3.3 it is easy to show that the Hilbert geometry on B is isometric to .Rn ; k kH /. In [22] Foertsch and Karlsson showed that the only Hilbert geometry isometric to a normed space is the one on an open simplex. See [11] for an early version. Thus, the following result holds. Theorem 3.4. An n-dimensional Hilbert geometry is isometric to a normed space if and only if its domain is an open n-dimensional simplex. In that case it is isometric to .Rn ; k kH /.

3.2 Polyhedral cones In this subsection we shall see how we can use Birkhoff’s version of Hilbert’s metric on polyhedral cones to show that the Hilbert geometry on the interior of a polytope with m facets can be isometrically embedded into the normed space .Rm.m1/=2 ; kk1 /, where kzk1 D maxi jzi j is the supremum norm. It turns out the polytopal Hilbert geometries are the only ones that can be isometrically embedded into a finite-dimensional normed space, see [6], [15], [16], [21]. To prove these results we need to recall some basic concepts concerning polyhedral cones.

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Recall that a closed cone C in Rn is called a polyhedral cone if it is the intersection of finitely many closed half-spaces, i.e., if there exist finitely many linear functionals 1 ; : : : ; k such that C D fx 2 Rn W i .x/ 0 for all i g. A subset F of a polyhedral cone C is called an (exposed) face if there exists a linear functional 2 C such that F D fx 2 C W .x/ D 0g. A face F of C is called a facet if dim F D dim C 1. Furthermore, it is a well-known fact from polyhedral geometry, that if C is a polyhedral cone in Rn with m facets and C B is nonempty, then there exist m linear functional 1 ; : : : ; m such that C D fx 2 Rn W

i .x/

0 for all i g:

and each i defines a facet of C . The facet defining functionals, 1 ; : : : ; m , of a polyhedral cone C correspond to the extreme rays (1-dimensional faces) of the dual cone C of C . If we take u 2 C B , then we can normalize each facet defining functional i so that i .u/ D 1. In that case we have Eu D f 1 ; : : : ; m g. Now using Lemma 3.1 we can show the following result regarding isometric embeddings from [53]. Theorem 3.5. If P is an n-dimensional polytope in Rn with m facets, then the Hilbert geometry on P B can be isometrically embedded into .Rm.m1/=2 ; k k1 /. Proof. Let C D f.x; / 2 Rn R W x 2 P and 0g. Then C is a polyhedral cone in RnC1 D Rn R with nonempty interior and m facets. Denote the facet defining functionals of C by 1 ; : : : ; m . Furthermore, let †B D f.x; s/ 2 C B W s D 1g. So, by Theorem 2.2 the Hilbert geometry .P B ; ı/ is isometric to .†B ; d /. Now define ‰ W †B ! Rm.m1/=2 by ‰ij .x/ D log

i .x/ j .x/

for all 1 i < j m and x 2 †B :

It follows from Lemma 3.1 that M.x=y/ D maxi i .x/= i .y/ for all x; y 2 †B . So, d.x; y/ D log M.x=y/M.y=x/ i .x/ j .y/ D log max i;j i .y/ j .x/ i .x/ i .y/ log D max log i;j j .x/ j .y/ ˇ ˇ ˇ i .x/ i .y/ ˇˇ ˇ log D max ˇ log ˇ; 1i<j m j .x/ j .y/ which shows that d.x; y/ D k‰.x/ ‰.y/k1 for all x; y 2 †B . Thus, ‰ is an isometry of .†B ; d / into .Rm.m1/=2 ; k k1 /. The following result was essentially proved by Foertsch and Karlsson [21]. Theorem 3.6. A Hilbert geometry embeds isometrically into a finite-dimensional normed space if and only if its domain is the interior of a polytope.

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Proof. By the previous theorem it remains to show that a non-polytopal Hilbert geometry cannot be isometrically embedded into any finite-dimensional normed space. For the sake of contradiction, assume that .; ı/ is a non-polytopal Hilbert geometry and that h is an isometry from .; ı/ into a finite-dimensional normed space .V; k k/. The following result by Karlsson and Noskov [31] concerning the behavior of the Gromov product, 2.x j y/p D ı.x; p/ C ı.y; p/ ı.x; y/; will be useful. Suppose that p 2 and .xk / and .yk / are sequences in such that xk ! x 2 @, yk ! y 2 @, and the straight line segment Œx; y 6 @, then there exists a constant R > 0 such that lim sup .xk j yk /p R: k!1

This result will be used to prove the following claim, from which the contradiction will be derived. Claim. If there exist 1 ; : : : ; m 2 @ such that the straight line segment Œi ; j 6 @ for all i ¤ j , then there exist v1 ; : : : ; vm 2 V with kvi k D 1 for all i and kvi vj k 2 for all i ¤ j . To prove the claim let p 2 be fixed. Obviously the mapping x 7! h.x/ h.p/ is also an isometric embedding of .; ı/ into .V; k k/. So, we may as well assume that h.p/ D 0. Now for i D 1; : : : ; m and 0 t < 1 let zi .t / D .1 t /p C t i . Note that the mapping t 7! ı.p; zi .t// is continuous and lim t !1 ı.p; zi .t // D 1. So, for each i and n 2 N there exists 0 < ti;n < 1 such that ı.p; zi .ti;n // D n. For simplicity we write zin D zi .ti;n /. From the result by Karlsson and Noskov it follows that there exists a constant M > 0 such that ı.zin ; zjn / 2n M for all i ¤ j and n N0 , where N0 is a sufficiently large integer. Define uni D n1 h.zin /. Then for each i ¤ j and n N0 we have M 1 1 kuni ujn k D ı.zin ; zjn / .2n M / D 2 : n n n But also kuni k D n1 kh.zin / h.p/k D n1 ı.p; zin / D 1. As the unit sphere in V is n compact, we can find a subsequence .ui k / converging to some vi 2 V for each i . Clearly the limits v1 ; : : : ; vm satisfy kvi k D 1 for all i , and kvi vj k 2 for all i ¤ j , which completes the proof of the claim. To obtain a contradiction we will now show that there exist infinitely many points 1 ; 2 ; : : : 2 @ such that Œi ; j 6 @ for all i ¤ j , which, by the claim, violates the compactness of the unit sphere in V . Without loss of generality we may assume that is contained in a finite-dimensional x of is not a polytope, the polar of , x vector space W and 0 2 . As the closure, , x x D f 2 W W .x/ 1 for all x 2 g;

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x has infinitely many extreme points. As the exposed is also not a polytope. Thus, points are dense in the extreme points, see [67], there exist infinitely many exposed x . Now using the fact that x D x we can find 1 ; 2 ; : : : 2 points 1 ; 2 ; : : : 2 @ x n fi g. Clearly, @ such that i .i / D 1 for all i , and .i / < 1 for all 2 if i ¤ j and 0 < t < 1, then .1 t /i C t j 2 , as ..1 t /i C t j / D x and 0 < t < 1. Thus, the segment .1 t /.i / C t .j / < 1 for all 2 Œi ; j 6 @ for all i ¤ j , which completes the proof. Theorem 3.6 has been strengthened by Colbois andVerovic [15] to quasi-isometries, and by Bernig [6] and Colbois, Vernicos and Verovic [16] to bi-Lipschitz mappings.

3.3 Symmetric cones Let C B be a symmetric cone in .V; h j i/, and let e 2 C B denote the unit in the associated Euclidean Jordan algebra on V . An element c 2 V is called an idempotent if c 2 D c. It is said to be a primitive idempotent if c cannot be written as the sum of two non-zero idempotents. The set of all primitive idempotents in V is denoted by J.V /. A set fc1 ; : : : ; ck g is called a complete system of orthogonal idempotents if (1) ci2 D ci for all i, (2) ci cj D 0 for all i ¤ j , (3) c1 C C ck D e. The spectral theorem ([21], Theorem III.1.1) says that for each x 2 V there exist unique real numbers 1 ; : : : ; k , all distinct, and a complete system of orthogonal idempotents c1 ; : : : ; ck such that x D 1 c1 C C k ck : The numbers i are called the eigenvalues of x. The spectrum of x is denoted by .x/ D f W eigenvalue of xg; and we write C .x/ D maxf W 2 .x/g and

.x/ D minf W 2 .x/g:

It can be shown, see Theorem III.2.1 of [21], that x 2 C B if and only if .x/ .0; 1/. So, one can use the spectral decomposition, x D 1 c1 C C k ck , of x 2 C B , to define the unique square root of x by p p x 1=2 D 1 c1 C C k ck : Similarly, the functions x 7! log x and x 7! x t for t 2 R can be defined on C B . For x 2 V , the linear mapping P .x/ D 2L.x/2 L.x 2 /

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is called the quadratic representation of x. (Recall that L.x/ W V ! V is the linear map given by L.x/y D xy.) Note that P .x 1=2 /x D e for all x 2 C B . It can also be shown that P .x 1 / D P .x/1 for all x 2 C B . In the example of the Euclidean Jordan algebra on Symn the reader can verify that P .A/B D ABA. It is known, see [21], Proposition III.2.2, that if x 2 C B , then P .x/ 2 Aut.C /, and hence M.P .x/w=P .x/y/ D M.w=y/ and m.P .x/w=P .x/y/ D m.w=y/ for all w 2 V and x; y 2 C B . For w 2 V and x 2 C B we write C .w; x/ D C .P .x 1=2 /w/

and

.w; x/ D .P .x 1=2 /w/:

The following formula for Hilbert’s metric on symmetric cones was derived by Koufany [34]. Theorem 3.7. If V is a Euclidean Jordan algebra with symmetric cone C B , then for w 2 V and x 2 C B , M.w=x/ D C .w; x/ and m.w=x/ D .w; x/: In particular, we have for x; y 2 C B ,

C .x; y/ : d.x; y/ D log .x; y/ Proof. Let z D P .x 1=2 /w 2 V and let z D 1 c1 C Ck ck be the spectral decomposition of z. Note that w C ˇx is equivalent to z D P .x 1=2 /w ˇP .x 1=2 /x D ˇe, since P .x 1=2 / 2 Aut.C /. As e D c1 C C ck , this inequality holds if and only if 0 C .ˇ 1 /c1 C C.ˇ k /ck , which is equivalent to ˇ C .z/ D C .w; x/. As M.w=x/ D inffˇ 2 R W w C ˇxg, we deduce that M.w=x/ D C .w; x/. In the same way it can be shown that m.w=x/ D .w; x/. In the example of the Euclidean Jordan algebra on Symn we find for A; B 2 …n .R/ that C .A; B/ D C .P .B 1=2 /A/ D maxf W 2 .B 1=2 AB 1=2 /g D maxf W 2 .B 1 A/g and .A; B/ D minf W 2 .B 1 A/g: Remark 3.8. If we combine Lemma 3.1 and Theorem 3.7 we find for x 2 V and y in a symmetric cone C B that C .x; y/ D max c2Ee

hx j ci hy j ci

and

.x; y/ D min

c2Ee

hx j ci ; hy j ci

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where Ee is the set of extreme points of †e D fx 2 C W hx j ei D 1g. It is known, see [21], Proposition IV.3.2, that Ee D fx 2 †e W x is a primitive idempotentg:

So, C .x; y/ D

max

c2†e \J.V /

hx j ci hy j ci

and

.x; y/ D

min

c2†e \J.V /

hx j ci : hy j ci

These equalities are closely related to the min-max characterization of the eigenvalues of the elements of a Euclidean Jordan algebra by Hirzebruch [26].

4 Non-expansive mappings on Hilbert geometries Many interesting examples of non-expansive mappings on Hilbert geometries arise as normalizations of order-preserving, homogeneous mappings on cones. For example, Bellman operators in Markov decision processes and Shapley operators in stochastic games are order-preserving and homogeneous mappings on RnC after a change of variables, see [5], [7], [51]. These mappings, f W RnC ! RnC , are of the form: n Y

fi .x/ D inf sup ri .˛; ˇ/ ˛2Ai ˇ 2B i

RnC .

P

j D1

p .˛;ˇ /

xj j

;

Here ri .˛; ˇ/ 0, j pj .˛; ˇ/ D 1 and 0 pj .˛; ˇ/ for 1 i n and x 2 1 for all ˛, ˇ, i , and j . The iterates of these operators are used to compute the value of Markov decision processes and stochastic games. Other interesting examples of nonlinear order-preserving mappings on cones are so-called decimation-reproduction operators in the analysis of fractal diffusions [46], [50], and DAD-operators in matrix scaling problems, see [49], [56]. In many applications it is important to understand the iterative behavior of such mappings, f W C B ! C B and of the normalized mappings, g W †B ! †B , given by g.x/ D

f .x/ .f .x//

for x 2 †B D fx 2 C B W .x/ D 1g;

where 2 .C /B . The fact that these mappings are non-expansive with respect to Hilbert’s metric is a very useful tool to analyze their dynamics. In the analysis of the dynamics of a non-expansive mapping f on a Hilbert geometry .; ı/ it is important to distinguish two cases: (1) f has a fixed point in , and (2) f does not have a fixed point in . In the first case, the limit points of each orbit of f lie inside , whereas in the second case all the limit points of each orbit of f lie inside @ by a result of Całka [14]. In the next subsection we will consider the first case.

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4.1 Periodic orbits Before we get started we recall some basic notions from the theory of dynamical systems. A point w 2 is called a periodic point of f W ! if the exists an integer p 1 such that f p .w/ D w. The smallest such p 1 is called the period of w. In particular, w 2 is a fixed point if f .w/ D w. The orbit of x 2 is given by O.x/ D ff k .x/ W k D 0; 1; 2; : : :g. We say that the orbit of x 2 converges to a periodic orbit if there exists a periodic point w of f with period p such that limk!1 f kp .x/ D w. For x 2 we define the !-limit set by x W f ki .x/ ! y for some subsequence ki ! 1g: !.xI f / D fy 2 Note that we allow the limit point y to be in @ even though f need not be defined there. The following result will play an important role. Theorem 4.1. If f W X ! X is a non-expansive mapping on a closed subset X of .Rn ; kk1 / and f has a fixed point in X , then every orbit of f converges to a periodic orbit whose period does not exceed maxk 2k kn . A proof and a discussion of the history of this result can be found in Chapter 4 of [42]. The upper bound given in Theorem 4.1 is currently the strongest known and was obtained by Lemmens and Scheutzow in [40]. It was conjectured by Nussbaum in [53] that the optimal upper bound is 2n , but at present this has been confirmed for n D 1; 2 and 3 only, see [48]. Combining Theorem 4.1 with the isometric embedding result in Theorem 3.5 we obtain the following corollary for non-expansive mappings on polytopal Hilbert geometries. Corollary 4.2. If .; ı/ is a polytopal Hilbert geometry with m facets and f W D ! D is a non-expansive mapping on a closed subset D of .; ı/ with a fixed point, then each orbit of f converges to a periodic orbit whose period does not exceed maxk 2k Nk , where N D m.m 1/=2. It is an interesting open problem to find the optimal upper bound for the possible periods of periodic points of non-expansive mappings on .; ı/ in case is the interior of an n-dimensional simplex. For the 2-simplex, it was shown in [39] that 6 is the optimal upper bound. It is believed that there exists a constant c > 2 such that the periods do not exceed c n if is an n-simplex, but this appears to be hard to prove. Remark 4.3. Corollary 4.2 has the following interesting geometric consequence: It is impossible to isometrically embed a Euclidean plane into any polytopal Hilbert geometry, as it is impossible to isometrically embed a rotation under irrational angle in such Hilbert geometries. Thus, the Euclidean rank of a poytopal Hilbert geometry is 1. This observation complements results by Bletz-Siebert and Foertsch [9], who conjectured that the Euclidean rank of any Hilbert geometry is 1.

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The following, more detailed, results exist for the possible periods of periodic points for order-preserving homogeneous mappings on polyhedral cones, see [1], [40], [41] and [42], Chapter 8. Theorem 4.4. If f W C ! C is a continuous, order-preserving, homogeneous mapping on a polyhedral cone C with nonempty interior and m facets, then the following assertions hold: 1. Every norm bounded orbit of f converges to a periodic orbit whose period does not exceed mŠ ; m mC1 b 3 cŠb 3 cŠb mC2 cŠ 3 where brc denotes the greatest integer q r. 2. In case C is an n-dimensional simplicial cone the set of possible periods of periodic points of f is precisely the set of integers k p for which there exist integers , and 1 q2 kn for some q1 and q2 such that p D q1 q2 , 1 q1 bk=2c 0 k n. 3. If f .v/ D v for some > 0 and v 2 C B , then each orbit of the normalized mapping g W †B ! †B given by g.x/ D

f .x/ .f .x//

for x 2 †B

converges to a periodic orbit whose period does not exceed the upper bound is sharp in case C is a simplicial cone.

m bm=2c

. Moreover,

In particular, we see that on the cone R3C , the set of possible periods of periodic points of order-preserving homogeneous mappings f W R3C ! R3C is f1; 2; 3; 4; 6g. So, it is impossible to have a period-5 point in that case. An example of a mapping on R3C with a period-6 orbit is the mapping f W R3C ! R3C given by 1 0 1 0 x1 .3x1 ^ x2 / _ .3x2 ^ x3 / f @ x2 A D @ .3x1 ^ x3 / _ .3x3 ^ x2 / A for x 2 R3C x3 .3x2 ^ x1 / _ .3x3 ^ x1 / which has x D .1; 2; 0/ as a period-6 point. Here a ^ b D minfa; bg and a _ b D maxfa; bg for a; b 2 R. In view of Corollary 4.2 it is interesting to ask the following question. For which Hilbert geometries .; ı/ do we have that the orbits of each non-expansive mapping f W ! with a fixed point in converge to periodic orbits? Obviously the answer is negative if is the interior of an ellipsoid. However, the following was shown in [39]. Theorem 4.5. If .; ı/ is a strictly convex Hilbert geometry and there exists no 2dimensional affine plane H such that H \ is the interior of an ellipsoid, then every

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orbit of a non-expansive mapping f W ! , with a fixed point in , converges to a periodic orbit. In fact, there exists an integer q 1 such that limk!1 f kq .x/ exists for all x 2 .

4.2 Denjoy–Wolff type theorems In this subsection we briefly discuss the behaviour of fixed point free non-expansive mappings on Hilbert geometries. A more detailed overview of this topic is given by A. Karlsson in Chapter 9 of this volume [30]. Recall that if f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/, then the attractor of f , [ Af D !.xI f /; x2

is contained in @ by Całka’s result [14]. In that case it is interesting to understand the structure of Af in @. This problem was considered by Beardon in [3], [4]. He showed that there is a striking resemblance between the dynamics of fixed point free mappings on Hilbert geometries and the dynamics of fixed point free analytic selfmappings of the open unit disc in the complex plane, which is characterized by the classical Denjoy–Wolff theorem [18], [71], [72]. Theorem 4.6 (Denjoy–Wolff). If f W D ! D is a fixed point free analytic mapping on the open unit disc D in C, then there exists a unique 2 @D such that limk!1 f k .x/ D for all x 2 D, and the convergence is uniform on compact subsets of D. Analytic self-mappings of the open unit disc are non-expansive under the Poincaré metric by the Schwarz–Pick lemma. In [4] Beardon noted that the Denjoy–Wolff theorem should be viewed as a result in geometry, as it essentially only depends on the hyperbolic properties of the Poincaré metric on D. In fact, he showed that the Denjoy– Wolff theorem can be generalized to fixed point free non-expansive mappings on metric spaces that possess sufficient “hyperbolic” properties. As a particular consequence of his results he obtained in [4] the following Denjoy–Wolff theorem result for strictly convex Hilbert geometries. Theorem 4.7. If f W ! is a fixed point free non-expansive mapping on a strictly convex Hilbert geometry .; ı/, then there exists a unique 2 @ such that limk!1 f k .x/ D for all x 2 , and the convergence is uniform on compact subsets of . Beardon’s arguments were sharpened by Karlsson in [29]. It is known that Beardon’s result does not hold for general Hilbert geometries. In fact, for the Hilbert geometry on the open 2-simplex, 2 , Lins [45] showed that if S is a convex subset of

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@2 , then there exists a fixed point free non-expansive mapping f W B2 ! B2 such that [ !.xI f /: SD x2

It was conjectured, however, by Karlsson and Nussbaum that the following is true, see [56]. Conjecture (Karlsson–Nusbaum). If .; ı/ is a Hilbert geometry and f W ! is a fixed point free non-expansive mapping, then there exists a convex set ƒ in @ such that !.x; f / ƒ for all x 2 It turns out that to prove the conjecture it suffices to show that there exists x 2 such that the convex hull of !.xI f / is contained in @, see [56]. At present there exist only partial results for Conjecture 4.2. To begin there exists the following result by Lins [45]. Theorem 4.8. If .; ı/ is a polytopal Hilbert geometry and f W ! is a fixed point free non-expansive mapping, then there exists ƒ @ such that !.xI f / ƒ for all x 2 . To prove this theorem Lins used the fact that a polytopal Hilbert geometry can be isometrically embedded into a finite-dimensional normed space. This property allowed him to show that for each x 2 there exists a horofunction h W ! R such that limk!1 h.f k .x// D 1. For general Hilbert geometries such a horofunction does not always exist, see Remark 3.2 of [44]. So, there seems to be no apparent way to generalize Lins’ arguments to the general case. The following partial result for Conjecture 4.2 is due Karlsson [29]. Theorem 4.9. If f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/ and there exists z 2 such that ı.f k .z/; z/ > 0; k k!1 then there exists a convex ƒ @ such that !.xI f / ƒ for all x 2 . lim

In [31] Karlsson and Noskov showed the following result, which says that the attractor Af of a fixed point free non-expansive mapping f on a Hilbert geometry .; ı/ is a star-shaped subset of @. Theorem 4.10. If f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/, then there exists 2 @ such that for y 2 Af the straight line segment Œy; is contained in @. The following counterpart to Theorem 4.9 was proved by Nussbaum [56].

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Theorem 4.11. If f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/ and there exists z 2 such that lim inf ı.f kC1 .z/; f k .z// D 0; k!1

then there exists a convex ƒ @ such that !.xI f / ƒ for all x 2 . Other Denjoy–Wolff type theorems for finite and infinite-dimensional Hilbert geometries can be found in [24], [42], [46], [56]. Despite numerous efforts the KarlssonNussbaum conjecture remains one of the most outstanding problems in the field.

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[10] F. F. Bonsall, Linear operators in complete positive cones. Proc. London Math. Soc. 8 (1958), no. 3, 53–75. [11] H. Busemann and B. B. Phadke, A general version of Beltrami’s theorem in the large. Pacific J. Math. 115 (1984), 299–315. [12] P. J. Bushell, Hilbert’s projective metric and positive contraction mappings in a Banach space. Arch. Ration. Mech. Anal. 52 (1973), 330–338. [13] P. J. Bushell, On the projective contraction ratio for positive linear mappings. J. London Math. Soc. 6 (1973), 256–258. [14] A. Całka, On a condition under which isometries have bounded orbits. Colloq. Math. 48 (1984), 219–227. [15] B. Colbois and P. Verovic, Hilbert domains that admit a quasi-isometric embedding into Euclidean space. Adv. Geom. 11 (2011), no. 3, 465–470.

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Part III

Developments and applications

Chapter 11

Convex real projective structures and Hilbert metrics Inkang Kim and Athanase Papadopoulos

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to affine spheres . . . . . . . . . . . . . . . . . . . . . . . . . . Parametrizations of real projective structures and the deformation spaces 4.1 Goldman’s parametrization . . . . . . . . . . . . . . . . . . . . . 4.2 Hitchin’s parametrization . . . . . . . . . . . . . . . . . . . . . . 4.3 Length spectra as parameters . . . . . . . . . . . . . . . . . . . . . 5 Strictly convex manifolds and topological entropy . . . . . . . . . . . . 6 Geodesic currents on strictly convex surfaces . . . . . . . . . . . . . . . 7 Compactification of the deformation space of convex real projective structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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307 309 315 319 320 325 326 327 329

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1 Introduction In what follows, RP n is the n-dimensional real projective space, that is, the set of lines through the origin in RnC1 , and An the n-dimensional affine space, considered as the complement of a hyperplane in RP n . A subset of An is convex if its intersection with each affine line is connected. Let be an open convex subset of An . Then is equipped with a canonical metric, called the Hilbert metric, whose definition we recall in Section 2 below. This metric is invariant by the group of projective transformations of RP n which preserve . Several interesting phenomena were discovered and several good questions arose recently concerning that metric. It is non-Riemannian except in the case where is an ellipsoid, but it shares several properties with the hyperbolic metric (that is, a Riemannian metric of constant negative curvature), especially if is strictly convex. There are also many differences. In the next sections, we will highlight some of these analogies and differences. We start by recalling very classical questions concerning convex sets equipped with their Hilbert geometry. From the observation that the boundary of the unit disk

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is a smooth circle, it was natural to ask whether there exist convex domains in the projective plane with less regular boundary which admit compact quotients by discrete subgroups of the automorphism group of the domain. This question is natural because the unit disk admits such quotients. This existence question was already addressed by Ehresmann in the 1930s [18], and it was answered affirmatively thirty years later by Kac and Vinberg [25]. In modern language, a convex domain in the projective plane which admits a cocompact action is termed divisible. An ellipse in RP 2 or an ellipsoid in RP 3 are well-known examples of convex domains. Their compact quotients are the 2- and 3-dimensional compact hyperbolic manifolds. The theory of hyperbolic manifolds (their construction, classification, their deformation and moduli spaces and their rigidity properties) was born in the works of Klein and Poincaré in the 1880s. This theory rapidly evolved into one of the most beautiful geometric theories that were developed during the twentieth century. Important questions concerning more general convex projective manifolds (quotients of divisible convex domains) were addressed by Benzécri in his thesis [6] (1960). This thesis contains some foundational work on the subject. To find conditions under which a convex domain is divisible is a difficult matter, as is the general question of existence of lattices in semisimple Lie groups. In most cases, the boundary of such a domain is nowhere analytic. Benzécri proved that if the boundary is C 2 , then the original convex set is an ellipsoid, which makes it identified with the hyperbolic space, more precisely, with the Cayley–Klein–Beltrami model of that space. It also follows from the work of Benzécri and from later works that the boundary @ of a divisible convex domain is either a conic (in which case the projective structure arises from a hyperbolic structure) or this boundary is nowhere C 1C for some > 0. Explicit examples of quotients of projective manifolds obtained by reflections along convex polyhedra in the sphere were obtained by Vinberg in the early 1970s. These constructions may be considered as projective analogues of Poincaré’s examples of hyperbolic manifolds that use reflections along convex polyhedra in hyperbolic space. A convex real projective manifold is the quotient of a convex set RP n by a discrete group of projective transformations, and the structure is termed strictly convex if is strictly convex. The moduli space of convex projective structures generalizes the Teichmüller spaces of hyperbolic surfaces. There are many interesting recent developments on convex projective manifolds and their deformations. Hitchin [23] discovered a component in the space of representations of fundamental groups of surfaces in SL.n; R/, for every n 2, which contains Teichmüller space, and whose elements have several properties in common with Teichmüller space. He called these components Teichmüller components, and today such a component is called a Hitchin component. In the case n D 3, such components consist of real projective structures on surfaces. Johnson and Millson [20] showed that there are non-trivial continuous deformations of higher-dimensional hyperbolic structures through strictly convex projective structures. Conversely, one might expect that if a compact manifold admits a strictly convex projective structure, then it is a deformation of a hyperbolic structure. But Benoist [3] constructed in dimension four an example of a manifold

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which admits a strictly convex real projective structure but no hyperbolic structure. Later, Kapovich [26] generalized the method to show that such examples exist in any dimension 4. In this chapter, we study some relations between hyperbolic geometry and differential projective geometry. Some of the natural questions that appear in this setting are the following. • To what extent does Hilbert geometry generalize hyperbolic geometry? • What are the relations and the common properties between spaces of deformations of convex projective structures and Teichmüller spaces? • Are there compactifications of deformation spaces of convex projective manifolds that are analogous to compactifications of Teichmüller spaces? • How does the hyperbolic behavior of geodesic flows of Hilbert manifolds generalize the hyperbolic behavior (and in particular the Anosov theory) of negatively curved Riemannian manifolds? The exposition is by no means complete and thorough. In some cases, we just record the results together with some references known to us, hoping to arouse the reader’s interest in these questions. The hyperbolic-geometry aspects of Hilbert geometry, from the dynamical point of view, are more thoroughly discussed in Chapter 7 of this volume [17]. We would like to thank Sarah Bray, Bill Goldman, Ludovic Marquis and Marc Troyanov for valuable comments and corrections on a preliminary version of this chapter.

2 Preliminaries We recall some basic notions of projective geometry on manifolds. The automorphism group of RP n , denoted by PGL.n; R/ – the group of projective transformations – is the quotient of the linear group GL.n C 1; R/ by the action of the nonzero scalar transformations. It is sometimes convenient to work on the sphere Sn , which is a double-sheeted cover of RP n . The sphere is equipped with the induced projective structure whose automorphism group (also called the projective automorphism group) is the group SL˙ .n C 1; R/ of real .n C 1/ .n C 1/ matrices of determinant ˙1. This is also the group of volume-preserving affine transformations of RnC1 . The projective lines (or, more simply, the lines) in RP n are the projections of the great circles of the sphere Sn by the canonical map Sn ! RP n . This set of lines is preserved by the group of projective transformations. The projective lines play simultaneously the role of lines of a geometry defined in the axiomatic sense, and the role of geodesics for the Hilbert metric, as we shall recall below.

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Let be a subset of RP n . We say that is convex if the intersection of with any line of RP n is connected. We say that is a properly convex (or a proper convex) subset of RP n if it is convex and contained in the complement of a hyperplane. An affine patch in RP n is the complement of a hyperplane. An affine patch can be seen as the affine n-dimensional space An . Let be a properly convex subset of RP n . The convexity of as a subset of projective space (that is, the intersection of with each projective line is connected) is equivalent to the convexity of as a subset of affine space (that is, the intersection of with each affine line is connected). Let be a properly convex subset of RP n contained in an affine patch A. We equip A with a Euclidean norm j j. We shall also denote by j j the associated Euclidean distance. The space A plays now the role of a Euclidean space Rn in which sits. We present a brief summary of some notions associated to the Hilbert geometry of that we shall use in this chapter. For x ¤ y 2 , let p, q be the intersection points of the line xy with @ such that p, x, y, q are in this order. The Hilbert distance between x and y is defined by d .x; y/ D

jp yjjq xj 1 log : 2 jp xjjq yj

The value of d .x; y/ does not depend on the choice of the Euclidean metric j j on An . For x D y, we set d .x; y/ D 0. This metric coincides with the familiar hyperbolic metric is the case where @ is an ellipse (in dimension 2) or ellipsoid (in dimension 3). This metric of the ellipse or ellipsoid is the so-called “projective model”, or Cayley–Klein–Beltrami model of hyperbolic geometry. The Hilbert metric is Finsler, and it is not Riemannian unless @ is an ellipsoid. The Finsler norm is given, for x 2 and a vector v in the tangent space of at x, by kvkx D

1 1 1 C jvj 2 jx p j jx p C j

(2.1)

where p ˙ are the intersection points with @ of the oriented line in defined by the vector v based at x and where j j is our chosen norm on the affine patch. This norm is reversible, that is, it satisfies kvkx D k vkx . The Finsler metric associated to this Finsler norm is the metric on defined by taking the distance between two arbitrary points to be the infimum of the lengths of C 1 paths joining them, where the length of a path is defined by integrating the norms of tangent vectors using Formula (2.1). It is an easy exercise to show that this Finsler metric on is the Hilbert metric. The (projective) automorphism group of is the group of projective transformations of RnC1 that preserve . In other words, we have Aut./ D ff 2 PGL.n; R/ j f ./ D /g: The Hilbert metric of is invariant by the group Aut./. A recent result by Walsh described in Chapter 5 of this volume ([50]) says that the isometry group Aut.; d /

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of the Hilbert metric d coincides with the group Aut./ except in a few special cases where Aut.; d / is an order-two extension of Aut./. We now associate a Borel measure on using the Finsler structure. The construction is analogous to the one used in Riemannian geometry. We follow the presentation of Marquis in [40]. Let Vol be a Lebesgue measure on A normalized by Vol.fv 2 A W jvj < 1g/ D 1. We define a measure on by setting, for each Borel subset A A, Z d Vol.x/ ; .A/ D A Vol.Bx .1// where Bx .1/ D fv 2 Tx W kvkx < 1g, the norm k k being the one given by (2.1). This measure turns out to be the Hausdorff measure induced by the Hilbert metric [9]. In particular, it is independent of the choice of the Euclidean norm of A. It is called the Busemann volume, and also the Hilbert volume. It is invariant by the action of Aut./. From the definition, we have the following, for any two convex domains 1 2 of Rn : 1 2 (1) kvk x kvkx for every tangent vector v based at a point x in 1 ;

(2) d2 .x; y/ d1 .x; y/ for every x and y in 1 ; (3) Bx1 .1/ Bx2 .1/ for every x in 1 ; (4) 2 .A/ 1 .A/ for any Borel set A in 1 . For more details on the notion of measure associated to a general Finsler metric, we refer the reader to [1] and [9]. We now consider projective structures on manifolds. Definition 2.1. A real projective structure on an n-dimensional differentiable manifold M is a maximal atlas with values in the n-dimensional projective space RP n whose transition functions are restrictions of projective automorphisms of RP n . Equipped with such a structure, the manifold M becomes a real projective manifold. We shall sometimes delete the adjective “real” since all the projective structures we consider in this chapter are real. An isomorphism between two n-dimensional projective manifolds is a homeomorphism between the underlying manifolds which, in each projective chart, is locally the restriction of an element of the projective transformation group of RP n . To each real projective structure is associated, by the general theory of G-structures of Ehresmann [18], a developing map and a holonomy homomorphism. The developing map arises from the attempt to define a global chart for the structure. It is constructed by starting with a coordinate chart around a given point and trying to extend it by analytic continuation. There is an obstruction for doing so if the manifold is not simply connected, and in fact, the only obstruction is the fundamental group of M . (When we come back to the point we started with, by analytically continuing

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along a nontrivial path, we generally end up with a different germ of a chart into RP n than the one we started with.) Thus, instead of obtaining a map from our manifold M to RP n , we end up with a map z ! RP n ; dev W M z is the universal covering of M . The holonomy homomorphism is an injective where M homomorphism hol W 1 .M / ! PSL.n C 1; R/ satisfying dev B D hol. / B dev: This developing map is completely determined by its restriction to an arbitrary open subset of M . When we change the initial coordinate chart in the above construction, the resulting map differs from the previous one by post-composition by an element of PGL.n; R/. The result is that although the developing map and the holonomy homomorphism depend on some choices (namely, the choice of the initial chart), there are nice transformation formulae relating the various maps and homomorphisms obtained. In particular, the holonomy homomorphism is well defined up to a conjugation by an element of PGL.n; R/. In the case where the developing map is a homeomorphism onto its image, we can z /=hol.1 .M //. We refer the reader to the paper by Ehresmann [18] write M D dev.M for the general theory of developing maps and holonomy representations associated to geometric structures. Benzécri [6], Kuiper [32] and subsequently Koszul [30] considered thoroughly the case of real projective structures. Thurston, starting in the 1970s, included this theory as an important part of the general theory of geometrization of low-dimensional manifolds, see [48]. Goldman [21], motivated by ideas of Thurston, developed the theory of moduli spaces of projective structures on surfaces. In their paper [47], Sullivan and Thurston give an example of a projective structure on the torus whose developing map is not a covering of projective space. Talking about the sources of differential projective geometry, one has also mention the work of Chern on the Gauss–Bonnet formula and characteristic classes that motivated several later works [13]. In what follows, we shall use the more restrictive notion of convex real projective structure. This is the case where the developing map sends homeomorphically the z of M onto a convex subset of some Rn sitting in RP n as the comuniversal cover M plement of an affine hyperplane. It turns out that such a structure is more manageable than a general real projective structure, and in particular, one can introduce the Hilbert metric into the playground. There is a lot of classical and more recent activity on convex projective structures and their deformation spaces. In what follows, we shall use the terminology convex to mean properly convex. There is a characterization of such a structure, which we can also take as a definition:

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Definition 2.2. A convex real projective manifold M is an object of the form = where is a convex domain in RP n containing no projective line and a discrete group of the projective automorphism group Aut./ of . We call M strictly convex if is strictly convex i.e., if is convex and @ does not contain any line segment. The relation between Definitions 2.1 and 2.2 is made through the natural identifications between and the universal cover of M and between and the fundamental group of M . In dimension two, we have more knowledge about convex projective structures, and in particular there is the following in Goldman [21] (Proposition 3.1): Proposition 2.3. Let M be a projective structure on a surface. Then, the following three properties are equivalent: (1) M is projectively equivalent to a quotient = where is a convex open subset of RP 2 and a discrete group of projective transformations of RP 2 which acts freely and properly discontinuously; z ! RP 2 is a diffeomorphism onto a convex subset (2) the developing map dev W M 2 of RP ; (3) every path in M is homotopic relative endpoints to a geodesic path (that is, a path which in coordinate charts is contained in a line of RP 2 ). We shall be interested in quotients of convex sets, and we make right now the following definition: Definition 2.4. An open (properly) convex set is said to be divisible if there exists a discrete subgroup Aut./ such that = is compact. There is a natural identification between two convex real projective manifolds 1 = 1 and 2 =2 , defined by the condition that there exists a projective transformation g of RP n such that g.1 / D 2 and g1 g 1 D 2 . This equivalence relation is used in the definition of the moduli space and the deformation space of convex real projective structures. We are mainly interested in the deformation space of convex projective structures on surfaces, which, like the Teichmüller space of hyperbolic (or of conformal) structures, is a space of equivalence classes of marked convex projective structures. We recall the concept of marking, which originates in Teichmüller theory. We start with a fixed topological surface S0 . Definition 2.5. A marked convex projective structure on S0 is a pair .f; S / where S is a surface homeomorphic to S0 equipped with a convex projective structure, and f W S0 ! S is a homeomorphism. A marked convex projective structure on S0 induces a convex projective structure on the base surface S0 itself by pull-back. Conversely, a convex projective structure on

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S0 can be considered as a marked convex projective structure, by taking the marking to be the identity homeomorphism of S0 . Definition 2.6. The deformation space of convex projective structures on S0 is the set of equivalence classes of pairs .f; S/, where S is a convex projective surface homeomorphic to S0 and where two pairs .f; S / and .f 0 ; S 0 / are considered to be equivalent if there exists a projective homeomorphism f 00 W S ! S 0 that is homotopic to f 0 B f 1 . Equivalently, we can define the deformation space of convex projective structures on a surface (or, more generally, on a manifold) as the space of homotopy classes of convex projective structures on that surface (or manifold). This space is equipped with a natural topology arising from the C 1 topology on developing maps. This topology is Hausdorff (see [21], p. 793). We shall say more about the deformation space of real projective structures on surfaces in §4. The basic elements of the theory of convex real projective structures on closed surfaces and their deformations are due to Kuiper [32], Kac–Vinberg [25] and Benzécri [6], and a complete theory has been developed (including the case of surfaces with boundary) by Goldman [21]. We shall elaborate on Goldman’s parametrization of the deformation space in §4. For any convex real projective manifold M D = , the Hilbert metric on descends to a metric on M called the Hilbert metric of M . For a strictly convex real projective structure M D = , as in the hyperbolic case, there exists a unique Hilbert geodesic in each homotopy class of a loop, unless this loop represents a parabolic element of the fundamental group, in which case the curve is homotopic to a puncture (or cusp) of S . Lemma 2.7. For any convex real projective surface S of finite type and of negative Euler characteristic (the surface may have geodesic boundary components and cusps), the area of S with respect to the Hausdorff measure induced by the Hilbert metric is uniformly bounded below independently of the topology of S . The proof uses a pair of pants decomposition of S. We recall that a (topological) pair of pants in S is an embedded surface which is homeomorphic to a sphere with three holes, where a hole is either a puncture or a boundary component, such that the following two conditions hold: • Each boundary component of P is a simple closed curve in S which is not homotopic to a point or to puncture of S. • There is no embedded annulus in S whose two boundary components are the union of boundary component of P and a boundary component of S. Note that the holes of P might be boundary components of S, and they can also be punctures, and the latter are considered as boundary curves of length zero.

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A topological pair of pants decomposition P of S is a union of disjoint simple closed curves in that surface such that the closure of each connected component of the complement of P in S is a topological pair of pants in S . Any surface of negative Euler characteristic admits a topological pair of pants decomposition. It is easy to see, using an Euler characteristic argument, that for a closed surface Sg of genus g 2, there are 2g 2 pairs of pants in a pair of pants decomposition. Given a surface equipped with a metric, a geodesic pair of pants decomposition is a topological pair of pants decomposition in which each curve which is not homotopic to a puncture is a closed geodesic. On any surface with finitely generated fundamental group equipped with a hyperbolic metric or with a Hilbert metric, any topological pair of pants decomposition is homotopic to a geodesic pair of pants decomposition. Furthermore, for hyperbolic metrics and for strictly convex structures, the closed geodesics in each free homotopy classes are unique, so every topological pair of pants decomposition is homotopic to a unique geodesic pair of pants decomposition. We now sketch the proof of Lemma 2.7. Proof. Take a geodesic pair of pants P in S. One can decompose P into two ideal triangles. Therefore, it suffices to give a bound for the area of an ideal triangle. Let T be a lift of an ideal triangle in with three ideal vertices p1 , p2 , p3 . Choose three projective lines P1 , P2 , P3 containing p1 , p2 , p3 disjoint from . They form a triangle 4 containing . Now by a comparison argument, it suffices to lower bound the area of T in 4. This is an exercise in projective geometry; see [14]. There are several papers where the reader can find concise introductions to the basics of the modern theory of projective geometry. We refer to the survey by Benoist [4] and to the sections on preliminaries in the thesis of Marquis [40]. Let us note that in studying divisible convex sets of finite co-volume, we do not lose a lot if we restrict ourselves to strictly convex sets. The following theorem is due to Marquis [41]. Theorem 2.8. Let be a proper open convex subset of RP 2 . If is not a triangle and admits a finite-volume quotient surface, then is strictly convex.

3 Relation to affine spheres An affine sphere is a smooth hypersurface in RnC1 characterized by the condition that its affine normal lines meet at a common point. The family of affine spheres is invariant under the group of affine transformations of RnC1 . In fact, it is the simplest interesting such family. Thus, it is not surprising that affine spheres are useful in the study of affine, and also of projective structures. (Recall that affine geometry is projective geometry where a hyperplane at infinity in RP n has been fixed.) Affine spheres also appear

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naturally in the solutions of certain PDEs, namely, the Monge–Ampère equations, and also in the study of hyperbolic surfaces. In fact, it is known that on a compact hyperbolic surface equipped with a cubic differential, there is a unique associated equiaffine metric called the Blaschke metric. (An object is called equiaffine if it is invariant by the group of volume-preserving affine transformations.) This, together with the Cheng–Yau classification of complete hyperbolic affine spheres, gives a new parametrization of the space of real projective structures on the surface and of the Hitchin component of the representations of the fundamental group of the surface into SL.3; R/, see [23], [33], [36]. In this section, we review some intricate relations between affine spheres and Hilbert metrics. We shall also refer to the relation between cubic differentials and affine spheres in §4.2 below. Let M RnC1 be a transversely oriented smooth hypersurface with a trivial bundle E D M RnC1 . Choose a transverse vector field over M so that E D TX M ˚ L where TM is the tangent bundle to M and L a trivial line bundle over M spanned by . If r is the standard affine flat connection on RnC1 , its restriction on E gives the following equations (Gauss and Weingarten): rX Y D DX Y C h.X; Y /; rX D S.X / C .X / for any tangent vector fields X and Y to M . The equations are obtained by splitting at each point x 2 M the tangent space to RnC1 at that point to TMx ˚ L. Here, D is a torsion-free connection on TM , h a symmetric 2-form on TM , S a shape operator and a 1-form. If M is locally strictly convex, i.e., if it can be written locally as the graph of a function with positive definite Hessian, then there exists a unique transverse vector field such that (1) D 0, (2) h is positive definite, and (3) jdet.Y1 ; ; Yn ; /j D 1 for any h-orthonormal frame Yi of TM . (see Proposition 2.1 of [5]). The vector field is called the affine normal, D the Blaschke connection, and h the affine metric on M . These are equiaffine notions. Definition 3.1. A hypersurface M in RnC1 is said to be an affine sphere with affine curvature 1 if the shape operator satisfies S D Id. In more geometric terms, an affine sphere in RnC1 is a smooth hypersurface characterized by the fact that its affine normal lines meet at a common point, called the center of the affine sphere (which could be at infinity, and in that case the affine sphere is said to be improper). The affine normal field is an affine invariant of the surface and the condition of being an affine sphere is therefore affinely invariant. This makes the family of affine spheres invariant under the group of affine transformations of RnC1 . Examples of affine spheres include ellipsoids and quadric hypersurfaces. There are

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two kinds of proper locally convex affine spheres, the hyperbolic, all of whose normals point away from the center, and the elliptic, all of whose normals point towards the center. An affine sphere is not necessarily affinely equivalent to a Euclidean sphere and, in fact, there are infinitely many non-equivalent affine spheres. See [38], [43] and [39] for surveys on affine spheres. Another family of objects which is well-known to be invariant under the group of affine transformations is the family of straight lines. Affine spheres were introduced at the beginning of the twentieth century by Ti¸ ¸ teica, and they were studied later on by Blaschke, Calabi, Cheng–Yau and others; see the survey by Loftin [38] for some historical background. The definition of the affine metric using the invariance of the affine normal was derived by Blaschke [7]. There are relations, discovered by Blaschke and by Calabi, between the theory of affine spheres and the real Monge–Ampère equations, and there is also a relation between affine spheres and the theory of convex real projective structures. The work of Cheng–Yau on affine spheres, combined with work of Wang [51] on PDEs in the setting of affine differential geometry, was also used by Labourie and Loftin to parametrize equivalence classes of representations in SL.3; R/ by cubic differentials on a Riemann surface (see §4.2 below). The work of Cheng–Yau on the Monge–Ampère equations associates to each properly convex subset of RnC1 an affine sphere in this space. Our aim in the rest of this section is to give an idea of how Hilbert geometry fits into this picture, in particular through the following two results: • For any proper convex set , the construction of affine spheres by Cheng and Yau leads naturally to a Riemannian metric on which is bi-Lipshitz equivalent to the Hilbert metric at the level of norms, cf. Proposition 3.3 below. • In the case of a strictly convex real projective surface, there is a comparison between the Hilbert volume on that surface with the affine volume (see Corollary 3.4). In both cases, the constants that appear in the comparison (that of the norms and that of the volumes) are universal. We briefly recall the construction by Cheng and Yau of an affine sphere in RnC1 associated to properly convex subset of this space. A cone C RnC1 is a subset which is invariant by the action of the positive reals by homotheties. A convex cone C RnC1 is a cone which is (the closure of) the inverse image of a convex set in RP n by the canonical projection RnC1 n f0g ! RP n . To any bounded open convex subset Rn , there is an associated convex cone C./ D ft.1; x/jx 2 ; t > 0g RnC1 which is a connected component of the inverse image of a convex set by the natural map RnC1 n f0g ! RP n with fiber RC . A group of projective transformations which acts properly discontinuously on also acts on the cone C ./. This is a consequence of the fact that any representation of a discrete group into PSL.n C 1; R/ lifts to a representation into the group SL˙ .n C 1; R/ of invertible .n C 1/ .n C 1/ matrices whose determinant is ˙1.

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Cheng and Yau [12] proved the following, which solved a conjecture of Calabi [11]: Theorem 3.2. If C RnC1 is an open convex cone containing no lines, then there exists a unique embedded hyperbolic affine sphere H whose center is the origin, which has affine curvature 1, and which is asymptotic to the boundary of C . The fact that C contains no lines is equivalent to the fact that the convex set is proper. The affine invariants of H are invariants of . The projection map induces a homeomorphism between H and . To the bounded open convex set Rn is associated the affine sphere defined as ²

HD

³

1 .1; x/ j x 2 u.x/

where u is the unique convex solution of the real Monge–Ampère equation detD 2 u D .1=u/nC2 satisfying [email protected] D 0: The projection map RnC1 n f0g ! RP n induces a diffeomorphism between H and , and the affine metric h on H induces a Riemannian metric on , still denoted by h and called the affine metric. This metric gives rise to a measure h on . Now we have two measures on , one is h , coming from the affine metric and the other is , coming from the Hilbert metric. Benoist and Hulin proved in [5] that the affine metric is bi-Lipschitz equivalent to the Hilbert metric: Proposition 3.3 (Benoist–Hulin). There exists a constant c > 0 such that for any properly convex set , x 2 and X 2 Tx , 1 kX kF kXkh ckX kF ; c where the subscript F denotes the Finsler norm of the Hilbert metric and h the affine metric. Benoist and Hulin deduce this result from the cocompactness of the action of SL .n C 1; R/ on the set of pairs .x; / and the continuous dependence on .x; / of both affine and Hilbert metrics. The result for Hilbert metrics is contained in Benzécri’s thesis, [6], Chapter V, where the author introduces several spaces he calls body spaces and form spaces (“espaces de corps” and “espaces de formes”). One of these spaces is, for n 1, the space of pairs .x; /, where RP n is a properly convex open subset and x is a point in . It is equipped with the Hausdorff topology. More precisely, the topology on the set of pairs .x; / is the product topology, where ˙

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on the first factor the topology is induced by the canonical metric on projective space (which is the quotient metric of the canonical metric on the sphere) and on the second factor the topology is induced by the Hausdorff distance on the compact subsets of projective space. (To use precisely the last notion, one needs to replace a subset by its closure.) Benzécri considers then the natural quotient of this space by the action of the group SL˙ .n C 1; R/ and he proves that this quotient is compact and metrizable (see [6], Théorème 2, p. 309). This implies the following: Corollary 3.4 ([5] Proposition 2.6). There exists a universal constant C (that depends only on the dimension) such that for any convex real projective manifold M , we have, for any Borel subset B of M , 1 VolH .B/ VolA .B/ C VolH .B/; C where H denotes the Hilbert volume and A the affine volume. In particular we have, in the case where M is a finite volume quotient, 1 VolH .M / VolA .M / C VolH .M /: C Thus, for a convex real projective manifold, having finite affine volume and having finite Hilbert volume are equivalent properties. One should mention that in the special case of surfaces and under the additional assumption that the boundary of the convex set is smooth, Loewner and Nirenberg solved the Monge–Ampère equation and they also constructed a Riemannian metric on the convex set which is invariant by projective transformations. By the uniqueness of the solution to the Monge–Ampère equation, this metric is the same as the one constructed by Cheng and Yau using affine spheres. The relation between the structure equations of affine spheres and RP 2 structures is explained in detail in the survey [39] by Loftin and McIntosh. The asymptotic affine sphere and an invariant Riemannian metric associated to a Hilbert geometry are also mentioned in Chapter 8 of this volume [42] (§4.1 and 4.2). We shall say more on affine spheres in §4.2 below.

4 Parametrizations of real projective structures and the deformation spaces The classification of convex real projective structures on surfaces of nonnegative Euler characteristic is due to Kuiper, cf. [31] and [32]. In this section, we describe parameter spaces for these structures. We start, in the first subsection, with Goldman’s parametrization, which was inspired by Thurston’s exposition of the Fenchel–Nielsen parametrization of Teichmüller space associated to a pants decomposition of a hyper-

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bolic surface. The parametrization consists of the length and the twist parameters of the pants curves. The Teichmüller space of a surface sits naturally as a subset of the deformation space of real projective structures, and Goldman’s parameters are a generalization of the Fenchel–Nielsen parameters. We then describe Hitchin’s parametrization, which also generalizes the parameters for Teichmüller space, when this space is considered as a connected component of the character variety of the fundamental group of the surface in SL.2; R/. Hitchin’s parameter space is a component of the character variety of representations of the fundamental group of the surface in SL.n; R/. Hitchin’s work gave rise to several other works by various authors, and it is also related to several questions addressed in this chapter. We shall mention some of these relationships below. In the final subsection, we mention a set of parameters due to the first author of this chapter that use the length spectra of the Hilbert metrics. This is also a generalization of a classical parametrization of hyperbolic structures by geodesic length spectra. It is an interesting question to study more carefully the structure of the parameter spaces and their nature (analytic, algebraic, etc.)

4.1 Goldman’s parametrization Our goal in this subsection is to give a brief description of Goldman’s parameters for the deformation space of convex real projective structures on surfaces of negative Euler characteristic. Recall that by using the Cayley–Klein–Beltrami model of hyperbolic geometry, a hyperbolic structure (in the sense of a Riemannian metric of constant curvature equal to 1) on a closed surface S of negative Euler characteristic is a special case of a convex real projective structure. This is a structure of the type = where Rn RP n is the interior of an ellipse and a subgroup of projective transformations of RP n that preserve . For any topological surface S of negative Euler characteristic, a classical and useful set of parameters for the space of isotopy classes of hyperbolic structures on S (that is, for the Teichmüller space of S ) is provided by the Fenchel–Nielsen coordinates associated to pair of pants decompositions. These parameters consist of the set of lengths of the pants curves of this decomposition, rendered geodesic for the hyperbolic structure, together with the twist parameters along these curves that measure the way in which the pairs of pants are glued together. (One needs a convention to measure the twists, whereas the length parameters are intrinsic.) In the case of a closed surface of genus g 2, an Euler characteristic argument shows that the number of curves in any pants decomposition of S is 3g 3. Thus, the length and twist parameters of S make a total of 6g 6 parameters, which is indeed the dimension of the Teichmüller space of S . These parameters are the so-called Fenchel–Nielsen parameters. For convex real projective structures on surfaces, there are analogues of the Fenchel– Nielsen parameters associated to a pair of pants decomposition, and we now describe

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them briefly. This parametrization of the space of equivalence classes (the so-called deformation space) of convex real projective structures on a closed surface of genus g 2 was obtained by Goldman [21], who showed that this deformation space, D.S /, is homeomorphic to an open cell of dimension 16.g 1/. In this projective setting, the pants curves are made geodesic with respect to the Hilbert metric (in the coordinate charts, the curves are made affine straight lines). The length of a simple closed geodesic in a given homotopy class is uniquely defined, and this set of lengths is part of the parameters associated to a pair of pants decomposition. However, in the setting of projective structures, the other parameters of the Fenchel–Nielsen coordinates are more complicated to describe that in the case of Teichmüller space; they are described in terms of 3 3 matrices that represent the curves on the surface. In what follows, we shall give an idea of these parameters. We recall that the fundamental group of a surface equipped with a convex real projective structure acts freely and properly discontinuously on the convex set which is the image of the associated developing map. Thus, instead of talking about parameters for the equivalence classes of convex real projective structures on a given closed surface S, one can talk about parameters for the equivalence classes of properly convex open subsets of RP 2 equipped with a properly discontinuous free action of a group isomorphic to the fundamental group 1 .S /. In this way, the deformation space can be viewed as an open subset of the character variety Hom.1 .S/ ! PSL.3; R//=PSL.3; R/ of representations of the fundamental group 1 .S / into the Lie group PSL.3; R/ and where the quotient is by the action of PSL.3; R/ on the space of representations by conjugation. The Teichmüller space T .S / becomes a subspace of D.S /. In the general case of a compact surface S of finite type with n boundary components with negative Euler characteristic .S/, denoting by D.S / the deformation space of convex real projective structures on S , Goldman proves the following (see [21], Theorem 1): Theorem 4.1. The space D.S/ is an open cell of dimension 8 .S /, and the map which associated to each convex projective surface S the germ of its structure near @S is a fibration of D.S/ over an open 2n-cell with fiber an open cell of dimension 8 .S / 2n. We shall explain below the behavior of the projective structures near the boundary. In the rest of this subsection, we give a brief description of Goldman’s parameters. The parameters are based on pairs of pants decompositions of the surface. In what follows, we shall assume that the pairs of pants decompositions that we consider are all geodesic. The parametrization of convex real projective structures on Sg is then done by first describing the structures on individual pairs of pants and then gluing, as we usually do in Teichmüller theory for the parametrization of hyperbolic structures. It turns out however that in the case of general real projective structures, the parameters associated to the pair of pants and to their gluing are more complex than in the case of hyperbolic structures.

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In this way, one is led to the question of understanding convex projective structures on pairs of pants, and this naturally requires the study of convex structures on surfaces with boundary. Recall that a convex projective structure on a surface S is a maximal atlas with values in RP 2 whose transition functions are restrictions of projective automorphisms of RP n . We need to consider convex projective structure on surfaces with boundary. This is also defined by an atlas with values in RP 2 , and we furthermore require that for each open set U in S which is the domain of a chart satisfying U \ @S 6D ; and for each arc in U which is in @S , the image of this arc by the chart map is contained in a projective line of RP 2 . The boundary components therefore become closed geodesics, and we require that each such closed geodesic has a geodesically convex collar neighborhood whose holonomy has distinct positive eigenvalues. Such structures are sometimes called structures “with standard convex projective collar neighborhood”. The image by the holonomy map of a loop representing a boundary component, which is well defined up to conjugacy, is a matrix in SL.3; R/ which has three distinct and positive real eigenvalues whose product is equal to 1. Such a matrix is termed by Goldman positive hyperbolic. The matrix in SL.3; R/ associated to such a boundary component is well defined up to conjugacy and it plays the role of the Fenchel–Nielsen length parameter in the hyperbolic setting. Thus, we have a Fenchel–Nielsen type length parameter associated to a closed geodesic which is two-dimensional in the case of convex projective structures. One can glue projective structures on surfaces with boundary with standard convex projective collar neighborhoods by identifying the collar neighborhoods of boundary components using projective isomorphisms. Goldman proves in [21] that if we glue in this way a finite number of convex projective structures on surfaces with boundary which all have negative Euler characteristic, then the resulting projective structure is convex. On each pair of pants, a convex real projective structure is determined by 3 2 C 2 parameters. Here, 3 is the number of boundary curves of the pair of pants, and near each boundary curve, the projective structure is determined by the two real parameters which we mentioned above. Also, whereas a hyperbolic structure on a pair of pants is completely determined by the “length parameters” associated to the boundary components (which are the lengths of these boundary components, including boundary component of length zero, which correspond to cusps), a general convex real projective structure on a pair of pants is not determined by the sole length parameters associated to the boundary, but there are two more extra real parameters involved, called “twist parameters”. To be more precise, recall first that the projective transformations of RP 2 can be represented by 3 3 matrices. These matrices act on the vector space R3 , and their action on RP 2 is the quotient action. The whole discussion can be reduced then to considerations on matrices and their actions on R3 , or, more precisely, on the set of lines through the origin of that vector space (which is the projective space RP 2 ). Note also that the group acting by matrices on R3 is in fact the group SL˙ .3; R/ and not PSL.3; R/, but we shall not worry here about the difference between the two groups.

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Thus, the holonomy of each closed geodesic on the surface S is positive hyperbolic. In particular, the set of parameters for SL.3; R/-conjugacy classes of such a matrix is an open 2-cell. Goldman, in [21], gives three equivalent sets of coordinates for the space of conjugacy classes of such matrices, and he makes a complete study of the dynamics of the action of a transformation of RP 2 representing such an element. We are now interested in the automorphisms of RP 2 that preserve . For a hyperbolic element of Aut./, the attractive and repulsive fixed points belong to the boundary of the convex set . A lift to of the geodesic in the quotient surface S is preserved by the action of the corresponding affine transformation, and that action is indeed hyperbolic in the sense that it is a translation along that geodesic, with an attractive and a repelling fixed point as endpoints. A positive hyperbolic isometry representing a boundary component of a pair of pants in the decomposition has three distinct eigenvalues 1 > 2 > 3 and it is conjugate to a diagonal matrix with these eigenvalues, 1 , 2 , 3 , in that order, on the diagonal. The action on RP 2 of such a matrix has three fixed points. Using the coordinates of R3 , these points correspond to the line passing through .1; 0; 0/, which is an attracting fixed point, the line passing through .0; 1; 0/, which is a saddle point, and the line passing through .0; 0; 1/, which is repelling. Now the three lines in RP 2 passing through the pairs of such points divide the space into four triangles which are invariant by the action of the given diagonal matrix. Goldman in [21] describes in detail the action of this matrix on these triangles. In the above matrix representation, the first parameter for the projective structure on the pair of pants which is associated to a closed curve which is the boundary of a pair of pants is log 13 , which is the Hilbert metric length of the closed geodesic. The other parameter is 3 log 2 . (We are following Goldman’s exposition, and the factor 2 is a convenient normalization.) These are the six parameters associated to the boundary curves of a pair on pants. The extra two parameters are called by Goldman “interior parameters”. In conclusion, we have the following proposition: Proposition 4.2 (Goldman [21]). The deformation space of RP 2 structures on a pair of pants is an open cell of dimension 8. When one glues two convex projective structures on pairs of pants along simple closed curves, there are two new parameters involved, associated to a simple closed curve C obtained after gluing. One of these parameters is called the twisting parameter, and the other one is called the vertical twist parameter. The two parameters combined are the analogues of the single Fenchel–Nielsen twist parameter of the case of hyperbolic surfaces. In fact, in the case where the projective structure is a hyperbolic structure, the twisting parameter is the usual twist parameter. More precisely, given a diagonal hyperbolic isometry .1 ; 2 ; 3 / representing C , the matrices 2 1 3 2 t 3 e 3 t 0 0 e 0 0 2 6 7 t D 4 0 1 0 5 ; Ot D 4 0 e 3t 0 5 1 0 0 e t 0 0 e 3 t

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commute with the diagonal matrix .1 ; 2 ; 3 / where 1 , 2 , 3 are the three eigenvalues which we described above and the parameters associated to these two matrices are used to glue two structures along C . Here, t is the twisting parameter and O t is the vertical parameter. Suppose that two pairs of pants P1 and P2 are glued along C . Once a projective structure on P1 is given, the two parameters corresponding to the length and 2 for C are already determined for P2 , and the twisting and the vertical parameters are needed to glue along C . In any case, the total parameters are 8.2g 2/, since there are 2g 2 pairs of pants and 8 parameters on each pair of pants. The vertical parameter gives rise to a so-called bulging deformation of the projective structure. This deformation is associated to a measured geodesic lamination on a hyperbolic surface, and it deforms the underlying RP 2 structure of that surface. The bulging deformation is an extension of the earthquake deformation (which itself is the extension of the Fenchel–Nielsen deformation from closed geodesics to measured geodesic laminations) and of the bending deformation in PSL.2; C/ (which is a complexification of the Fenchel–Nielsen deformation) to the context of convex real projective structures. The bulging deformation was introduced by Goldman in [21], who also wrote a recent paper on that subject [22]. Let us recall this deformation. For a closed hyperbolic surface S and a closed geodesic on S, after conjugation, we may assume that 2

./ D t0

e t0 4 D 0 0

3 0 0 1 0 5; t0 0 e

t0 > 0;

as an element of SL.3; R/. Now let t be the one-parameter subgroup generated by . /. If is an ellipse so that S D = .1 .S //, the eigenspaces Rv1 , Rv2 , Rv3 corresponding to e t , 1, e t respectively, define three points C , 0 , in RP 2 where ˙ are the attracting and the repelling fixed points of . / on @, and 0 is outside of . Consider a triangle 4 passing through these three points with left and right vertices corresponding to C , , and top vertex to 0 . Then the dynamics of . / on 4 is from the right vertex to the left and top vertices, and from the top vertex to the left vertex. Inside 4, the orbits of t are arcs of conics tangent to 4, one of which is the segment C of @ from to C . See Figure 1.2 in [21]. The time-t earthquake map of S along is given by the partial right Dehn twist along , which amounts to moving the right hand side of the lifts of by the amount t in . This can be realized by conjugating the action of the right hand side of S n D S1 [ S2 (if is separating) by t , .1 .S1 // h. /i t .1 .S2 // t1 , and correspondingly, using an HNN extension for the non-separating case. Obviously this earthquake deformation does not change the domain . To deform the domain, we perform the bulging deformation which we already mentioned. We want to replace C by another conic which is an orbit of t tangent to 4. This can be

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realized by conjugating the right hand side of S n (if is separating) by 2 1 3 e 3 t 0 0 2 6 7 Ot D 4 0 0 5; e 3t 1 0 0 e 3 t and correspondingly using an HNN extension for the non-separating case. What O t does to the domain is stretching it in the 0 direction, which entails to move the boundary arc C to one of the conics traced by t outside if t > 0, to one inside if t < 0. When t ! ˙1, the domain degenerates to the one containing edges of 4. By Theorem 2.8, the degenerate structures have infinite Hilbert area. Geometrically, the degenerate structure has an infinitely long cylinder attached along . The following is an open question: Question. Is the topological entropy function decreasing to some non-zero number in the above bulging deformation? And what is that number?

4.2 Hitchin’s parametrization Hitchin parametrized a specific component, called the Hitchin component, of the character variety of representations of 1 .S/ in SL.n C 1; R/ [23]. This is the component that contains the representations of hyperbolic structures, and it has some properties which are analogous to those of the Teichmüller space, which is a component of the character variety of representations in SL.2; R/. In the case n D 2 and if a complex structure on S is fixed, Hitchin identified the Hitchin component with the vector space H 0 .S; K 2 ˚ K 3 / of holomorphic quadratic and cubic differentials over the Riemann surface S. Here K is the canonical line bundle of S. In his work, Hitchin used the techniques of Higgs bundles, which are holomorphic vector bundles equipped with so-called “Higgs fields” that appeared in earlier works of Hitchin and of Simpson in their study of Teichmüller space. The Hitchin component for n D 2 (that is, representations in SL.3; R/)) was described again by Labourie [33] and Loftin [36] independently using affine spheres, a notion which we considered in Section 3. Both Labourie and Loftin showed that the space of equivalence classes of convex real projective structures on a closed oriented surface of genus 2 is parametrized by the space of conformal structures equipped with holomorphic cubic differentials. Using the theorem of Riemann–Roch gives another proof of the result of Goldman that this space of equivalence classes is an open cell of dimension 16.g 1/ (cf. Theorem 4.3). The parametrization is by a fiber bundle whose base (Teichmüller space) has dimension 3g 3 and each fiber has dimension 5g 5. This parametrization has also the advantage of equipping the deformation space of convex real projective structure with a natural complex structure. Labourie made in [33] the relation between these cubic differentials and those which appear in Hitchin’s parametrization.

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The proof of the correspondence established by Labourie and Loftin is based on the works of Cheng–Yau [12] that we mentioned above and the work of Wang [51]. It starts with the fact that S can be written as the quotient = where is a bounded convex domain in R2 and a subgroup of SL.3; R/ acting properly discontinuously on , and it makes use of the fact that can be canonically identified with the affine sphere H asymptotic to the open cone C R3 that sits over it, cf. §3. The group can be lifted to a group acting linearly on RnC1 and preserving the affine sphere. The natural projection C ! induces a diffeomorphism between H and . The affine metric associated to the affine sphere H induces a Riemann surface structure on = , and the cubic differential on this Riemann surface is essentially obtained by taking the difference between the Levi-Civita connection of the affine metric on H and the Blaschke connection of H . This follows from the work of Wang [51] who related convex projective structures on a surface to holomorphic data. Wang worked in the setting of affine differential geometry. He gave a condition in terms of the affine metric for a two-dimensional surface to be an affine sphere that involves conformal geometry. See also Labourie [33] and Loftin [37]. The work of Labourie and Loftin has been extended to the case of noncompact surfaces of finite Hilbert volume by Benoist and Hulin [5]. For the relation between cubic differentials and the differential geometry of surfaces, we refer the reader to [39].

4.3 Length spectra as parameters Let M be a manifold equipped with a metric g and let be the set of free homotopy classes of simple closed curves on S . The marked length spectrum of .M; g/ is the function on which associates to each element the infimum of the lengths of closed curves in that free homotopy class. In the above definition, the adjective marked refers to the fact that we are not only considering the set of lengths associated to the elements of , but we keep track of each element of with its associated length parameter. In what follows, all length spectra that we consider are marked, and we shall sometimes denote marked the length spectrum by the term length spectrum. The first author of this chapter showed in [27] that a normalized version of the length spectrum for the Hilbert metric is a set of parameters for convex projective structures on surfaces, up to dual structures. Here a dual structure is the real projective x f0gg. structure induced from the dual cone D fw W hw; vi > 0 for all v 2 (Note that length spectra cannot distinguish between dual structures.) This result is an analogue of the fact that the projectivization of length spectra of hyperbolic structures on a surface of finite type can be used as parameters for the Teichmüller space of that surface. Thurston used these parameters on Teichmüller space in his construction of his boundary whose elements are projective equivalence classes of measured foliations (or, equivalently, of measured laminations), see [49] and [19]. The analogous result for convex projective structures on the surface was also used in order to define a boundary for the deformation space of these convex structures. In fact, Kim’s parameters are

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expressed in terms of logarithms of eigenvalues of hyperbolic 3 3 matrices; see [27] for the details. We shall get back to this matter in §6.

5 Strictly convex manifolds and topological entropy We already mentioned that strictly convex real projective structures have several properties which are similar to properties of negatively curved Riemannian manifolds. Let us give a few examples. Benoist [2] proved the following fundamental theorem about strictly convex projective manifolds, relating the regularity of the boundary of the universal covering to the large-scale geometry of a group that divides it. Theorem 5.1. Let be a divisible convex set, divided by a group . Then the following are equivalent: (1) .; d / is Gromov-hyperbolic; (2) is strictly convex; (3) @ is C 1 ; (4) The geodesic flow on = is Anosov. Let us make a few comments on these statements. Although the Hilbert metric is generally non-Riemannian, there is a natural notion of geodesic flow associated to the projective manifold = , which is the quotient by of the geodesic flow on . The geodesic flow of is defined on the unit tangent bundle T1 ./ ' .T n f0g/=RC . This is the space of pairs .x; / where x is a point in and is a vector of length one based at x (remember that for strictly convex manifolds, the projective lines are exactly the Euclidean lines) with respect to the Hilbert metric. The geodesic flow t W T1 ./ ! T1 ./ is then obtained by moving the pair .x; / unit length in the direction specified by . The quotient flow, denoted by the same name, t W T1 .=/ ! T1 .= /, is the geodesic flow associated to = . We recall that a C 1 flow f t W W ! W on a Riemannian manifold W is said to be Anosov if there is a splitting of the tangent space at each point v into three subspaces Tv W D E u .v/ ˚ E s .v/ ˚ R .X / where E u , E s have positive dimension, E u expanding under the flow, E s contracting and X the generator of the flow. A typical Anosov flow is the geodesic flow on the unit tangent bundle of a manifold of negative curvature. Anosov flows have interesting dynamical properties, for instance, the union of periodic orbits are dense. The definition of an Anosov flow can be transcribed in a straightforward way to the setting of the Finsler manifolds and =.

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We also recall that Gromov hyperbolicity is a property of the large-scale geometry of a metric space. In the case of a geodesic metric space (that is, a metric space in which distances between points are realized by length of curves joining them), the property says that triangles are uniformly ı-thin, that is, that there exists a ı > 0 such that for any triangle, any side is contained in the ı-neighborhood of the two others. See [24] and [15]. It follows from Theorem 5.1 that convex real projective structures on a given closed manifold M are either all strictly convex or they are all non-strictly convex. The topological entropy of a flow t W W ! W on a compact manifold W with distance function d is defined as follows. For all t 0, let d t be the function on W W defined by d t .x; y/ D max d. s .x/; s .y// 0st

for any two points x and y in W . It is not difficult to see that d t satisfies the properties of a distance function. For any > 0 and t 2 R, we consider coverings of W by open sets of diameter less than with respect to the distance d t and we let N. ; t; / be the minimal cardinality of such a covering. The topological entropy of is then defined as htop . / D lim lim sup 1t log N. ; t; / : !0

t!1

For a strictly convex compact manifold M , one can consider the associated geodesic flow on the unit tangent bundle T1 M of M . The strict convexity ensures that in the local projective charts, any geodesic is contained in a projective line, and therefore there are unique geodesics between any two given points. Benoist started the study of this flow in [2] and he showed that it is Anosov and topologically mixing, generalizing properties which were known to hold for the geodesic flow associated to a hyperbolic manifold. We recall that a flow f t on a space M is said to be topologically mixing if for any two open sets A and B in M , there exists a real number t0 such that for every t > t0 , we have f t .A/ \ B 6D ;. An Anosov flow is not necessarily topologically mixing and vice versa. There are several interesting global questions concerning the long-term behaviour of orbits and more generally the dynamics and the ergodic theory of Anosov flows. Some of these questions are considered by Crampon [16] who continued the study initiated by Benoist. Crampon showed the following result on entropy: Theorem 5.2. Let t be the geodesic flow of the Hilbert metric on a strictly convex projective compact manifold M of dimension n. Its topological entropy satisfies htop . t / .n 1/; with equality if and only if M is a hyperbolic manifold with constant curvature 1.

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Some of the results of Crampon are developed in detail in Chapter 7 of this volume [17]. The regularity of the boundary of the convex set plays an important role in the study of the geodesic flow.

6 Geodesic currents on strictly convex surfaces We recalled in §4.3 that it was proved in [27] that if two strictly convex compact real projective manifolds have the same length spectrum with respect to the Hilbert metric, then they are projectively equivalent, up to dual structures. Thus, the length spectrum can be used as a parameter space for real projective structures up to dual structures. The length spectrum, and more generally, the set of geodesics in the manifold, or the space of geodesics in the universal cover equipped with the action of the fundamental group, can be studied from various points of view: dynamical, measure theoretical, etc. We shall consider more particularly the case where the manifold is two-dimensional. We start by recalling the notion of geodesic current. Let S D = be a closed surface of genus at least two equipped with a strictly convex real projective structure. A geodesic current on S is a D 1 .S /-invariant Borel measure on the set [email protected] @ n Diag/=Z2 ; where Diag is the diagonal set and where Z2 acts by interchanging the coordinates. Equivalently, a geodesic current is an invariant transverse measure for the geodesic flow on the unit tangent bundle of S. The unit tangent bundle T1 S of S can be seen as the quotient of the set of bi-infinite geodesics by the action of . The term “geodesic” denotes here the lines in the sense of projective geometry, but we also recall that in the case where is strictly convex (and in fact, it suffices that @ does not contain two affinely independent non-empty open segments), the set of projective geodesics coincides with the set of geodesics (in the sense of distance-minimizing curves) for the associated Hilbert metric. The equivalence between the above two definitions of a geodesic current is based on the fact that from any two distinct points in @ there is a unique projective geodesic in having these points as endpoints. The invariant transverse measures for the geodesic flow of S are also the -invariant invariant transverse measures of the geodesic flow on . There are methods for obtaining invariant measures for geodesic flows and we shall mention the Bowen–Margulis measure below. In the setting of Riemannian manifolds of negative curvature, there are classical methods for constructing transverse measures for geodesic flows, and the theory of such transverse measures is well developed. The methods have been adapted by several authors to the case of Hilbert geometry. For more information on this subject, we refer the reader to the chapter by Crampon in this volume [17].

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Let W 1 .S/ ! SL.3; R/ be a holonomy representation associated to a real projective structure on S. We now explain a method of constructing geodesic currents which originates in the work of Ledrappier [34]. We start with the fact that for a strictly convex projective structure, the boundary @ is C 1 , and we recall the definition of a Busemann cocycle B in this context, cf. [27]. For a fixed base point o 2 and for 2 @, choose a unit speed geodesic ray r.t / with r.0/ D o and r.1/ D , and let B .o; y/ D lim .d .y; r.t // t / : t!1

The map t 7! d .y; r.t// t is non-increasing and bounded from below, therefore the above limit exists. This function is called the Busemann function associated to the geodesic ray r. Since the geodesic ray between a point in and a point in @ is unique, one can think of the Busemann function as a function on @ that depends on the choice of a basepoint in . The function changes by an additive constant when we change the basepoint. When is the unit disc (i.e. the hyperbolic space), the value of B .o; y/ is the signed distance between the two horospheres based at and passing by o and by y. One also talks about the Busemann cocycle B .o; r 1 o/ associated to a point 2 @; the reason for this terminology is the cocycle property expressed in Equation (6.1) below. We refer the reader to [44] for a systematic presentation of several properties of Busemann functions. The group , being isomorphic to the fundamental group of a closed surface of genus 2, is hyperbolic in the sense of Gromov (meaning that its Cayley graph with respect to some – or, equivalently to any – finite generating system, equipped with the word metric, is a hyperbolic geodesic metric space). Such a group has a well-defined Gromov boundary. Such a boundary carries a natural Hölder structure, see [24] and [15]. We shall also use the canonical identification between the two boundary spaces @ and @. Since the surface S is closed, each element 2 1 .S / ' is hyperbolic, that is, it acts on without fixed point leaving invariant a unique geodesic of that space and acting as a translation along that geodesic, with one attractive and one repelling fixed point in @, the two endpoints of the invariant geodesic. The translation length of , denoted by `./, is defined as `./ D inf d .x; .x//: x2

Checking the translation length of a group element is generally a pleasant exercise. The attracting fixed point at infinity of is denoted by C . For each 2 @, consider a Busemann cocycle .; / 7! B .o; 1 o/;

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where o 2 is a fixed base point. Then B .o; 11 01 o/ D B1 .1 o; 01 o/ D B1 .o; 01 o/ C B1 .1 o; o/ D B1 .o; 01 o/ C B .o; 11 o/: Hence, if we set c.; / D B .o; 1 o/, we have c.0 1 ; / D c.0 ; 1 / C c.1 ; /:

(6.1)

This map c W @ ! R is a Hölder cocycle, that is, besides the cocycle property given by the preceding equation, c.; / is a Hölder map for every 2 . The period of c at is defined to be `c ./ D c.; C /: The reason for this terminology is that two Hölder cocycles are cohomologous if and only if they have the same periods (see Theorem 1.a of [34]). In our case, `c ./ D B C .o; 1 o/ D `. / D `. 1 / D `c . 1 /: Hence the set of periods of c is just the length spectrum of the real projective structure with respect to the Hilbert metric. Such a cocycle c (that is, a cocycle satisfying the property `c ./ D `c . 1 /), is said to be even. The exponential growth rate of a Hölder cocycle c is defined as hc D lim sup s!1

log #fŒ 2 Œ W `c . / sg s

where Œ denotes the conjugacy class of . In [34], it is shown that if 0 < hc < 1 then there exists an associated Patterson–Sullivan measure on @, i.e., a probability measure on @ such that d 1 ./ D e hc c. ;/ : d In the case at hand, hc is just the growth rate of lengths of closed geodesics in the Hilbert metric. We recalled the definition of the topological entropy of a flow in §5. Now we need the notion of volume entropy of a metric. This is a measure of the asymptotic growth rate of volumes of metric balls. We recall that the Hilbert metric on the convex set equips this set with a notion of volume, viz., the Hausdorff measure associated to the metric. It is called the Hilbert volume (see §2). This notion of volume descends to the quotients of by properly discontinuous actions of groups of projective transformations. We choose a point x in . The volume entropy, hvol of the metric is then defined as 1 hvol D lim log vol.B.x; r// r!1 r

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if this limit exists, where B.x; r/ denotes the open ball of center x and radius r. The geodesic flow on the unit tangent bundle of a strictly convex real projective surface is C 1 and Anosov. Therefore a classical theorem of Bowen [10] applies to conclude that the topological entropy hvol is equal to the growth rate of lengths of closed geodesics. More precisely, if N.T / denotes the number of closed geodesics in = and htop the topological entropy of the flow, we have N.T / '

e htop T : htop T

It is shown in [16] that 0 < hvol hhyp D 1 where hhyp is the topological entropy of a hyperbolic structure. Now we summarize some general facts about the theory of Hölder cocycles and Patterson–Sullivan measures associated to real projective structures. Let be a properly discontinuous action of a group of projective transformations on a convex set . A family of finite Borel measures fx gx2 defined on @ is said to be an ˛-conformal density (or a conformal density of dimension ˛) for if any two metrics in this family are equivalent (that is, if they have the same measure-zero sets) and if they satisfy the following properties: (1) (2)

dy ./ D e ˛B .x;y/ where B .x; / dx that B .x; x/ D 0 for all x; y 2 ; dx ./ dx

D

dx . 1 / d 1 x

is the Busemann function based at such

for all x; y 2 and 2 ;

(3) x D x for all x 2 and 2 . d

Condition (1) expresses the fact that the Radon–Nikodym cocycles dyx ./ of the family of measures fx gx2 are expressed in terms of the Busemann cocycles of the convex set . We refer the reader to the paper [45] of Patterson and [46] of Sullivan for the original ideas behind the introduction of conformal densities. Given an ˛-conformal density fx gx2 on @, the measure d U.; / D d Ux .; / D e 2ı. /. ;/x dx ./dx ./ is a -invariant measure on @ @ which is independent of x 2 , where .; /x denotes the quantity .; /x D B .x; z/ B .x; z/ for any z in and where ı./ is equal to the volume entropy of the associated Hilbert metric. (The quantity .; /x is also related to the Gromov product of and with basepoint x.) Finally d Udt is a geodesic flow invariant measure on T1 M , where M D = , and it is called a Bowen–Margulis measure. In the classical theory of dynamical systems, the Bowen–

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Margulis measure is an invariant measure for a hyperbolic system (in particular, for an Anosov flow). In the case of a mixing Anosov flow (like the geodesic flow of a compact hyperbolic manifold), the Bowen–Margulis measure maximizes entropy. In this sense, it provides a very good description of the complexity of the dynamical system. There is a construction of the Bowen–Margulis measure using the Patterson– Sullivan techniques and we shall talk about this below. Since we are interested in the group itself rather than , a Patterson–Sullivan measure is d o 1 .x/ D e hc c. ;x/ do for some Hölder cocycle c. Henceforth we will omit the base point o. Two measures and 0 are equivalent if and only if two associated cocycles c and c 0 have the same periods [34]. The Patterson–Sullivan geodesic current associated to the Hölder cocycle c is d mc .x; y/ D e 2hc .x;y/o d.x/d.y/: In [34], it is shown that there is a 1-1 correspondence between Hölder cocycles and Patterson–Sullivan geodesic currents. From now on, we denote by G the set of Patterson–Sullivan geodesic currents defined by Hölder cocycles. The space of geodesic currents, as a space of measures, is equipped with a natural weak topology of convergence on continuous functions. Note that if is a dual structure of , since has the same marked length spectrum as , the cocycle defined by is the same as the one defined by . By associating a Hölder cocycle c D B. / .o; / and the Patterson–Sullivan geodesic current to the projective structure , we obtain a map from the moduli space D of strictly convex structures on S P W D ! G: This map is 1-1 on Teichmüller space and 2-1 elsewhere [27]. The preimage of a point consists in two dual projective structures. The interested reader will find more details in [29].

7 Compactification of the deformation space of convex real projective structures Extensive work has been done on the compactification of Teichmüller space and of Riemann’s moduli space, and more recently on that of the character varieties of representations of fundamental groups of surfaces into various Lie groups. Several possible compactifications and boundary constructions of Teichmüller space have been obtained. Some of them use hyperbolic geometry (Thurston’s compactification, the horofunction boundary construction, etc.), other compactifications use complex structures (Teichmüller’s compactification, the Bers compactification, etc.), and others use

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algebraic geometry (e.g. the Morgan–Shalen compactification). There are also combinatorial compactifications. Each compactification captures some essential properties of the space. It is then natural to try to construct compactifications of moduli spaces of projective structures. A compactification of the deformation space of strictly convex real projective structures on a manifold has been constructed by the first author in [28]. It is related to Hilbert geometry in the sense that it uses the length spectrum of this metric, in the same way Thurston’s geometric compactification Tx of Teichmüller space T uses the hyperbolic length spectrum. The compactification of the deformation space D of strictly convex real projective structures is however more mysterious than Thurston’s compactification of Teichmüller space. Loftin developed in [37] another compactification of D which is based on holomorphic cubic differentials on degenerate (noded) surfaces. Since Teichmüller space T is a subspace of D, one can naturally expect that the compactification of D obtained by using the length spectrum should include the Thurston boundary consisting of projective measured laminations or, equivalently, of actions of the fundamental group of the surface on R-trees. The compactification in [28] is done in terms of the geometry of X D SL.3; R/=SO.3/ by regarding a real projective structure as a holonomy representation W 1 .S / ! SL.3; R/. A boundary point of the compactification is then either a reducible representation or a limit representation which acts on the asymptotic cone of X. Specifically, if i W 1 .S / ! SO.2; 1/ SL.3; R/ is a sequence of hyperbolic structures which converges to a projective lamination in Thurston’s compactification, the limit action of i converges to the affine building R T , where T is the real tree dual to the measured lamination . In this way, the space of projective measured laminations appears naturally as a subset of the boundary of the deformation space of strictly convex real projective structures. On the other hand, the geodesic currents introduced in Section 6 are natural generalizations of measured laminations. Hence we may also use the space G of geodesic currents to compactify D. On G , one can define an intersection form i W G G ! RC ; so that i.˛; ˇ/ is the total mass of the product measure ˛ ˇ on the set of pairs of transversal geodesics on S . This is a natural generalization of the intersection form i.1 ; 2 / when 1 and 2 are measured geodesic laminations. The geodesic current m associated to the Hilbert metric m of a strictly convex real projective structure is properly normalized as: i.m ; m / D

Area.m/; 2

like in the hyperbolic case [8]. This compactification by geodesic currents satisfies the following (see [29]):

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Theorem 7.1. The two compactifications of the space of strictly convex real projective structures on a closed surface S, the one defined via the marked length spectrum and the other via geodesic currents, are naturally homeomorphic. Sketch of proof. Look at the following diagram: L

/ P .R / DE OC EE EE EE E ` P P EEE E" PG,

where is the set of conjugacy classes of elements in 1 .S / and L is the marked length spectrum map relatively to the Hilbert metric. The map ` is defined using the periods of the corresponding Hölder cocycles. The diagram commutes, and the map ` is injective, hence the compactifications of the images of L and P P are homeomorphic. Corollary 7.2. Thurston’s compactification Tx by projective measured laminations is contained in P P .D/. Proof. When restricted to T , if i mi ! ˛ in G and the projective structures mi diverge, then by the properness of the map T ! G , i ! 0 and i.˛; ˛/ D lim i.i mi ; i mi / D lim 2i i.mi ; mi / D lim 2i 2j .S /j D 0: i !1 i!1 i!1 2 Hence ˛ must be a measured lamination since measured laminations are characterized by self-intersection number being zero. In this way, we see again Thurston’s compactification sitting inside the compactification of real projective structures. We believe that whenever the Hilbert metric area of a diverging sequence is bounded above, the sequence will converge to a projective measured lamination. Acknowledgement. The first author gratefully acknowledges the partial support of NRF grant (2010-0024171). The second author is supported by the French ANR project FINSLER.

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[10] R. Bowen, Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30. [11] E. Calabi, Complete affine hyperspheres I. In Symposia Mathematica, Vol. X, Academic Press, London 1972, 19–38. [12] S. Cheng and S. Yau, On the regularity of the Monge–Ampère equation [email protected] [email protected] @sxu / D F .x; u/. Comm. Pure Appl. Math. 30 (1977), 41–68. [13] S.-S. Chern, On the curvatura integra in a Riemannian manifold. Ann. of Math. (2) 46 (1945) 674–684. [14] B. Colbois and C. Vernicos, L’aire des triangles idéaux en géométrie de Hilbert. Enseign. Math. (2) 50 (2004), no. 3, 203–237. [15] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes: Les groupes hyperboliques de Gromov. Lecture Notes in Math. 1441, Springer-Verlag, Berlin 1990. [16] M. Crampon, Entropies of strictly convex projective manifolds. J. Mod. Dyn. 3 (2009), no. 4, 511-547. [17] M. Crampon, The geodesic flow of Finsler and Hilbert geometries. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 161–206. [18] Ch. Ehresmann, Sur les espaces localement homogènes, Enseign. Math. 35 (1936), 317–333. [19] A. Fathi, F. Laudenbach, and V. Poénaru (eds.), Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979). [20] D. Johnson and J. Millson, Deformation spaces associated to compact hyperbolic manifolds. In Discrete groups in geometry and analysis (New Haven, Conn., 1984), Progr. Math. 67, Birkhäuser, Boston 1987, 48–106.

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[21] W. Goldman, Convex real projective structures on compact surfaces. J. Differential Geom. 31 (1990), 791–845. [22] W. Goldman, Bulging deformations of RP 2 -manifolds. Preprint, arXiv:1302.0777. [23] N. Hitchin, Lie groups and Teichmüller space. Topology 31 (1992), 449–473. [24] M. Gromov, Hyperbolic groups. In Essays in group theory, Publ. Math. Sci. Res. Inst. 8, Springer-Verlag, New York 1987, 75–263. [25] V. Kac and E. Vinberg, Quasi-homogeneous cones. Mat. Zametki 1 (1967), 347–354; English transl. Math. Notes 1 (1968), 231–235. [26] M. Kapovich, Convex projective structures on Gromov-Thurston manifolds. Geom. Top. (2007), 1777–1830. [27] I. Kim, Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds. J. Differential Geom. 58 (2001), 189–218; erratum ibid. 86 (2010), 189. [28] I. Kim, Compactification of strictly convex real projective structures, Geom. Dedicata. 113 (2005), 185–195. [29] I. Kim, Deformation of strictly convex real projective structures via geodesic currents, Patterson–Sullivan measures, and cubic differentials. In preparation. [30] J.-L. Koszul, Variétés localement plates et convexité. Osaka J. Math. 2 (1965), 285–290. [31] N. H. Kuiper, Sur les surfaces localement affines. In Géométrie différentielle, Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, CNRS, Paris 1953, 79–87. [32] N. H. Kuiper, On convex locally-projective spaces. In Convegno Internazionale di Geometria Differenziale (Italia, 20–26 Sett. 1953), Edizioni Cremonese, Roma 1954, 200–213. [33] F. Labourie, Flat projective structures on surfaces and cubic holomorphic differentials. Pure Appl. Math. Q. 3 (2007), 1057–1099. [34] F. Ledrappier, Structure au bord des variétés à courbure négative. In Semin. Theor. Spectr. Geom. 13, Année 1994–1995, Université de Grenoble I, Institut Fourier, Saint-Martind’Hères 1995, 97–122. [35] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations. In Contributions to analysis, a collection of papers dedicated to Lipman Bers, Academic Press, New York 1974, 245–272. [36] J. Loftin, Affine spheres and convex RP 2 manifolds, Amer. J. Math. 123 (2) (2001), 255–274. [37] J. Loftin, The compactification of the moduli space of convex RP 2 -surfaces I. J. Differential Geom. 68 (2004), 223–276. [38] J. Loftin, Survey on affine spheres. In Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM) 13, Internat. Press, Somerville, MA, 2010, 161–191. [39] J. Loftin and I. McIntosh, Cubic differentials in the differential geometry of surfaces. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume V, European Mathematical Society, to appear. [40] L. Marquis, Les pavages en géométrie projective de dimension 2 et 3, Thèse, Université de Paris-Sud, Orsay 2009.

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[41] L. Marquis, Surface projective convexe de volume fini. Ann. Inst. Fourier 62 (2012), no. 1, 325–392. [42] L. Marquis, Around groups in Hilbert geometry. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 207–261. [43] K. Nomizu and T. Sasaki, Affine differential geometry. Cambridge Tracts Math. 111, Cambridge University Press, Cambridge 1994. [44] A. Papadopoulos, Metric spaces, convexity and nonpositive curvature. 2nd edition, IRMA Lect. Math. Theor. Phys. 6, European Mathematical Society, Zürich 2014. [45] S. J. Patterson, Lectures on measures on limit sets of Kleinian groups. In Analytical and geometric aspects of hyperbolic space, Symp. Warwick and Durham, 1984, London Math. Soc. Lecture Note Ser. 111, Cambridge University Press, Cambridge 1987, 281–323. [46] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. [47] D. Sullivan and W. P. Thurston, Manifolds with canonical coordinate charts: some examples. Enseign. Math. (2) 29 (1983), no. 1–2, 15–25. [48] W. P. Thurston, The geometry and topology of three-manifolds. Mimeographed notes, Princeton University, 1976. [49] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417–431. [50] C. Walsh, Gauge-reversing maps on cones, and Hilbert and Thompson isometries. Preprint, arXiv:1312.7871. [51] C. P. Wang, Some examples of complete hyperbolic affine 2-spheres in R3 . In Global differential geometry and global analysis (Berlin 1990), Lecture Notes in Math. 1481, Springer-Verlag, Berlin 1991, 271–280.

Chapter 12

Weil–Petersson Funk metric on Teichmüller space Hideki Miyachi, Ken’ichi Ohshika and Sumio Yamada

Contents 1 2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Funk metric in Rd and its representations . . . . . . . . . . Weil–Petersson geometry . . . . . . . . . . . . . . . . . . . . . . 3.1 Teichmüller space . . . . . . . . . . . . . . . . . . . . . . 3.2 Weil–Petersson metric and geodesic completions of .T ; d / . 3.3 Tx as a convex subset in D.Tx ; / . . . . . . . . . . . . . . . 3.4 The Weil–Petersson Funk metric F2 . . . . . . . . . . . . 4 Properties of the Funk metric F2 . . . . . . . . . . . . . . . . . . 4.1 Density of strata in the Weil–Petersson visual sphere . . . . 4.2 Non-degeneracy of the weak metric F2 . . . . . . . . . . . 4.3 Weil–Petersson Funk topology and Weil–Petersson topology 4.4 Mapping class group invariance and the translation distances 5 Three Funk-type metrics on Teichmüller spaces . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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339 340 341 341 342 343 344 345 345 347 347 348 348 350

1 Introduction Given a closed surface †g of genus g > 1, the Deligne–Mumford–Mahler compactified moduli space Mxg of complex structures on † has been actively studied. It is a projective variety, that is, an algebraic subset of the complex projective space. There is a natural metric called Weil–Petersson metric on both moduli and Teichmüller spaces. The Weil–Petersson completion Tx of the Teichmüller space T is the space whose quotient space by the action of the mapping class group of † is the compactification of the moduli space. In 1937, Teichmüller showed that the space T is diffeomorphic to an open ball of dimension 6g 6, the number of parameters first claimed by Riemann. By going from moduli space to Teichmüller space, we lose the structure of projective variety, and we gain instead the structure of a “convex” set. Indeed the Teichmüller space Tg is a very intriguing and important convex set naturally arising from the theory of moduli of Riemann surfaces.

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In this chapter, we will make the Weil–Petersson convex structure precise, and will introduce the Funk-type distance function, called the Weil–Petersson Funk metric [30]. In particular, this approach provided an answer to a problem raised by A. Papadopoulos [13] “Realize Teichmüller space as a bounded convex set somewhere and study the Hilbert metric on it”, even though the resulting convex set is not bounded, where the Hilbert metric in question is the arithmetic symmetrization of the Weil– Petersson Funk metric. Additionally, a set of new results is presented concerning the properties of the Weil–Petersson Funk metric, juxtaposed with another set of results recently obtained by K. Fujiwara [9]. Those properties are interesting as they suggest further interactions between the Weil–Petersson geometry and the curve complex geometry. In the last section, we will give a short survey of other Funk-type metrics defined on Teichmüller spaces, further emphasizing the convex-geometric approach to Teichmüller theory. We hope that this will demonstrate the ubiquity of the Funk-type metrics associated with convex bodies naturally arising in various contexts. The authors thank Scott Wolpert for his interest and suggestions on this work. We also thank Athanase Papadopoulos and Marc Troyanov for inviting us to contribute to this volume as well as for their valuable comments.

2 The Funk metric in Rd and its representations Let be an open bounded convex subset in a Euclidean space .Rd ; d / where d is the standard Euclidean metric. We set the presentation in [14] as our reference for the Funk and Hilbert metrics of , and we also refer to the first part of the paper [29] and Chapter 2 of the present volume. There are three different descriptions of the Funk metric. The first one is the original definition: d.x; b.x; y// ; F1 .x; y/ D log d.y; b.x; y// where for x 6D y in , the point b.x; y/ is the intersection of the boundary @ with the Euclidean ray fx Ct xy W t > 0g from x though y and where xy is the unit tangent vector in Rd pointing from x to y. When x D y, we set F .x; y/ D 0. The second description is the variational interpretation of the above value using the geometry of supporting hyperplanes: F2 .x; y/ D sup log 2P

d.x; / ; d.y; /

where P is the set of all supporting hyperplanes of . This is given in [30]. In the literature, a variational characterization of Hilbert metrics appears in the work of Birkhoff and Nussbaum ([12]), where the supporting hyperplanes are treated as elements of the dual space, but the Funk metric was not mentioned there.

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Finally, the Finsler structure p;x ./ is given by the following function (the Minkowski functional) on vectors in each tangent space to at x: p;x ./ D sup

2P

h .x/; i ; d.x; /

where is the unit vector in Tx perpendicular to, and directed toward . This is a weak norm on each tangent space which is defined in such a way that the Funk distance is described as the infimum of length of curves: Z b p;.t / .P .t // dt; F3 .x; y/ D inf

a

the infimum being taken over all the piecewise C 1 curves satisfying .a/ D x and .b/ D y. For any convex domain Rd , the three quantities F1 .x; y/, F2 .x; y/, F3 .x; y/ are all equal to each other, and we set F .x; y/ WD F1 .x; y/ D F2 .x; y/ D F3 .x; y/ for every x and y in .

3 Weil–Petersson geometry 3.1 Teichmüller space Let † be a compact smooth surface with negative Euler characteristic. The Teichmüller space T D T .†/ of † is the quotient space of the space M1 of finite-area metrics of constant curvature 1 on † by the group Diff 0 † of diffeomorphisms of † homotopic to the identity. Each metric determines a conformal structure on † and the Teichmüller space of † is canonically identified with the space of marked conformal structures on the interior of †. The Weil–Petersson metric on the Teichmüller space T is the L2 metric on the surface † for deformation tensors of the hyperbolic metric G, Z hh1 ; h2 iWP D hh1 .x/; h2 .x/iG.x/ dG .x/: †

Here, the deformation tensors h1 , h2 are traceless and divergence-free with respect to G, which insures that the deformation of G preserves the constant curvature condition as well as the L2 -perpendicularity to the diffeomorphism fibers. We denote by dWP .p; q/ the Weil–Petersson distance between the points p and q in T induced from the Riemannian structure of the Weil–Petersson metric. It is known ([20], [7]) that .T ; dWP / is an incomplete metric space. We denote by .Tx ; dWP / the metric completion of .T ; dWP /.

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In [27], [28], the third author has studied the geometry of the Weil–Petersson completed Teichmüller space Tx of the surface †. It was shown there that the metric completion Tx is a CAT(0) space [2], that is, a non-positively curved metric space in the sense of Alexandrov. The completion Tx has a stratification by its boundary sets. We will discuss the stratification in the next section. The above definition of the Weil–Petersson metric came later; it was introduced by Fischer and Tromba in the 1970s. Historically the tangent space at p 2 T , consisting of deformation tensors of the hyperbolic metric p, is canonically identified with the space of harmonic Beltrami differentials ²

HB.p/ D

³

'N j ' is a holomorphic quadratic differential for p ; p

where p is the conformal factor of the hyperbolic metric representing p with respect to an isothermal coordinate system. The Weil–Petersson metric is a Riemannian metric on T defined by the scalar product Z hv1 ; v2 ip D v1 .z/v2 .z/ x .z/ dxdy; †

for v1 ; v2 2 HB.p/ and z D x Ciy. The Weil–Petersson metric is Kähler with respect p to the canonical complex structure on T , which corresponds to multiplication by 1 on holomorphic quadratic differentials.

3.2 Weil–Petersson metric and geodesic completions of .T ; d/ In the Weil–Petersson geometric setting, given a closed surface † of genus g > 1, the Weil–Petersson metric completion ([27], [28]) Tx is identified with a stratified space known as the augmented Teichmüller space [1], [11], S Tx D 2C .†/ T ; where C .†/ is the curve complex of †, which can be identified as the index set of nodal surfaces † specifying the location of the nodes. The stratum T is the Teichmüller space of the nodal surface † . Let be the set of homotopy classes of non-contractible, non-peripheral simple closed curves on †, i.e. the vertex set of C.S /, namely D f 2 C .†/ j jj D 1g: Here j j stands for the number of nodes in † . Hence, each element in represents a single node. In [27], [28], the third author showed that each frontier stratum Tx is embedded totally geodesically into Tx , which is a natural extension of a preceding result [21] by S. Wolpert which states that T is Weil–Petersson geodesically convex. In [29], a new space was introduced which can be viewed as a Weil–Petersson geodesic completion, called the Teichmüller–Coxeter complex D.Tx ; /. The space is a development of the original space Tx by a Coxeter

Chapter 12. Weil–Petersson Funk metric on Teichmüller space

343

group W generated by reflections across the frontier strata fTx g, a quotient space of W Tx by an equivalence relation. Here stands for a simple complex of groups [2]. It was shown by Wolpert [23] that two intersecting strata of the same dimension meet at a right angle (in the sense of the Alexandrov angle between Weil–Petersson geodesics), making the development D.Tx ; / a so-called cubical complex [2], [8]. This feature then is used to show that the development is also a CAT(0) space. The Coxeter complex setting allows to view Teichmüller space as a convex set in an ambient space D.Tx ; /, bounded by a set of complex-codimension one “supporting hyperplanes” fD.Tx ; / j 2 g of the frontier stratum fTx g with each representing a single node. Every boundary point is contained in at least one of the set of supporting hyperplanes fD.Tx ; / j 2 g. In this picture, each D.Tx ; / is a totally geodesic set, metrically and geodesically complete, and when D.Tx1 ; / and D.Tx2 ; / intersect along D.Tx1 [2 ; /, they meet at a right angle. One can view the translates of fD.Tx ; / j 2 g by the action of the Coxeter group W as forming a right-angled grid structure in D.Tx ; /, whose lattice points are the orbits by the Coxeter group W of the set fTx j j j D 3g 3g with indexing the maximal set of nodes on the surface.

3.3 Tx as a convex subset in D.Tx ; / In this setting, for each in , one can consider a half space, namely the set H in D.Tx ; /, containing Tx and bounded by D.Tx ; /. We note the fact obtained by Wolpert [23] that the Weil–Petersson metric completion Tx is the closure of the convex hull of the vertex set fTx j j j D 3g 3g, which suggests an interpretation of the Teichmüller space as a simplex. We can summarize the above discussion as S T Tx D 2 H with @Tx D.Tx ; /; where every boundary point b 2 @Tx belongs to D.Tx ; / for some in . Each half space H is bounded by the “supporting hyperplane” D.Tx ; /. 3.3.1 Geometry of the nearest point projection. We have the following general facts about the nearest point projection to a complete convex subset of a CAT.0/ space. Proposition ([2], II.2.4). Suppose that X is a CAT.0/ space, and that C is a complete convex subset of X. Then there exists a well-defined nearest point projection W X ! C satisfying the following. (1) For each x 2 X, the projection .x/ of x satisfies d.x; .x// D d.x; C /. (2) If x 0 lies on the geodesic segment Œx; .x/, then .x 0 / D .x/. (3) For x 62 C and y 2 C and y ¤ .x/, then †.x/ .x; y/ =2, where † denotes the Alexandrov angle.

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(4) The projection map x 7! .x/ is a retraction of X to C . We are interested in the case where X D Tx and C D Tx , as each Tx lies in Tx as a complete convex set. We use the symbol … to denote the nearest point projection W Tx ! Tx . Actually there is another nearest point projection, which we also denote by … from D.Tx ; / to D.Tx ; /. The former projection … W Tx ! Tx is the restriction of … W D.Tx ; / ! D.Tx ; /. It follows from the proposition that for each point x 2 Tx D.Tx ; /, there exists a nearest point projection … .x/ 2 D.Tx ; /, and that the Weil–Petersson geodesic x… .x/ meets D.Tx ; / perpendicularly, its length uniquely realizing the distance inf y2Tx d.x; y/ D d.x; … .x//. Also note that when x 2 T Tx , the foot … .x/ lies in T Tx , due to the fact that the frontier sets intersect perpendicularly in terms of the Weil–Petersson Alexandorov angle. In other words, when the projection is restricted to T , as … jT W T ! Tx , we have .… jT /1 ŒT D T for any 2 , and .… /1 ŒTx n T \ T D ;. It follows that for x 2 T and y 2 Tx with y ¤ … .x/, we have †… .x/ .x; y/ D =2, in light of the statement .3/ of the above proposition. It was shown in [9] that the Weil–Petersson geodesic segment x… .x/ is also the F2 -geodesic of infinite F2 -length provided that the stratum Tx is the closest to x among all the strata.

3.4 The Weil–Petersson Funk metric F2 We now transcribe the Euclidean Funk geometry as well as its compatible Finsler structure in the previous section to the Weil–Petersson setting. We exhibited three equivalent ways of writing down the Funk distance, which we called F1 ; F2 and F3 . In the Weil–Petersson setting, these definitions a priori differ from each other, and they are related by inequalities. We define the Weil–Petersson Funk metric F2 on T as F2 .x; y/ D sup log 2

d.x; Tx / ; d.y; Tx /

where d is the Weil–Petersson distance defined on Tx . That this is indeed a metric will be shown below, and in particular it will be convenient for that to use the formula given in Equation 3.1 below. The fact that F2 satisfies the triangle inequality will be proved using that second formula, and the fact that it is non-negative and non-degenerate is the content of Theorem 4.4. In order to make the analogy with the Euclidean setting more obvious, and in order to make clearer the viewpoint that Teichmüller space T is a convex body in the ambient space D.Tx ; / with supporting hypersurfaces fD.Tx ; g2 , we can instead define the metric as d.x; D.Tx ; // F2 .x; y/ D sup log : (3.1) d.y; D.Tx ; // 2

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Chapter 12. Weil–Petersson Funk metric on Teichmüller space

This follows immediately from the content of Subsection 3.3.1. Note that F2 .x; y/ satisfies the triangle inequality, for F2 .x; y/ C F2 .y; z/ D sup log 2

d.x; D.Tx ; // d.y; D.Tx ; // C sup log d.y; D.Tx ; // 2 d.z; D.Tx ; //

sup log 2

D sup log 2

d.y; D.Tx ; // d.x; D.Tx ; // C log d.y; D.Tx ; // d.z; D.Tx ; //

d.x; D.Tx ; // D F2 .x; z/: d.z; D.Tx ; //

As in the Euclidean case, the equality holds when the stratum D.Tx ; / achieving the value of the supremum for x and y coincides with the one for y and z. However, those strata are no longer characterized in terms of Weil–Petersson geodesic rays, and this prevents us from stating that the Weil–Petersson geodesics are Funk distance minimizers. Recall that in the Euclidean setting, the relevant supporting hypersurfaces are the ones containing the boundary points b.x; y/ and b.y; z/ where the rays hit the boundary set @. We also remark that there is no definite statement for the convexity of the Funk distance F2 .x; y/ in the y variable, unlike the Euclidean counterpart.

4 Properties of the Funk metric F2 4.1 Density of strata in the Weil–Petersson visual sphere For x 2 T , we denote by ;x the shortest Weil–Petersson geodesic from x to Tx . Recall that the endpoint of the Weil–Petersson geodesic segment is the nearest point projection … .x/ of x. We call … .x/ 2 Tx the foot of ;x . Theorem 4.1 (Density). For any point x 2 T , the set of Weil–Petersson geodesic rays f ;x g 2 is dense in the unit tangent sphere at x, where each ray, parameterized by arc-length, is uniquely represented by a unit tangent vector. Proof. The following argument is due to Wolpert [24]. Let c W Œ0; 1/ ! Tx be any Weil–Petersson geodesic ray issued from x. We shall show that c can be approximated arbitrarily closely by rays i ;x for a sequence of strata fi g with i 2 . An arc-length-parameterized Weil–Petersson geodesic ray in the Teichmüller space is said to be forward geodesic flow transitive, transitive for short, if its projection to the moduli space Mg lifted to its unit tangent bundle U TMg has dense image. By Burns–Masur–Wilkinson [5], any Weil–Petersson geodesic ray can be approximated by transitive geodesics. Let fci W Ji ! T g be a sequence of arc-length parameterized

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geodesics converging uniformly on every compact set to the geodesic ray c. There is a decreasing sequence of positive numbers i converging to zero such that both the distance between ci .0/ and c.0/ and the distance between the unit tangent vector of ci at ci .0/ and that of c at c.0/ are less than i , the latter distance being measured using the parallel transport along the geodesic connecting ci .0/ and c.0/. Since each ci is transitive, there exist a sequence of diverging positive numbers fti g, y 2 T and 2 , such that the tangent vector of ci at ti is sufficiently close to a translate of the tangent vector v0 of ;y at y by an element gi of the mapping class group Mod.†/ so that the geodesic arc i connecting ci .ti / to gi .… .y// (D …gi ./ .gi y/) forms an angle greater than =2 i with Tgi . / and the tangent vectors at ci .ti / of i and of ci form an angle less than i . Now, consider the geodesic arc connecting x and ci .ti /, which we denote by ˛i . Since the tangent vectors of c at c.0/ D x and of ci at ci .0/ are within distance i , by the CAT.0/ property, there is a sequence ıi converging to 0 such that the angle formed by ˛i and ci at ci .ti / is less than ıi , which implies that the angle formed by ˛i and i at ci .ti / is less than

i C ıi . On the other hand, as ci converges to c uniformly on any compact set, the CAT.0/ geometry, applied to the geodesic triangle 4.x; ci .ti /; c.i // where c.i / is the projection of ci .ti / on c, implies that the angle between the tangent vectors of c and of ˛i at x is less than ıi . We consider the quadrilateral formed by ˛i , i , gi . /;x and the geodesic segment …gi ./ .x/ …gi ./ .gi y/ on Tgi ./ . Then using the CAT.0/ property again, we see that the angle formed by gi ./;x and ˛i at x is less than i C ıi . Therefore, the angle formed by c and gi ./;x at x is less than i C 2ıi . This shows that c is approximated by rays in f ;x g2 arbitrarily closely. Remark 4.2. K. Fujiwara in [9] independently obtained a similar result, where it was shown that the set of Weil–Petersson geodesic rays ;x translated by the pseudoAnosov elements of the mapping class group is dense in the unit tangent sphere. Remark 4.3. Instead of taking the nearest point projections … W T ! Tx with 2 (or equivalently jj D 1), one can consider the projections … W T ! Tx with j j D 3g 3. For such a , Tx Tx , which is equal to T , is a zero-dimensional stratum representing a nodal surface † with maximal number of nodes, so that the nodal surface is a union of thrice-punctured spheres. A result by J. Brock [4] states that in the Weil–Petersson visual sphere VT .x/, of each point x 2 T , the Weil–Petersson geodesic rays ;x are dense. Here, the visual sphere VT .p/ at p is the set of all finite and infinite unit-speed geodesic rays emanating from p, and it is homeomorphic to the unit tangent sphere at p, where the correspondence is by assigning the unit tangent vector at p to each geodesic. As each … W T ! T is a projection to a point, the proof follows much more readily than that of the density theorem we have stated, which uses the heavy machinery of the Weil–Petersson geodesic flow ergodicity.

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Chapter 12. Weil–Petersson Funk metric on Teichmüller space

4.2 Non-degeneracy of the weak metric F2 Having the knowledge of the density of rays ;x with 2 in the visual sphere VT .x/, we can proceed to show: Theorem 4.4. The Funk metric F2 is non-negative and non-degenerate. Proof. Given a pair of distinct points x; y 2 T , we apply Theorem 4.1 to the geodesic arc connecting them. We then see that for any > 0, there is a simple closed curve 2 such that d.y; ;x / < . Take y 0 2 ;x with d.y; y 0 / < . Note that … .y 0 / D … .x/. Since the development D.Tx / is CAT.0/, d.… .x/; … .y// D d.… .y 0 /; … .y// d.y 0 ; y/ : Therefore, d.x; y/ C d.y 0 ; Tx / d.x; y 0 / C d.y 0 ; Tx / d.x; Tx / > : d.y 0 ; Tx / C 2 d.y 0 ; Tx / C 2 d.y; Tx / Therefore, if we take sufficiently small so that 3 < d.x; y/, we have F2 .x; y/ log

d.x; y/ C d.y 0 ; Tx / d.x; Tx / log > 0; d.y; Tx / 2 C d.y 0 ; Tx /

(4.1)

and the statement is proved.

4.3 Weil–Petersson Funk topology and Weil–Petersson topology It is well known that on the Teichmüller space T , the metric topologies (namely the topology whose open sets are realized as unions of the metric balls) with respect to the Teichmüller distance and the Weil–Petersson distance are equivalent. As for the comparison between the Weil–Petersson metric topology and the Weil–Petersson Funk metric topology, we have the following qualitative statement. Proposition 4.5. The identity map W .T ; d / ,! .T ; F2 / is locally Lipschitz. Proof. Let K be a compact set in T . Then, there is an M D M.K/ > 0 such that d.x; Tx / M for all x 2 K and all 2 . Let x; y be points in K. For any 2 , we have d.y; Tx / C d.x; y/ d.x; y/ d.x; y/ d.x; Tx / ; log log 1 C C log M d.y; Tx / d.y; Tx / d.y; Tx / for some constant C D C.K/ > 1. By taking supremum over 2 , we have C d.x; y/. Therefore the identity map is C =M -Lipschitz in K. F2 .x; y/ M

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4.4 Mapping class group invariance and the translation distances By definition, the Weil–Petersson Funk metric, F2 is invariant under the mapping class group action as the mapping class group Map.†/ is a subgroup of the permutation group of : F2 .x; y/ D F2 .gx; gy/ for all g 2 Map.†/. Namely each element g of Map.†/ is an F2 -isometry. Thurston [18] and Bers [3] classified the elements of the mapping class group into three categories; elliptic, reducible, and irreducible/pseudo-Anosov. For a metric space .X; d / and an element g of the isometry group Isom.X /, we define the translation length ı.g/ by ı.g/ D inf d.x; gx/: x2X

Fujiwara showed [9] that for each pseudo-Anosov mapping class g and a point x 2 T , there exists E > 0 such that for all n ¤ 0, log jnj E F2 .x; g n x/ log jnj C E; while F2 .x; g n x/ is bounded if g is a Dehn twist. Furthermore if g is a pseudo-Anosov mapping class, there exists Q > 0 such that for all n ¤ 0, log jnj Q ı.g n / log jnj C Q: For sufficiently large n, ı.g n / > 0 and the infimum is achieved by some x 2 T . If is a Dehn twist, ı.g n / D 0 for each n.

5 Three Funk-type metrics on Teichmüller spaces In this section, we make a comparison among three metric structures defined on Teichmüller spaces. The first is the Teichmüller metric, which is defined as follows. Definition 5.1. Let ŒG1 and ŒG2 be two conformal structures (uniformized by hyperbolic metrics Gi ) on †. The Teichmüller distance between ŒG1 and ŒG2 is given by 1 dT .ŒG1 ; ŒG2 / D inf log K; 2 f where the infimum is taken over the set of all K > 0 such that there exists a K-quasiconformal homeomorphism f W .†; ŒG1 / ! .†; ŒG2 / that is isotopic to the identity map. Here the Teichmüller space is regarded as the space of conformal structures, where the uniformization is not relevant.

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Kerckhoff [10] showed that the Teichmüller distance can be alternatively defined as dT .ŒG1 ; ŒG2 / D

Ext ŒG1 . / 1 ; sup log 2 2 Ext ŒG2 . /

where Ext ŒG ./ is the extremal length of the homotopy class of simple closed curves in †. Recall [15] that the extremal length of is defined as 1=Mod† . / where Mod† . / is the supremum of the moduli of the topological cylinders embedded in † with core curves in the class . Secondly, Thurston’s asymmetric metric [19] is defined as follows. Definition 5.2. Let G1 and G2 be the two hyperbolic metrics on †. A distance function can be defined as ` .G1 / T .G1 ; G2 / D sup log : ` .G2 / 2 It is called Thurston’s asymmetric metric. This quantity was shown by Thurston to be equal to the following number L.G1 ; G2 / D inf Lip./; Id†

where the infimum is taken over all diffeomorphisms in the isotopy class of the identity, and Lip./ is the Lipschitz constant of the map : dG2 ..x/; .y// : dG1 .x; y/ x¤y2†

Lip./ D sup

For this reason, the quantity T .G1 ; G2 / is sometimes called Thurston’s Lipschitz metric. Thurston in his paper ([19], Chapter 4) emphasizes the underlying convex geometry for the metric. In particular, the space of measured geodesic laminations is embedded into the cotangent space TG T for a fixed point G in T as the boundary set of a convex body, where the embedding is given by the differential of the logarithm of the geodesic length function of measured geodesic laminations d log length W PL.†/ ! TG T ; where the map only registers the projective classes of measured geodesic laminations, for one is taking the logarithmic derivative of the length. Now recall the new metric we have introduced above: The Weil–Petersson-Funk metric F2 on T is defined as F2 .x; y/ D sup log 2

for x and y in T .

d.x; Tx / d.y; Tx /

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Now the analogy among the Teichmüller metric, the Thurston metric and the Weil– Petersson-Funk metric is clear: For each of them we have an embedding ‚ W T ! RC where the target space has a weak metric

²

³

x d.x; y/ D sup max log ; 0 y

for x D .x /2 , and each of the three Funk-type metrics is the pulled-back metric ‚ d defined on T T . Note that the weak metric d is the Funk metric for the convex set RC in R . Lastly we remark that the infinite-dimensional orthant RC with its Hilbert distance d.x; y/ is known to be isometric to an infinite-dimensional weak Minkowski space .R ; p/, namely a linear space with a weak Minkowski norm p, where the map is given by fx g 2 7! flog x g2 . (Compare also with Theorem 6.1 of Chapter 4 in this volume for a similar result in Hilbert geometry.) Hence the three different embeddings of the Teichmüller space T into the orthant RC induce isometric embeddings of the Teichmüller space into the infinite-dimensional weak Minkowski space .R ; p/ with respect to the Hilbert metrics obtained by taking the arithmetic symmetrization of the three Funk-type metrics. This observation provides an interesting starting point for a comparative study of the three Funk-type metrics on the Teichmüller space: the Teichmüller metric, the Thurston metric and the Weil–Petersson-Funk metric. Acknowledgement. This work is partially supported by MEXT KAKENHI 21540177 (H.M.), by JSPS KAKENHI 22244005 (K.O.) and by JSPS KAKENHI 24340009 (S.Y.).

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K. Fujiwara, Geometry of the Funk metric on Weil–Petersson spaces. Math. Z. 274 (2013), 647–665.

[10] S. Kerckhoff, The asymptotic geometry of Teichmüller space. Topology 19 (1981), 23–41. [11] H. Masur, The extension of the Weil–Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43 (1976), 267–304. [12] R. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 75 (1988), no. 391. [13] A. Papadopoulos, Problem 13 in Problem Session “Teichmüller Theory”. In Oberwolfach Reports Vol. 7, European Mathematical Society, Zürich 2010, 3085–3157. [14] A. Papadopoulos and M. Troyanov, Weak Finsler structures and the Funk weak metric. Proc. Cambridge Philos. Soc. 147 (2009), 419–437. [15] A. Papadopoulos and G. Théret, On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, European Mathematical Society, Zürich 2007, 111–204. [16] B. B. Phadke, A triangular world with hexagonal circles. Geom. Dedicata 3 (1975), 511–520. [17] H. Royden, Automorphisms and isometries of Teichmüller space. In 1971 Advances in the theory of Riemann surfaces, Ann. Math. Studies 66, Princeton University Press, Princeton, N.J., 1971, 369–383 [18] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417–431. [19] W. P. Thurston, Minimal stretch maps between hyperbolic surfaces. Preprint, arXiv:9801039v1. [20] S. Wolpert, Noncompleteness of the Weil–Petersson metric for Teichmüller space. Pacific J. Math. 61 (1975), 573–577. [21] S. Wolpert, Geodesic length functions and the Nielsen problem. J. Differential Geom. 25 (1987), 275–296. [22] S. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space. In Surveys in differential geometry, Vol. VIII, Surv. Differ. Geom. 8, International Press, Cambridge, MA, 2003. [23] S. Wolpert, Behavior of geodesic-length functions on Teichmüller space. J. Differential Geom. 79 (2008), 277–334. [24] S. Wolpert, Personal communication. [25] S. Wolpert, The Weil–Petersson metric geometry. In Handbook of Teichmüller theory (A. Papadopoulos ed.), Volume II, European Mathematical Society, Zürich 2009, 47–64. [26] S. Wolpert, Families of Riemann surfaces and Weil–Petersson geometry. CBMS Regional Conf. Ser. in Math. 113, Amer. Math. Soc., Providence, RI, 2010. [27] S. Yamada, Weil–Petersson completion of Teichmüller spaces and mapping class group actions. Preprint, arXiv:0112001.

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[28] S. Yamada, On the geometry of Weil–Petersson completion of Teichmüller spaces. Math. Res. Lett. 11 (2004), 327–344. [29] S. Yamada, Weil–Petersson geometry of Teichmüller–Coxeter complex and its finite rank property. Geom. Dedicata 145 (2010), 43–63. [30] S.Yamada, Convex bodies in Euclidean and Weil–Petersson geometries. Proc. Amer. Math. Soc. 142 (2014), 603–616 [31] S. Yamada. Local and global aspects of Weil–Petersson geometry. In Handbook of Teichmüller theory (A. Papadopoulos ed.), Volume IV, European Mathematical Society, Zürich 2012, 43–111.

Chapter 13

Funk and Hilbert geometries in spaces of constant curvature Athanase Papadopoulos and Sumio Yamada

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 On analogies between the Euclidean and the non-Euclidean geometries 3 The Funk metric in hyperbolic and in spherical geometries . . . . . . . 4 Properties of the Funk metric . . . . . . . . . . . . . . . . . . . . . . 5 The Finsler structure of the Funk metric . . . . . . . . . . . . . . . . . 6 An example of a hyperbolic Funk metric . . . . . . . . . . . . . . . . 7 The cross ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Hilbert metric in Hn and in S n . . . . . . . . . . . . . . . . . . . 9 Projective transformations as isometries of Hilbert metrics . . . . . . . 10 Hilbert metrics as models of the hyperbolic plane . . . . . . . . . . . . 11 Examples of Hilbert metrics . . . . . . . . . . . . . . . . . . . . . . . 12 On Hilbert’s Problem IV . . . . . . . . . . . . . . . . . . . . . . . . . 13 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

353 354 358 360 361 363 364 366 367 368 369 370 370 377

1 Introduction It is well known that convex sets and their properties can be studied not only in Euclidean space but also in other metric spaces and in particular in the spaces of constant curvature: the sphere S n (or more conveniently the upper-hemisphere U n S n ) and the hyperbolic space Hn . The Funk and Hilbert metrics of convex subsets of S n and Hn can be defined in a way which is parallel to the way they are defined in the Euclidean space Rn . In particular, in each of these spaces of constant curvature, there is an invariant under the so-called projectivity (or perspectivity) transformation group which generalizes the cross ratio defined on real projective space. We developed this point of view in our papers [21] and [22] and we highlighted there some analogies and some differences between the three constant curvature settings, Rn , S n and Hn . The goal of the present chapter is to give a comprehensive survey of

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these results and to work out a few examples. We also provide a historical perspective for this theory. This study can be included in the setting of a generalized form of Hilbert’s Problem IV.

2 On analogies between the Euclidean and the non-Euclidean geometries The analogy we describe in this chapter between the Euclidean, spherical and hyperbolic worlds is in the lineage of several observations that were made by various people along several decades (even centuries), stressing on formal analogies between results and formulae in the three constant curvature geometries. In this section, we review such analogies. This will motivate our introduction of the non-Euclidean Funk and Hilbert metrics. We start with the observation that it is remarkable that one can pass from certain trigonometric formulae of Euclidean geometry to corresponding formulae in spherical (respectively hyperbolic) geometry by replacing side lengths by the sines (respectively the hyperbolic sines) of these lengths or by similar formal transformations. The simplest and most well-known example of such an occurrence is the following result known as the sine rule.1 Consider, in any one of the three geometries of constant curvature 0, 1 or 1, a triangle ABC with sides a, b, c opposite to the vertices A, B, C . We have, in Euclidean geometry, a b c D D ; sin A sin B sin C in spherical geometry, sin b sin c sin a D D ; sin A sin B sin C and in hyperbolic geometry, sinh b sinh c sinh a D D : sin A sin B sin C From these formulae, one can get other sets of formally analogous formulae in the three geometries, since many trigonometric formulae can be obtained by using the sine rule alone. As an example, the length ` of the side of a regular n-gon inscribed in a circle of radius R is given by the following: in Euclidean geometry, ` D R sin ; 2 n 1 These results and other results stated below are valid for spaces of constant curvature 0, 1 and 1. For spaces of different constant curvature, a normalization factor is needed. For the proof, and for other trigonometric formulae in hyperbolic trigonometry we refer the reader to [1], where the proofs are given in a model-free setting. The advantage of such a setting is that the proofs can be done at the same time in the spherical and in the hyperbolic geometry cases.

Chapter 13. Funk and Hilbert geometries in spaces of constant curvature

in spherical geometry, sin

355

` D .sin R/ sin ; 2 n

and in hyperbolic geometry, ` D .sinh R/ sin : 2 n Another formula with similar patterns in the three geometries is the one of the for circumference of the circle of radius r. In the Euclidean plane it is r, on the sphere it is 2 sin r and in the hyperbolic plane2 it is 2 sinh r. These formulae can be obtained by taking limits in the preceding formulae for n-gons as the number of sides tends to infinity. In the appendix, we review some other results of this kind. Finally, let us mention the famous relation known as Ptolemy’s Theorem, which gives a necessary and sufficient condition for a quadrilateral to be inscribed in a circle. In this context, a circle is a geometric circle, i.e. it is an equidistant locus from a point. This result says that a quadrilateral with side lengths a, b, c, d and diagonal lengths e and f is inscribed in a circle if and only if we have, in Euclidean geometry, sinh

ef D ac C bd; in spherical geometry, sin e sin f D sin a sin c C sin b sin d; and in hyperbolic geometry, sinh e sinh f D sinh a sinh c C sinh b sinh d where a, b, c, d denote the doubles of the side lengths and e, f denote the doubles of the diagonal lengths (see Figure 1). There are interesting generalizations of these relations in the papers [30] and [31] by Valentine and in the paper [7] by Guo and Sönmez. The following is a brief list of observations that were made by various people concerning the analogies between the three geometries. The Alsatian mathematician Johann Heinrich Lambert (1728–1777), in his Theorie der Parallellinien, [25] and [12], written in 1766, made the observation that certain properties of hyperbolic geometry (which, for him, was hypothetical) can be obtained from corresponding properties of spherical geometry by replacing certain distances p by the same distances multiplied by the imaginary number 1. In fact, if we take as a model of spherical geometry a sphere of radius r in the 3-dimensional Euclidean space,3 then, by a classical theorem attributed to Albert Girard (1595–1632), the area 2 This

formula in the case hyperbolic geometry was known to Lobachevsky and it is contained in several of his memoirs; see e.g. [13], p. 34. It is also contained in a letter by Gauss to Schumacher, dated July 31, 1831. 3A sphere of radius r has constant curvature 1=R2 . We recall however that Lambert did not use this language since he did not have the notion of curvature.

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a

b

f

e d

c

Figure 1

of a triangle on that sphere is a multiple of the angular excess; more precisely, it is equal to the quantity (2.1) r 2 .˛ C ˇ C / where ˛, ˇ, are the three angles of the triangle (in radians). p If in Equation 2.1 for the area we replace the radius r by an “imaginary radius” 1r, we obtain r 2 .˛ C ˇ C / D r 2 . ˛ ˇ /; which is the area of triangle of angles ˛, ˇ, in a hyperbolic space of constant curvature. Lambert declares at this occasion that one “is almost led to the conclusion that the third hypothesis4 occurs on a sphere of imaginary radius”. In the 19th century, spherical and hyperbolic geometry formulae were commented by F. A. Taurinus (1794–1874) (a contemporary of Gauss and Lobachevsky), who pointed out in 1825 simple passages between hyperbolic and spherical trigonometry using transformations similar to those we mentioned. In his memoir, which is also called Theorie der Parallellinien [25], he obtained fundamental trigonometric formulae for hyperbolic geometry (which was, from his point of view, like it was for Lambert, purely hypothetical), again by working on a sphere of “imaginary radius”, and he called the geometry that he so obtained the logarithmic-spherical geometry (“logarithmisch-sphärische Geometrie”) 4 The “third hypothesis”, in Lambert’s memoir, is the hypothesis under which the angle sum in some triangle, or, equivalently, in any triangle, is less than two right angles. Lambert made this hypothesis at some point in his memoir Theorie der Parallellinien with the aim of deriving a contradiction. He did not find the contradiction he was seeking for but his development of the subject constitutes a substantial part of the field of hyperbolic geometry. We refer the reader to [12] for a critical edition with a French translation and a mathematical commentary of Lambert’s work.

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Lobachevsky, in his Elements of geometry [14], in his Geometrical Researches on the Theory of Parallels [15], in his Pangeometry [13] and in other works, noted an analogous passage between the formulae of hyperbolic trigonometry and those of spherical trigonometry which consists in replacing, in the trigonometric equations of spherical geometry, p p the edge lengths a, b, c of a triangle by the imaginary quantities p in these equations the terms a 1, b 1, c 1, and at the same time, replacing p p sin ….a/, cos ….a/ and tan ….a/ by 1= cos a, 1 tan a and 1= 1 sin a respectively. (Here, … denotes Lobachevsky’s angle of parallelism function, see [14], [15] and the exposition in [1].) Beltrami, in his Saggio di interpretazione della geometria non-euclidea [2], showed that the trigonometric formulae for the pseudo-sphere (which is a model he had constructed for the hyperbolic plane) can be obtained from those ofpthe usual sphere by considering the pseudo-sphere as a sphere of imaginary radius 1, and he attributed this observation to E. F. A. Minding and to D. Codazzi. Let us quote Beltrami from [2] (Stillwell’s translation [26], p. 18): By an observation of Minding (Vol. XX of Crelle’s Journal), the ordinary formulae for spherical triangles are converted into those for geodesic triangles on a surface p of constant negative curvature by inserting the factor 1 in the ratio of the side to radius and leaving the angles unaltered, which amounts to changing the circular functions involving the radius into hyperbolic functions. For example, the first formula of spherical trigonometry cos becomes cosh

b c b c a D cos cos C sin sin cos A R R R R R

c b c b a D cosh cosh C sinh sinh cos A: R R R R R

Here, in the case of spherical geometry, R denotes the radius of the sphere, which means that we take the sphere of constant curvature R2 , and in the case of hyperbolic geometry, the plane has constant curvature R2 . We note that it is because of such a “duality” between the trigonometric formulae for spherical and for hyperbolic geometry that Beltrami chose, for hyperbolic geometry, the name “pseudo-spherical geometry”. Klein, in his Über die sogenannte Nicht-Euklidische Geometrie [11] made a similar observation (Stillwell’s translation [26], p. 99): The trigonometric formulae that hold for our measure result from the formulae of spherical trigonometry by replacing sides by sides divided by ci .

Story, in his paper On the non-Euclidean trigonometry [27], working in Klein’s setting of projective geometry, showed that some trigonometric formulae of hyperbolic geometry can be deduced from those of spherical geometry by replacing each length by the corresponding length divided by 2ik and each angle by the corresponding angle divided by 2ik 0 , where k and k 0 are two constants which can be imaginary and which are related to Klein’s constants in [11].

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Coxeter discovered a passage between formulae for volumes of hyperbolic polyhedra and formulae for volumes of spherical polyhedra, see [5]. In this work, Coxeter made a relation between computations of Lobachevsky in hyperbolic geometry and formulae discovered by Schläfli on the sphere. Thurston, in his notes on geometry (see [28] and [29]) continues this tradition of using the term “sphere of imaginary radius” to describe hyperbolic space. This refers usually to the so-called hyperboloid model of n-dimensional hyperbolic geometry embedded in .n C 1/-dimensional Euclidean space, a model that bears a close analogy with the n-sphere embedded in the same Euclidean space. Indeed, whereas the sphere in this model is the space of unit norm vectors for the quadratic form q.x/ D x02 C x12 C C xn2 , the hyperbolic space is a connected component of the space of vectors of norm i for the quadratic form q.x/ D x02 C x12 C xn2 . In both models, lines (and more-generally, k-dimensional planes, for 1 k n 1) are the intersections with the model with planes (or .k C 1/-planes) passing through the origin, etc.

3 The Funk metric in hyperbolic and in spherical geometries We start by some remarks on convexity. In hyperbolic n-space Hn , convexity is defined as in Euclidean space: we say that a set Hn is convex if for any two points in , the geodesic segment joining them is contained in . To define convexity for an open subset of the sphere S n , one has to take into account the fact that geodesic segments between two points are not unique. Even if we restrict ourselves to geodesics of shortest length, there may be two distinct such geodesics joining some pairs of points of the sphere, namely, diametrically opposite points. For this reason, we shall always assume that all subsets of S n that we consider are contained in an open hemisphere. Another way of proceeding would be to work in elliptic space (the quotient of the sphere by its order-two canonical involution) rather than on the sphere itself, but we prefer to work on the sphere. In any case, and modulo this complication, a convex set in the sphere (or in projective space) is a subset whose intersection with any line (that is, any great circle) is connected. In the rest of this section, will be an open convex subset of one of the three constant curvature spaces, Rn , S n or Hn . We do not assume that is bounded but we shall always assume that is not equal to the whole space Rn or Hn and, in the case of the sphere, we shall assume that is contained in an open hemisphere. At some point below, we shall assume, for reasons that will become clear, that our convex subset S n has diameter < =2 with respect to the angular metric of the sphere. We shall specify this whenever needed. We now define the Funk metric. Let us first consider the case of the hyperbolic space.

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Given two distinct points x and y in a convex set in Hn or S n , we denote by R.x; y/ D fexpx .txy / j t > 0g the geodesic ray starting at x and passing through y. Here xy is the unit vector at x tangent to the arc-length parameterized geodesic in Hn or S n connecting x to y. The ray R.x; y/ is the image of the exponential map ft 7! expx txy ; t 0g. Note that for the convex sets under consideration, each exponential map restricted to the component of exp1 x ./ containing the origin of the tangent space is a diffeomorphism onto . (This point is only relevant in the spherical case because of the existence of cut loci.) In the three geometries (Euclidean, hyperbolic and spherical), a hyperplane is a complete totally geodesic codimension-one subspace, and an (open) half-space is a connected component of the complement of a hyperplane in one of these spaces. (Note that on the sphere S n , hyperplanes are codimension-one great spheres and that half-spaces are called hemispheres.) n n n T Any convex set in one of the three geometries R , H or S can be represented as .b/2P H.b/ where H.b/ is a half-space bounded by a hyperplane .b/ tangent to @ at a boundary point b. We call these totally geodesic submanifolds .b/ supporting hyperplanes of . We denote by P the set of all supporting hyperplanes of and by P .b/ P the set of supporting hyperplanes of at the boundary point b 2 . Definition 3.1 (The hyperbolic Funk metric). For a pair of points x and y in Hn , the Funk (asymmetric) distance from x to y is defined as 8 d.y; b.x; y// for x ¤ y. This implies that the Funk distance F .x; y/ is always non-negative. Given a point x in the convex set S n and a hyperplane .b/ supporting at a boundary point b 2 @, we have d.x; .b// =2. The Funk metric on a convex subset S n is an asymmetric metric and it is also given by the following variational formula: F .x; y/ D sup log 2P

sin d.x; / sin d.y; /

for x 6D y.

4 Properties of the Funk metric Assume as before that is an open convex subset of any one of the three constant curvature geometries, let d denote the distance function in that geometry and let F .x; y/ denote the Funk metric of . The following propositions either follow easily from the definitions, or they are proved in [22]. Proposition 4.1. The Funk metric is unbounded. Proposition 4.2. We have the following: (1) The d -geodesics of are also Funk geodesics.

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(2) All the Funk geodesics of are d -geodesics if and only if is strictly convex, that is, if and only if the boundary of does not contain any nonempty open d -segment. We recall that a path s W Œa; b ! .X; d / in a metric space .X; d / is said to be geodesic if for any a < t1 < t2 < t3 < b, the equality d.s.t1 /; s.t2 // C d.s.t2 /; s.t3 // D d.s.t1 /; s.t3 // is satisfied. The reader should remember that in writing such an equality, when the metric is not symmetric (as in the case of the Funk metric), the order of the variables in d.; / must be respected. A geodesic ray is a geodesic s W Œ0; 1Œ! .X; d / whose restriction to any segment is a geodesic path. Proposition 4.3. The image of any d -geodesic segment starting at a point x in and ending at a point on @ is the image of an F -geodesic ray. Proposition 4.4. The Funk metric F separates points (that is, we have F .x; y/ > 0 () x 6D y) if and only if is bounded. Proposition 4.5. If is bounded, then, we have, for every x in and for every sequence xn in , F .x; xn / ! 0 () d.x; xn / ! 0 () F .xn ; x/ ! 0: Having a geodesic metric space, it is natural to ask whether the distance function to a point is convex, this being a kind of non-positive curvature property. In [22] we prove the following: Proposition 4.6. In spherical geometry, the Funk metric F .x; y/ defined on a convex set S n is convex in the y-variable. In hyperbolic geometry, the statement does not hold. In Euclidean geometry, the Funk metric F .x; y/ is convex in the y-variable, see [32].

5 The Finsler structure of the Funk metric In each of the three geometries, there is an infinitesimal linear structure for the Funk metric F , which provides a description of this metric as a Finsler metric. Recall that a family of linear functionals p;x defined on the family of tangent spaces Tx of a manifold is called a Finsler structure for a metric d on if the distance d is equal to the infimum of the lengths of all the piecewise C 1 paths W Œ0; 1 ! with .0/ D x; .1/ D y, that is, Z 1 p;.t / .P .t //dt: F .x; y/ D inf

0

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We now describe the Finsler norm of tangent vectors to for the Funk metric. We first recall the Euclidean setting. In this setting, the Funk metric is induced by the tautological Finsler structure in the sense of [19]. The norm function (also called the Minkowski functional) on each tangent space is given by the following: p;x ./ D sup

2P

kk d.x; T .x; ; //

where T .x; ; / is the point where the geodesic ray in Rn from x with initial velocity intersects the hyperplane . In this formula, it is easy to see that the supremum is achieved when the point T .x; ; / coincides with a boundary point b 2 @, namely 2 P .T .x; ; //. Using similarity of Euclidean triangles, this can be written as p;x ./ D sup

2P

h; i d.x; /

where is the unit tangent vector at x with direction opposite to the gradient vector of the functional d.; /. In the hyperbolic setting, the value of the Minkowski functional for the Funk metric F is given by p;x ./ D sup

2P

kk tanh d.x; T .x; ; //

(5.1)

h .x/; iHn tanh d.x; /

(5.2)

or, equivalently, by p;x ./ D sup

2P

where, as in the Euclidean case, T .x; ; / is the point where the geodesic ray in Hn from x with initial tangent vector intersects the hyperplane and .x/ is the unit tangent vector at x whose direction is opposite to the one of the gradient vector of the functional d.; / at x. We note that the gradient vector field’s integral curves are the geodesics which meet the supporting hyperplane perpendicularly. The details are given in [22]. Note that in either representation it is easy to see that the Minkowski functional is convex in 2 Tx Rd , since the functional p;x is convex (in fact it is linear) in for each fixed 2 P , and since by taking the sup over , convexity is preserved. We recall that in the Euclidean setting, the Funk metric for a convex set corresponds z in the sense to a linear structure which we called the tautological Finsler structure of [19]: z D f.x; / 2 Rn j x 2 and x C 2 g: It is called tautological because the fiber over each point x of is the set itself, with the point x as origin. In the hyperbolic setting, to display such a property, we need to incorporate the exponential map expx W Tx Hn ! Hn . Namely, the Finsler structure

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363

corresponds to the following subset of the total space of the tangent bundle T Hn : z D f.x; / 2 T Hn j x 2 and expx 2 g; where the fiber over each point x of is the set exp1 x ./, with the point x identified z is given as the origin of the tangent space Tx Hn . The Finsler norm of elements of by the Minkowski functional p;x . The Minkowski functional formula for the spherical Funk metric is analogous to that of the hyperbolic case; it is simply obtained by replacing the hyperbolic sine and cosine functions by the usual sine and cosine functions respectively. We will omit its explicit description here. As a tautological Finsler structure, it corresponds to the following subset of the total space of the tangent bundle to the sphere: z D f.x; / 2 T S n j x 2 and expx 2 g:

6 An example of a hyperbolic Funk metric We take to be a half-plane in H2 . In the upper half-plane model, and up to an isometry of this hyperbolic plane, we can regard the positive y-axis as the unique supporting hyperplane for (which in this case is a line), which we call . Thus the hyperbolic half-plane is described in the complex coordinate z as fz W Im.z/ > 0; Re.z/ > 0g H2 , see Figure 2. Now we consider the nearest point projection of a point p in the first quadrant onto the line and we calculate the value of sinh d.p; /.

f .p/

p

Figure 2

The nearest point projection of p to is called the foot of p on and denoted by f .p/. The hyperbolic distance between p and f .p/ is the length of the hyperbolic geodesic connecting these two points, which in this model is an arc A of a Euclidean circle centered at the origin, since the circle meets the x-axis and the y-axis perpendicularly.

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Denoting the polar coordinates of p by .r; /, we have f .p/ D .r; =2/. Let

W t 7! .r cos t; r sin t / be the parametrized geodesic segment A. Then the norm of the tangent vector k .t /0 k to the path at t is computed using (1) .t /0 D .r sin t; r cos t /; (2) the hyperbolic metric tensor ds 2 D .dx 2 C dy 2 /=y 2 ; (3) y D r sin t . This gives k .t /0 k D 1= sin t . Thus the hyperbolic length of the arc A is Z =2 Z =2 1 0 dt D log j tan. =2/j: k .t / k dt D L.A/ D sin t Note that as 0 < < =2, L.A/ is indeed positive. Hence we have sinh d.p; / D sinh L.A/ D

1 t .p/2 ; 2t .p/

where tan. .p/=2/ D t .p/. From this, we can compute the Funk distance F .p; q/ between two points p and q in H2 . If the ray R.p; q/ from p through q does not hit the boundary line of , then F .p; q/ D 0. If the ray R.p; q/ from p through q hits at some point b.p; q/, then

t .q/ 1 t .p/2 F .p; q/ D log t .p/ 1 t .q/2

or, equivalently, by using elementary geometry,

1 m.q/2 m.p/ F .p; q/ D log 1 m.p/2 m.q/

where m.p/ is the absolute value of the slope of the Euclidean line segment connecting p and f .p/

7 The cross ratio We define the cross ratio for four collinear ordered (and thus distinct) points A1 , A2 , A3 , A4 , in the Euclidean case, by ŒA2 ; A3 ; A4 ; A1 e WD

A2 A4 A3 A1 ; A3 A4 A2 A1

in the hyperbolic case, as ŒA2 ; A3 ; A4 ; A1 h WD

sinh A2 A4 sinh A3 A1 ; sinh A3 A4 sinh A2 A1

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and in the spherical case, by ŒA2 ; A3 ; A4 ; A1 s WD

sin A2 A4 sin A3 A1 ; sin A3 A4 sin A2 A1

where in each case Ai Aj stands for the distance (in the given space) between the pair of points Ai and Aj , which is equal to the length of the line segment joining them. In the spherical geometry case, we assume that the four points are contained in an open hemisphere. We note that in the modern formulation, the cross ratio is defined using the algebraic length, which makes it a signed quantity. We do not need this fact here. In [22], we showed that these three cross-ratios are the manifestation of the same entity; they are obtained from each other via projection maps between familiar representations of the three geometries in RnC1 . More specifically, consider the sphere S n as the set of unit vectors in RnC1 with respect to the Euclidean norm 2 x12 C C xn2 C xnC1 D1

and the hyperbolic space Hn as the set of “vectors of imaginary norm i ” with xnC1 > 0 in RnC1 with respect to the Minkowski norm 2 x12 C C xn2 xnC1 D 1:

These models of the two constant curvature spaces are called “projective”. The geodesics in the curved spaces are realized as the intersection of the unit sphere with the two-dimensional subspace of RnC1 through the origin of this space. In [22], we proved the following: Theorem 7.1 (Spherical case). Let Ps be the projection map from the origin of RnC1 sending the hyperplane fxnC1 D 1g RnC1 onto the open upper hemisphere U n of S n . Then Ps preserves the values of the cross ratios; namely, for a set of four ordered pairwise distinct points A1 , A2 , A3 , A4 aligned on a great circle in the upper hemisphere, we have ŒPs .A2 /; P2 .A3 /; Ps .A4 /; Ps .A1 /s D ŒA2 ; A3 ; A4 ; A1 e : Theorem 7.2 (Hyperbolic case). Let Ph be the projection map from the origin of RnC1 sending the unit disc of the hyperplane fxnC1 D 1g RnC1 onto the hyperboloid Hn RnC1 . Then the map Ph preserves the cross ratios; namely, for a set of four ordered pairwise distinct points A1 , A2 , A3 , A4 aligned on a geodesic in the upper hyperboloid, we have ŒPh .A2 /; Ph .A3 /; Ph .A4 /; Ph .A1 /h D ŒA2 ; A3 ; A4 ; A1 e : We also mention the following: Proposition 7.3. The Euclidean, spherical and hyperbolic cross ratios are projectivity invariants.

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We elaborate on the meaning of this statement in the case of spherical geometry: consider four ordered distinct line segments l1 , l2 , l3 , l4 intersecting at a point A and let l be a geodesic line intersecting l1 , l2 , l3 , l4 at points A1 , A2 , A3 , A4 respectively (Figure 3). We assume as usual that all these line segments are contained in an open hemisphere, so that they are all distance minimizing paths between their endpoints. Then we have

2 2

2 2

sin A2 AA4 sin A3 AA1 sinh A2 A4 sinh A3 A1 D : sinh A3 A4 sinh A2 A1 sin A3 AA4 sin A2 AA1 Thus, the cross ratio depends only on the angles that the four lines make. The details are given in the appendix.

A4 A3 A2 A1

l1

l2

l3

l4

Figure 3

Proposition 7.3 can also be deduced from the perspectivity invariance of the Euclidean cross ratio and then using the projections Ph and Ps .

8 The Hilbert metric in Hn and in S n We start with the case of the hyperbolic space Hn . Let be again an open convex (as before, possibly unbounded) proper subset of Hn . Definition 8.1 (The hyperbolic Hilbert metric). The Hilbert metric of is defined by 1 F .x; y/ C F .y; x/ H.x; y/ D 2 1 sinh d.x; b.x; y// sinh d.y; b.y; x// D log 2 sinh d.y; b.x; y// sinh d.x; b.y; x// 1 D logŒx; y; b.x; y/; b.y; x/h : 2

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The pair .; H.x; y// satisfies all the axioms of a metric space except, perhaps, for the separation axiom (which says that if x 6D y then H.x; y/ > 0). The separation axiom is satisfied if and only if does not contain a bi-infinite geodesic of Hn . In particular, if is bounded, then H.x; y/ is a genuine metric. The hyperbolic geodesic segment connecting x and y is a Funk geodesic realizing both lengths F .x; y/ and F .y; x/. This implies that the hyperbolic geodesic line segment is also a Hilbert geodesic. Proposition 8.2 (Hilbert geodesics). For any convex subset of Hn , we have the following: (1) the geodesics for the hyperbolic metric are Hilbert geodesics; (2) the hyperbolic geodesics are the unique Hilbert geodesics joining their endpoints if and only if satisfies the following property: there does not exist two hyperbolic geodesic segments in @ of nonempty interior which span a 2-dimensional totally geodesic subspace in . The case of the spherical Hilbert metric, as the reader might have guessed, is treated in analogy with case of the hyperbolic Hilbert metric. We define the spherical Hilbert metric H.x; y/ as an arithmetic symmetrization of the Funk spherical metric, that is, we set 1 H.x; y/ D F .x; y/ C F .y; x/ 2 sin d.x; b.x; y// sin d.y; b.y; x// 1 D log 2 sin d.y; b.x; y// sin d.x; b.y; x// 1 D logŒx; y; b.x; y/; b.y; x/s : 2 Thus, the formula in the spherical case is obtained from that of the hyperbolic case by replacing the sinh function by the sine function. We assume here that the convex set is contained in an open hemisphere. Unlike the case of the Funk spherical metric, we do not need to assume that is contained in a subset of the sphere of diameter < =2, for the value H.x; y/ as defined is always non-negative. We will see below (Theorem 9.1) that all isometries of the Hilbert metric are induced by isometries of the hyperbolic metric.

9 Projective transformations as isometries of Hilbert metrics Proposition 7.3 saying that the generalized cross ratio is a projective invariant implies that a projective self-transformation of Rn , Hn or S n induced by a linear projective transformation of RnC1 , induces an isometry of the Hilbert metric H of a convex set in any one of these geometries; namely, the following two spaces are isometric

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z Hz / .; H / Š .;

z D ./. Clearly every isometry of the space forms Rn , Hn and S n is a where projective transformation. One can ask whether there are isometries of Hilbert metrics that are not induced by isometries of the underlying Riemannian manifolds. In Rn , the set of linear projective transformation is the general linear group GL.n; R/, a strictly larger group than the group of the linear isometries O.n/. From the fact, whose proof is given in the appendix (Proposition 13.5), that in hyperbolic geometry each projective transformation is an isometry of the space, we have the following: Theorem 9.1. Given a convex set in the hyperbolic space, a diffeomorphism is z H z / with z D ./ if and only an isometry of the Hilbert metric: .; H / Š .; if is the restriction of a hyperbolic isometry to . On the other hand, in the spherical case, a projective transformation of S n RnC1 can be regarded as an element of the general linear group GL.n C 1; R/. Therefore we have the following: Theorem 9.2. Given a convex set in S n , each element of GL.n C 1; R/ induces z H z / where z D ./. an isometry of the Hilbert metric: .; H / Š .; We note that by Theorems 7.1 and 7.2, which state that the projection maps Ph and Ps are natural with respect to perspectivities, each bounded convex set Hn corresponds to a convex set Ph1 ./ Rn with the hyperbolic Hilbert metric H isometric to the Euclidean Hilbert metric HP 1 ./ , and that each convex set 0 h

U n S n corresponds to a convex set Ps1 .0 / Rn with H0 isometric to HPs1 .0 / . This observation in effect says that among the three geometries of Rn , Hn and S n , the Hilbert geometries of bounded convex sets on each space are mutually identifiable. In light of this fact, the isometries of a Hilbert metric for a bounded convex set are always realized as that induced from the isometries of Hn via the identifications through the perspectivities Ps and Ph .

10 Hilbert metrics as models of the hyperbolic plane The projection maps Ps and Ph described in §7 between the Euclidean plane Rn D fxnC1 D 1g RnC1 , the upper hemisphere U n S n RnC1 and the hyperboloid Hn RnC1 are natural with respect to the cross ratios defined in each of these spaces n fxnC1 D 1g a disk of radius R > 0 centered (Theorems 7.1 and 7.2). Denote by DR at the origin of the Euclidean plane. By restricting the projection map Ps to the disk

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n DR , we obtain a map from this disk into the sphere S n . For R < 1, by restricting n , we obtain a map from this disk into the hyperboloid/hyperbolic space Hn . Ph to DR These maps are natural with respect the corresponding cross ratios, and therefore they n equipped with are isometries with respect to the Hilbert metrics. Since the disk DR its Hilbert metric is a model of hyperbolic geometry (the Beltrami–Klein model), we obtain in this way new models of hyperbolic space which sit inside hyperbolic space and inside the sphere respectively. We call such models generalized Beltrami–Klein models, since they are defined using the spherical and the hyperbolic cross ratios respectively. Thus we identify, in each of the positively/negatively curved spaces, a nested family of geodesic balls as models of the hyperbolic space. Note that the limit of these models in S n and in Hn as the radius R goes to =2 and 1 respectively, is the upper hemisphere U n and the entire space Hn whose Hilbert metrics, if we define them by the formula we used for proper open convex subsets, are metrics that are identically zero.

11 Examples of Hilbert metrics Let us first recall the Hilbert metric of a Euclidean planar example, described in detail by Phadke in the paper [23] (although the expression “Hilbert metric” is not used there). This is the case where is the quarter-plane fx > 0; y > 0g R2 . Denoting by .x1 ; x2 / and .y1 ; y2 / the coordinates of two points x and y in , we have, for x 6D y, the following formula for the Hilbert distance between them: ²ˇ ³ x1 ˇˇ ˇˇ x2 ˇˇ ˇˇ x1 y2 ˇˇ 1 ˇ H .x; y/ D max ˇlog ˇ; ˇlog ˇ; ˇlog ˇ : 2 y1 y2 y1 x2 Using our non-Euclidean cross ratios and the natural projections among the three geometries, we can write formulae for a quarter-plane in the hyperbolic plane H2 , and the tri-rectangular geodesic triangle of the upper hemisphere U S 2 using the obvious coordinates. Namely for the spherical case, they are the latitude values x1 , x2 of the images of the nearest point projection to a pair of longitude great circles separated by =2 in the upper hemisphere. In the hyperbolic case, the coordinates x1 , x2 are the distances measured from the vertex of the quarter plane to the feet of the nearest point projections to the pair of perpendicular geodesics bounding the quarter plane. In these spaces transported by the projection maps, we have ²ˇ ³ sin x1 ˇˇ ˇˇ 1 sin x2 ˇˇ ˇˇ sin x1 sin y2 ˇˇ ˇ H .x; y/ D max ˇlog ; log ; log ˇ ˇ ˇ ˇ ˇ 2 sin y1 sin y2 sin y1 sin x2 in the spherical setting, and ²ˇ ³ 1 sinh x1 ˇˇ ˇˇ sinh x2 ˇˇ ˇˇ sinh x1 sinh y2 ˇˇ ˇ H .x; y/ D max ˇlog ˇ; ˇlog ˇ; ˇlog ˇ 2 sinh y1 sinh y2 sinh y1 sinh x2 in the hyperbolic setting.

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We finally recall that by a special case of a result due to Nussbaum [4], de la Harpe [8] and Foertsch–Karlsson [6], the Hilbert metric of a convex open subset of R2 is isometric to a Minkowski normed space (that is, a translation-invariant metric on R2 arising from a norm) if and only if is a triangle.5 Since any triangle in the projective plane P 2 is projectively equivalent to the quarter-plane , the above formula in the Euclidean case applies to an arbitrary plane triangle. We conclude that the non-Euclidean analogues that we exhibit are formulae for non-Euclidean Hilbert metrics which are isometric to (linear) Minkowski metrics.

12 On Hilbert’s Problem IV Hilbert’s Fourth Problem, from the collection of mathematical problems he presented in 1900 at the Second International Congress of Mathematicians in Paris, is titled: “Problem of the straight line as the shortest distance between two points”. The problem asks for the characterization and the study of individual metrics (symmetric but also asymmetric) on subsets of Euclidean space for which the Euclidean straight lines are geodesic; see [10] and the survey [16] in this volume. It is natural to ask an analogue of Hilbert’s problem in the non-Euclidan context that we consider here, that is, to study the metrics on subsets of the sphere and of hyperbolic space for which the spherical and the hyperbolic straight lines respectively are geodesics. The projection maps defined in §7 between the Euclidean plane Rn , the upper hemisphere U n S n and the hyperbolic plane Hn send lines to lines, and one can capture the projective geometry in a manner almost identical to the one in the Euclidean situation, and in some sense the study of the Hilbert metrics on convex subsets of the non-Euclidean spaces of constant curvature can be reduced to the Euclidean one. However this does not hold in the case of Funk metrics.

13 Appendix In this appendix, we present the proofs of some results in the projective geometry of Hn and S n , in particular on the projective invariance of cross ratio. We start with a few results in the setting of hyperbolic geometry. The proofs are based on the sine rule and they mimic the proofs of corresponding results in Euclidean geometry. From our point of view, the fact that is most interesting is the formal analogy between the results and the formulas in the three cases: Euclidean, hyperbolic and spherical.

5 This result attributed to Nussbaum, de la Harpe, and Foertsch–Karlsson was known to Busemann 1967. See Chapter 1, p. 30 of this Handbook [17].

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Proposition 13.1. Let ABC be a hyperbolic triangle. We join A by a geodesic to a point D on the line BC . Then we have

1 1

sin DAC sin B sinh DC D : sinh BD sin BAD sin C A

B

C

D

Figure 4

Proof. The proof is independent of whether the point D is between B and C , or C is between D and B, or B is between C and D. Applying the sine rule in the triangle DAC , we have

1

sin CAD sinh DC D : sinh AD sin C Applying the sine rule in the triangle BAD, we have sin B sinh AD : D sinh BD sin BAD From the last two equations, we obtain

1

1 1

sin DAC sin B sinh DC D : sinh BD sin BAD sin C This proves Proposition 13.1. Proposition 13.2. Consider four ordered distinct geodesic lines l1 , l2 , l3 , l4 intersecting at a point A and let l be a geodesic line intersecting l1 , l2 , l3 , l4 at points A1 , A2 , A3 , A4 respectively. Then we have

2 2

2 2

sin A2 AA4 sin A3 AA1 sinh A2 A4 sinh A3 A1 D : sinh A3 A4 sinh A2 A1 sin A3 AA4 sin A2 AA1 Proof. We refer to Figure 5. Using Proposition 13.1, we have

2 2

sin A2 AA4 sin A3 sinh A2 A4 D : sinh A3 A4 sin A3 AA4 sin A2

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A4 A3 A2 A1

l1

l2

l3

l4

Figure 5

Using again Proposition 13.1, we have

2 2

sin A3 AA1 sin A2 sinh A3 A1 D : sinh A2 A1 sin A2 AA1 sin A3 This implies

2 2

2 2

sin A2 AA4 sin A3 AA1 sinh A2 A4 sinh A3 A1 D : sinh A3 A4 sinh A2 A1 sin A3 AA4 sin A2 AA1 This proves Proposition 13.2. Corollary 13.3 (Cross ratio invariance in hyperbolic geometry). Consider four ordered distinct geodesics l1 , l2 , l3 , l4 in the hyperbolic plane that are concurrent at a point A and let l be a geodesic that intersects these four lines at points A1 , A2 , A3 , A4 respectively. Then the cross ratio of the ordered quadruple A1 , A2 , A3 , A4 does not depend on the choice of the line l 0 . Proof. By Proposition 13.2, the cross ratio of the quadruple A1 , A2 , A3 , A4 depends only on the angles that are made by the lines l1 , l2 , l3 , l4 . The result of Corollary 13.3 is a hyperbolic analogue of the result saying that the Euclidean cross ratio is a projective invariant. We now present the theorem of Menelaus in hyperbolic geometry with a proof based on the sine rule.6 This theorem 6 Menelaus (2nd century A.D.) in his Spherics gave the theorem in the Euclidean and in the spherical cases. There are several proofs of that theorem, and the proof that we give is not the original proof given by Menelaus, since he did not have the sine rule at his disposal. In fact, Menelaus did not formulate his theorem in terms of sines, but of chords. The notion of sine was introduced by the Arabs in the ninth century, who also discovered the sine rule, in Euclidean and in spherical geometry. In fact, Menelaus deduces the proof for the spherical case from the proof in the Euclidean case, by reasoning in the sphere embedded in the three-dimensional Euclidean space, see [24]. This theorem of Menelaus is sometimes referred to – especially by historians of mathematics – as the sector figure theorem, because this is the way ancient authors (like Ptolemy) referred to it. Menelaus’

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will be used below in the proof of the triangle inequality for the hyperbolic Funk metric. Proposition 13.4 (Menelaus’ Theorem in the hyperbolic plane). Let ABC be a triangle in the hyperbolic plane and let A0 , B 0 , C 0 be three points on the lines containing the sides BC , AC , AB. Then, the points A0 , B 0 , C 0 are aligned if and only if we have the relation sinh AC 0 sinh BA0 sinh CB 0 D1 sinh AB 0 sinh BC 0 sinh CA0 Proof. The proof is analogous to the proof of the corresponding Euclidean theorem (see [20] in this volume). Here we just prove the “only if” direction in order to show how one can use the hyperbolic sine rule instead of proportionality in the Euclidean case. We consider the case of Figure 6. A

B0

C0

C

A0

B

Figure 6. The theorem of Menelaus says that three points A0 , B 0 , C 0 on the lines containing the sides BC , AC , AB of a triangle are aligned if and only they satisfy the relation in Proposition 13.4.

In the triangle AB 0 C 0 , the sine rule gives

2 D sinh AC : sinh AB 2 C B sin A sin AB 0 C 0 0

0

0

0

In the triangle BC 0 A0 , the sine rule gives

2 D sinh BA : sinh BC 2 sin B AC

sin B C 0 A0 0

0

0

0

In the triangle CA0 B 0 , the sine rule gives

1 D sinh CB : sinh CA 2 sin A BC sin CA0 B 0

0

0

0

0

Spherics has been conceived as the non-Euclidean counterpart of Euclid’s Elements. It survived only in Arabic texts. A German edition exists [18], based on a manuscript of Ibn ‘IrNaq (10th century), and a new critical English edition based on several other Arabic manuscripts is in preparation [24].

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2

2

2

Multiplying the three sides of the last three equations and using the fact that sin AB 0 C 0 D sin A0 B 0 C and sin B C 0 A0 D sin AC 0 B 0 , we obtain the desired equality 1D

sinh AC 0 sinh BA0 sinh CB 0 : sinh AB 0 sinh BC 0 sinh CA0

The spherical analogue of all these results can be done in a similar way; the function sinh should be replaced in this case by the function sin. We now give an alternative proof of the triangle inequality for the hyperbolic Funk metric F .x; y/. This will also provide an application of the non-Euclidean cross ratio and it will further illustrate the formal analogies between the formulae and the reasonings in the three constant curvature geometries. The proof uses a drawing (Figure 7) and Menelaus’ Theorem (cf. Proposition 13.4). We present it here, using the notation of [33]. Let x, y, z be three points in . In the case where the three points are collinear (that is, if they are on a common hyperbolic geodesic), we know that the triangle inequality is satisfied, and that it is an equality, that is, we have F .x; y/ C F .y; z/ D F .x; z/ if x, y, z lie in that order on the line. Assume now the three points are not collinear and let a, b, c, d , e, f the intersections with @ of the lines xz, yx and zy, with the orders of the various quadruples of collinear points represented in Figure 7. p

f a

a0

c y xg

d

z

b0

b

e Figure 7

By the cross ratio invariance in hyperbolic geometry (Proposition 13.3), we have sinh xb 0 sinh a0 g sinh xc sinh dy D sinh yc sinh dx sinh gb 0 sinh a0 x and

sinh gb 0 sinh a0 z sinh ye sinh f z D : sinh ze sinh f y sinh zb 0 sinh a0 g

Multiplying both sides of these two equations, we get sinh xb 0 sinh a0 z sinh dx sinh f y sinh xc sinh ye D : sinh yc sinh ze sinh zb 0 sinh a0 x sinh dy sinh f z

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By the Theorem of Menelaus (Theorem 13.4) applied to the triangle f a0 z, we have sinh a0 x sinh dx sinh f y : D sinh dy sinh f z sinh a0 z This gives

sinh xb 0 sinh xb sinh xc sinh ye D ; 0 sinh yc sinh ze sinh zb sinh zb

and the inequality is strict unless b D b 0 . From this the triangle inequality follows and at the same time we deduce that the inequality is strict for all x, y, z unless @ contains a hyperbolic segment. This gives the desired result. Proposition 13.5. A diffeomorphism W Hn ! Hn which sends each geodesic to a geodesic is an isometry of the hyperbolic space Hn . We thank Yukio Matsumoto for discussions which led to the following proof. Proof. When n D 2, by using the upper half-plane model of H2 , given a diffeomorphism sending a circle centered at a point on the x-axis to another circle centered on the x-axis, we claim that the map is an element of SL.2; R/, which is the isometry group of H2 . Consider a diffeomorphism F of the upper half-plane, fixing 1, 1 and 1. It is easy to see that in order to prove the claim, it suffices to show that the restriction of F to the x-axis, which we denote by f W R ! R, is the identity map. We first show that f is symmetric with respect to the origin of the x-axis, i.e. f .x/ D f .x/. Let us denote the semi-circle in the upper half-plane meeting the x-axis perpendicularly at a and b (a < b) by Ca;b . When a < b < c < d , we also denote the x-component of the intersection point of Ca;c and Cb;d by P .a; b; c; d /. By elementary algebraic calculations, we have P .1; a; 1; b/ D

1 C ab : aCb

It follows that 1 1 and P .b; 1; 0; 1/ D b b for b > 1. Because F maps the intersection of a pair of semi-circles to the intersection of the F -images of those semi-circles, we have P .1; 0; 1; b/ D

f .P .a; b; c; d // D P .f .a/; f .b/; f .c/; f .d //: Hence f .P .b; 1; 0; 1// D f

1 b

D P .f .b/; f .1/; f .0/; f .1// D

1 C f .b/f .0/ f .b/ C f .0/

and 1 C f . b1 /f .b/ D f .P .1; b1 ; 1; b// D f .0/ P f .1/; f b1 ; f .1/; f .b/ D f . b1 / C f .b/

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as we have P .1; b1 ; 1; b/ D 0. By eliminating f . b1 / in the two equations, we obtain f .b/ D f .b/ for b > 1, namely f W R ! R is symmetric outside Œ1; 1. Next, we show that for b > 1, f . b1 / D f . b1 /, implying the symmetry of f on .1; 1/. Choose R so that 1 < Œ1; 1,

R2 b

< R < b. Then from the symmetry of f outside

2 f .P .R; 0; R; b// D f Rb D P .f .R/; f .0/; f .R/; f .b// D

f .R/2 C f .0/f .b/ f .0/ C f .b/

is equal to 1 times

2 f .P .b; R; 0; R// D f Rb D P .f .b/; f .R/; f .0/; f .R// D

f .R/2 f .0/f .b/ ; f .0/ f .b/

concluding that f .0/ D 0. This in turn says that f .P .b; 1; 0; 1// D

1 1 1 C f .b/f .0/ D D f .b/ C f .0/ f .b/ f .b/

while f .P .1; 0; 1; b// D

1 C f .b/f .0/ 1 D ; f .b/ C f .0/ f .b/

thus implying f . b1 / D f . b1 /, proving the symmetry of f over the x-axis with respect to the origin. Utilizing the previous equation f

R2 b

D

f .R/2 C f .0/f .b/ f .0/ C f .b/

together with f .0/ D 0 and by letting R ! b, we have f .b 2 / D f .b/2 for b > 1 by the continuity of f , which implies that f .x/ D x ˛ for some ˛ > 0. Similarly one can show that f .x/ D x ˛ for 0 < b < 1. Now the equation

f

1 C ab aCb

D

1 C f .a/f .b/ f .a/ C f .b/

forces ˛ to be one. We conclude that f .x/ D x. In the following, we will consider the higher dimensional case. Let n 3. Given a point p in Hn and two distinct arc-length parameterized geodesics 1 ; 2 through p with their respective tangent vectors v1 ; v2 2 Tp Hn , the

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images . 1 / and . 2 / are geodesics distinct from each other and both passing through .p/. Let 12 be the two-dimensional subspace in Tp Hn spanned by v1 and v2 , and …12 be the image of 12 by the exponential map expp W Tp Hn ! Hn , which is a totally geodesically embedded copy of H2 . Consider a geodesic triangle whose vertices are p,

1 .1/ and 2 .1/. By the property that sends a geodesic to a geodesic, the geodesic triangle 4.p; 1 .1/; 2 .1// is sent to the geodesic triangle 4..p/; . 1 .1//; . 2 .1//, and the interior points of 4.p; 1 .1/; 2 .1// …12 are sent to the interior points of 4..p/; . 1 .1//; . 2 .1// (this follows from the fact that the convex hull of a geodesic triangle in Hn lies in the hyperbolic plane containing the three vertices). From this observation, we can deduce that the image of …12 by is identified with the image of d.12 / T.p/ Hn by the exponential map exp.p/ W T.p/ Hn ! Hn , once again a totally geodesically embedded copy of H2 . In short, the diffeomorphism sends a hyperbolic plane to a hyperbolic plane in the hyperbolic space Hn . Thus the restriction of to the hyperbolic plane …12 can be regarded as a diffeomorphism Q W H2 ! H2 which sends a geodesic to a geodesic. Now we know that such a map is a hyperbolic isometry of H2 , as was explained in the beginning of the proof. Hence the linear map d restricted to 12 Tp Hn preserves the norms and the angles. As the point p and v1 , v2 were chosen arbitrarily, we conclude that the map is a local isometry, which is also a global isometry in the current setting. Acknowledgement. This work is partially supported by the French ANR project FINSLER (A.P.) and by JSPS Grant-in-aid for Scientific Research No.24340009 (S.Y.).

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[22] A. Papadopoulos and S. Yamada, The Funk and Hilbert geometries for spaces of constant curvature. Monatsh. Math. 172 (2013), no. 1, 97–120. [23] B. B. Phadke. A triangular world with hexagonal circles. Geom. Dedicata 3 (1975), 511–520. [24] R. Rashed and A. Papadopoulos, Menelaus’ Spherics. Critical edition from the Arabic manuscript of al-HarawNı, with an English translation and a mathematical commentary, in preparation. [25] P. Stäckel and F. Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung zur Vorgeschichte der nicht-euklidischen Geometrie. An edition with introduction, comments and German translations of works on the parallel postulate by Wallis, Saccheri, Lambert, Gauss (notes and correspondence), Schweikart and Taurinus, B. G. Teubner, Leipzig 1895. [26] J. Stillwell, Sources of hyperbolic geometry. Hist. Math. 10, Amer. Math. Soc., Providence, RI; London Mathematical Society, London 1996. [27] W. E. Story, On the non-Euclidean trigonometry. Amer. J. Math. IV (1881), 332–335. [28] W. P. Thurston, The geometry and topology of three-manifolds. Princeton Lecture Notes, 1976 ca., available at MSRI, library.msri.org/books/gt3m/. [29] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, ed. by Silvio Levy, Princeton Math. Ser. 35, Princeton University Press, Princeton, NJ, 1997. [30] J. E. Valentine, An analogue of Ptolemy’s theorem and its converse in hyperbolic geometry. Pacific J. Math. 34 (1970) 817–825. [31] J. E. Valentine, An analogue of Ptolemy’s theorem in spherical geometry. Amer. Math. Monthly 77 (1970) 47–51. [32] S.Yamada. Convex bodies in Euclidean and Weil–Petersson geometries, Proc. Amer. Math. Soc. 142 (2014), no. 2, 603–616. [33] E. M. Zaustinsky, Spaces with nonsymmetric distance. Mem. Amer. Math. Soc. No. 34, 1959.

Part IV

History of the subject

Chapter 14

On the origin of Hilbert geometry Marc Troyanov

Hilbert geometry, which is the main topic of this Handbook, was born twelve decades ago, in the summer of 1894. Its birth certificate is a letter written by David Hilbert to Felix Klein, on August 14, 1894, sent from the village of Kleinteich near the city of Rauschen (today called Svetlogorsk), at the Baltic sea. The place was a fashionable vacation resort, situated some forty kilometers north of Hilbert’s hometown Königsberg (now Kaliningrad), where he was a Privatdozent.1 The mathematical part of this letter was published one year later in the Mathematische Annalen2 under the title Ueber die gerade Linie als kürzeste Verbindung zweier Punkte (On the straight line as shortest connection between two points) [6]. Before discussing the content of this letter, let us say a few words on the general situation of geometry in the XIXth century. At the dawn of the century, and despite numerous criticisms concerning the axioms, Euclid’s Elements were still considered as a model of mathematical rigor and an exposition of the “true” geometry. The 1820s saw on one hand the creation of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky, and on the other hand, a revival and a rapid development of projective geometry by Poncelet, Gergonne, Steiner, Möbius, Plücker and von Staudt. During this period, a controversy developed among supporters of the synthetic versus the analytic methods and techniques in the subject. In this context, the analytic methods are based on coordinates and algebraic relations for the description and the analysis of geometric figures, while the synthetic methods are based on the incidence relations between points, lines, planes and other loci. The key to unify both sides of the controversy came from the work of von Staudt [18], [19] who proved that numerical coordinates can be assigned to points on a line in a projective space through the use of purely synthetic methods and without using distances. Von Staudt’s theorem implies that the cross ratio of four aligned points can be defined synthetically, and thus it is invariant under collineations (that is, transformations preserving lines and incidence). Recall that the cross ratio of four points X, Y , Z, T is analytically defined as ŒX; Y; Z; T D 1 Hilbert 2 The

.x z/ .y t / ; .x t / .y z/

was appointed professor at Göttingen one year later. chief editor of this journal was Klein himself.

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where x; y; z; t 2 R [ f1g are coordinates on the given line, corresponding to the points X, Y , Z, T . The synthetic definition of the cross ratio is more elaborate and it is based on an iteration of the construction of the harmonic conjugate of a point with respect to a given pair of points (the harmonic conjugate corresponds to a cross ratio equal to 1 and can easily be constructed synthetically using complete quadrangles), together with ordering and density arguments. In fact, von Staudt was not able to fully prove his theorem since at his time the topological structure of the real numbers, in particular the completeness axiom, had yet to be clarified. This was done later in Dedekind’s short monograph Was sind und was sollen die Zahlen? (1888). A full exposition of von Staudt’s construction is given in Chapter VII of Veblen and Young’s book [16], see also Chapter VII of [15]. We refer to [10] and [17] for additional comments on von Staudt’s work. By the middle of the XIXth century, it was seen as a natural problem to try to define metric notions from projective ones, thus reversing the analytic way of seeing geometry. Von Staudt’s theorem allowed the use of the cross ratio in this adventure. The first step in this direction is usually attributed to Laguerre, who proved in 1853 that the Euclidean angle ' between two lines `, m through the origin in a plane is given by ˇ ˇ ˇ1 ˇ ˇ (1) ' D ˇ logŒX; Y; U; V ˇˇ; 2i where X 2 `, Y 2 m and U , V are points representing the “isotropic lines” of the plane3 . These are lines which are “orthogonal to themselves”, meaning that if U D .u1 ; u2 / and V D .v1 ; v2 /, then u21 C u22 D v12 C v22 D 0; or, equivalently, u2 D iu1 and v2 D iv1 . To explain Laguerre’s formula, one may assume x1 D y1 D u1 D v1 D 1; then x2 D tan.˛/, y2 D tan.ˇ/, u2 D i and v2 D i . We then have ŒX; Y; U; V D

.tan.˛/ i/ .tan.ˇ/ C i / D e2i.˛ˇ / : .tan.˛/ C i/ .tan.ˇ/ i/

Klein in [8], inspired by a formula published by Cayley4 in [3], observed later that Equation (1) also gives a projective definition of the distance in elliptic geometry, that is, the metric on RP n for which the standard projection Sn ! RP n is a local isometry. Here X and Y are arbitrary distinct points in RP n and U and V are points in CP n which are aligned with X and Y and belong to the quadric n X

u2i D 0:

(2)

iD0 3 In fact the formula in Laguerre’s paper does not explicitly involve the cross ratio, the present formulation seems to be due to F. Klein, see e.g. [9], p. 158. 4 Cayley’s formula did not involve the cross ratio; it is written in terms of homogenous coordinates.

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Klein also considered similar geometries where the quadric (2) is replaced by an arbitrary quadric, which, following Cayley’s terminology, he called In Pnthe absolute. 2 2 1871, Klein realized that choosing as the absolute the standard cone iD1 ui u0 D 0, the metric restricted to the interior of the cone is a model of Lobatchevsky’s geometry. See [2] for a historical discussion. In 1888, M. Pasch proposed a new axiomatic foundation of geometry based on the primitive notion of segment (or betweenness) rather than the projective notion of lines. Pasch’s geometry is often called ordered geometry and it greatly influenced Hilbert. Let us now return to Hilbert’s letter. The letter starts with a discussion of how geometry should be founded on three “elements” (in the sense of primitive notions) named points, lines and planes (Hilbert’s aim is a discussion of three-dimensional geometry). These elements should satisfy three groups of axioms, which he only shortly discusses. I. The first groups are axioms of incidence describing the mutual relations among points, lines and planes. These axioms state that two distinct points determine a unique line, that three non aligned points determine a unique plane, that a plane containing two distinct points contains the full line determined by those points, and that two planes cannot meet at only one point. Furthermore, every line contains at least two points, every plane contains at least three non aligned points and the space contains at least four non coplanar points. II. The second groups of axioms deals with ordering points on a line and these axioms have been proposed by M. Pasch in [12]. They state: between two distinct points A and B on a line, there exists at least a third point C , and given three points on a line, one and only one of them lies between the two others. Given two distinct points A and B, there is another point C on the same line such that B is between A and C . Given four points on a line, we can order them as A1 , A2 , A3 , A4 is such a way that if h < i < k, then Ai lies between Ah and Ak . If a plane ˛ contains a line a, then the plane is separated into two half-planes such that A and B belong to the same half-plane if and only if there is no point of the line a between A and B. III. The axiom of continuity. Let Ai be an infinite sequence of points on a line a. If there exists a point B such that Ai lies between Ak and B as soon as k i , then there exists a point C on the line a such that Ai lies between A1 and C for any i > 1 and any point C 0 with the same property lies between C and B. (We recognize an axiom stating that any decreasing sequence on a line that is bounded below has a unique greatest lower bound). Hilbert then claims that the von Staudt theory of harmonic conjugates can be fully developed on this axiomatic basis (in a “similar way” as in Lindemann’s book Vorlesungen über Geometrie). It therefore follows that coordinates can be introduced. More precisely, to each point, one can associate three real numbers x, y, z and each plane corresponds to a linear relation among the coordinates. Furthermore, if one interprets x, y, z as rectangular coordinates in the usual Euclidean space, then the

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points in our initial space correspond to a certain convex5 body in Euclidean space. Conversely, the points in an arbitrary convex domain represent points in our initial space: our initial space is built from the points in a convex region of the Euclidean space. The axiomatic framework set here by Hilbert is interesting in several aspects. The first group of axioms belongs to what is now called incidence geometry. Observe that Hilbert formulates them for a three-dimensional space – at that time it was unusual to discuss geometry in an arbitrary dimension n – but note also that this restriction reduces the complexity of the theory, because it is not so simple to define and discuss a general notion of dimension in pure incidence geometry. Also Desargues’ theorem (which plays a key role in the introduction of coordinates) need not be added as an axiom since it can be proved in this framework. The second group of axioms describes ordered geometry, see [15] and [14] for readable accounts of this topics and [13] for an impressive historical compilation of the literature since Pasch’s original work. Notice that the notion of line (and hence plane) can be defined on the sole base of the primitive notion of segment, but Hilbert found it convenient to have lines and planes as primitive notions also. Again, this reduces the complexity of the theory (and furthermore it is in Euclid’s spirit). In a more contemporary language, what Hilbert is claiming here is that an abstract space M consisting of points, together with a structure made of lines, planes and the betweenness relation (for points on a line) satisfying the said axioms is isomorphic to a convex region in R3 with the usual notions of lines and planes and the condition that C lies between A and B if and only if C 2 ŒA; B. It is not clear whether the axiomatics proposed in Hilbert’s letter is sufficient to rigorously prove this result, and the question is also not addressed in his Grundlagen [7]. However it is indeed the case that a synthetic axiomatic characterization of convex domains in Rn can be based on ordered geometry. A theorem of this type is given in W. A. Coppel’s book, who calls it the fundamental theorem of ordered geometry, see [4], p. 173. At this point of his letter, Hilbert stresses that arbitrary convex bodies also appear in Minkowski’s work on number theory. He then proceeds to define a notion of length for the segment AB in his general geometry, which he assumes is now realized as the interior of a convex region in R3 . He follows Klein’s construction and defines the distance between two distinct points A and B to be6

YA XB d.A; B/ D logŒX; Y; B; A D log ; YB XA

(3)

where X; Y are the two points on the boundary of the convex domain aligned with YA XB ; XA > 1 (if A and B in the order X; A; B; Y . Observe that d.A; B/ > 0 since AB A D B it is understood that d.A; B/ D 0). 5 Hilbert, following Minkowski, uses the expression nirgend concaven Körper, which means nowhere concave. In other words, the domain is convex, but not necessarily strictly convex. 6 The usual convention now is to divide this quantity by 2.

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Hilbert observes that the distance (3) depends on the given convex domain: if X approaches A and Y approaches B, then the distance d.A; B/ increases. In other words, a smaller convex domain gives rise to larger Hilbert distances. Hilbert then proceeds to give a proof of the triangle inequality. Here is the argument with Hilbert’s notation: From the invariance of the cross ratio with respect to the perspective at W , we have ŒU; V; C; A D ŒX 0 ; Y 0 ; D; A and ŒZ; T; B; C D ŒX 0 ; Y 0 ; B; D. Multiplying these identities gives ŒU; V; C; A ŒZ; T; B; C D ŒX 0 ; Y 0 ; D; A ŒX 0 ; Y 0 ; B; D D ŒX 0 ; Y 0 ; B; A ŒX; Y; B; A; which is equivalent to d.A; C / C d.C; B/ d.A; B/. W

Z V C X

X

A D

B

Y Y

U

T

Hilbert also observes that the triangle inequality degenerates to an equality when the three points are aligned with C between A and B. Furthermore, he discusses necessary and sufficient conditions for the equality in the triangle inequality. He shows that a non-degenerate triangle exists for which the sum of two sides is equal to the third if and only if there exists a plane which meets the boundary of the convex domain in two segments not on the same line (these would be the segments ŒV; T and ŒZ; U on the figure). He quotes, as a particularly interesting example, the case where the convex domain is bounded by a tetrahedron. As a final word, Hilbert points out that he always assumed the given convex body to be bounded, and this hypothesis implies that his geometry does not satisfy the Euclidean parallel postulate. Let us conclude with two remarks. We first mention that Hilbert came back at least at two occasions on the mathematical subjects discussed in his letter. First in his book

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on the foundation of geometry [7], in which he systematically develops the axiomatics of Euclidean 3-space, starting with the same axioms of incidence, order and continuity, to which he adds some axioms on congruence and the parallel postulate. The second occasion was the 1900 International Congress in Paris where he states the famous Hilbert problems. Problem IV concerns the construction and systematic treatment of the geometries in convex domains for which straight lines are the shortest, that is, the case where the triangle inequality is an equality in the case of three aligned points. Minkowski and Hilbert geometries are natural examples of such geometries. We refer to the discussion in the last chapter of this Handbook [11]. The second and last remark is about how Hilbert understood the word geometry. For Hilbert, a geometry was not conceived as an abstract metric space7 satisfying some specific axioms, but rather, as we saw, a geometry is a system made of points, lines and planes subject to a system of consistent interrelations wich are accepted as axioms. A notion of distance between points in a given geometry has then to be constructed from the given data and axioms and the properties of the distance, including the triangle inequality, is then a theorem that needs a proof rather than an axiom or a part of the initial definition. This is also in the spirit of Euclid’s Elements, where the triangle inequality is proved in Book 1, Proposition 20. The subject of metric geometry has been developed since the 1920s by a long list of mathematicians including Hausdorff, Menger, Blumenthal, Urysohn, Birkhoff, Busemann, Alexandrov, and others. Since the work of Gromov in the 1980s, metric geometry is seen as a topic of major importance in geometry. Acknowledgement. The author thanks A. Papadopoulos and K.-D. Semmler for carefully reading the manuscript and providing useful comments.

References [1]

H. Busemann, The geometry of geodesics. Academic Press, New York 1955, reprinted by Dover, Mineola, NY, 2005.

[2]

N. A’Campo and A. Papadopoulos, On Klein’s so-called non-Euclidean geometry. In Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics (ed. L. Ji andA. Papadopoulos), European Mathematical Society, Zürich, to appear.

[3]

A. Cayley, A sixth Memoir upon Quantics. Philosophical Transactions of the Royal Society of London 149 (1859), 61–90.

[4]

W. A. Coppel, Foundations of convex geometry. Austral. Math. Soc. Lect. Ser. 12, Cambridge University Press, Cambridge 1998

[5]

M. Fréchet, Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22 (1906), 1–72.

[6]

D. Hilbert, Über die gerade Linie als kürzeste Verbindung zweier Punkte. Math. Ann. 46 (1895), 91–96.

7 The

notion of abstract metric space first appeared in the 1906 thesis of M. Fréchet [5].

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[7]

D. Hilbert, Grundlagen der Geometrie. B.G. Teubner, Leipzig, 1899, various editions and translations.

[8]

F. Klein, Über die sogenannte Nicht-Euklidische Geometrie. Math. Ann. IV (1871), 573–625.

[9]

F. Klein, Vorlesungen über höhere Geometrie. Dritte Auflage, Grundlehren Math. Wiss. 22, Springer-Verlag, Berlin 1926.

[10] P. Nabonnand, La théorie des “Würfe” de von Staudt – une irruption de l’algèbre dans la géométrie pure. Arch. Hist. Exact Sci. 62 (2008), no. 3, 201–242. [11] A. Papadopoulos, Hilbert’s fourth problem. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 391–431. [12] M. Pasch, Vorlesungen über neuere Geometrie. Teubner, Leipzig 1882. [13] V. Pambuccian, The axiomatics of ordered geometry: I. Ordered incidence spaces. Expo. Math. 29 (2011), no. 1, 24–66. [14] W. Prenowitz, and M. Jordan, Basic concepts of geometry. Blaisdell Publishing Company, London 1965. [15] G. Robinson, The foundations of geometry. Math. Expositions 1, University of Toronto Press, Toronto, Ont., 1940. [16] O. Veblen and J. W. Young, Projective geometry. Vol. 1, Ginn and Co., London 1910. [17] J.-D. Voelke, Le théorème fondamental de la géométrie projective: évolution de sa preuve entre 1847 et 1900. Arch. Hist. Exact Sci. 62 (2008), no. 3, 243–296. [18] K. von Staudt Geometrie der Lage. Nürenberg 1847. [19] K. von Staudt Beiträge zur Geometrie der Lage. Nürenberg 1857.

Chapter 15

Hilbert’s fourth problem Athanase Papadopoulos

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 Precursors . . . . . . . . . . . . . . . . . . . 3 Hilbert’s Problem IV . . . . . . . . . . . . . . 4 Some early works on Hilbert’s Problem IV . . 5 Busemann’s approach and Pogorelov’s solution 6 Other works and other solutions . . . . . . . . 7 Further developments and perspectives . . . . Appendix I: Hilbert’s Problem IV . . . . . . . . . . Appendix II: Extract from the French version . . . References . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction On the 8th of August 1900, at the Second International Congress of Mathematicians held in Paris, David Hilbert delivered a lecture titled “The future problems of mathematics”, in which he presented a collection of open problems. Hilbert was forty-eight years old and was considered as one of the leading and most versatile mathematicians of his time. His list of problems turned out to be a working document for a large part of the mathematical research that was conducted in the twentieth century and it had a great influence on several generations of mathematicians. A first short paper based on that lecture appeared in 1900 in the newly founded Swiss journal L’Enseignement Mathématique [79], and after that several written versions of unequal lengths were published (see [68] for a commentary on these versions). This includes a paper published in the Bulletin of the American Mathematical Society ([80], 1901), containing a commented set of twenty-three problems.1 Today, several of these problems are 1 In the mid 1990s, a twenty-fourth problem was discovered in Hilbert’s massive files in Göttingen. This problem asks for “simpler proofs and criteria for simplicity”. The interested reader can refer to the reports in [118] and [117] on that intriguing problem that had remained hidden for almost a century. We also note that at the Paris 1900 congress, Hilbert presented only 10 problems. C. Reid, who recounts the story in [106], says (on p. 81): “In an effort to shorten his talk as Minkowski and Hurwitz had urged, Hilbert presented only ten problems out of a total of 23 which he had listed in his manuscript”.

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solved, but there is still very active research around them (the solved ones as well as the unsolved). As a general introduction to the Hilbert problems, we refer the reader to the two semi-popular books that appeared on the occasion of the anniversary of these problems: J. Gray’s The Hilbert Challenge [69] and B. H. Yandell’s The Honors Class: Hilbert’s Problems and Their Solvers [124]. The books [33] and [4] contain collections of more technical articles on the developments of these problems. Hilbert geometry, which is the topic of this Handbook, is closely connected with Hilbert’s Problem IV. The problem is titled “The problem of the straight line as the shortest distance between two points”. There are several interpretations of this problem and we shall discuss them below. Soon after Hilbert formulated it, the problem was reduced by G. Hamel – who was one of Hilbert’s student – to the case of metrics defined on convex subsets of Euclidean (or affine) spaces. If is a convex subset of a Euclidean space, then the restriction to of the ambient Euclidean metric is a metric on satisfying Hilbert’s requirement, that is, the (restriction to of the) Euclidean straight lines are geodesics for the restriction to of the Euclidean metric. One form of Hilbert’s problem asks for a characterization of all metrics on for which the Euclidean lines in are geodesics. Another natural question is to find a characterization of metrics on for which the Euclidean lines are the unique geodesics. One may put additional requirements on the metrics and try to characterize some special classes of metrics satisfying Hilbert’s requirements, such as geodesically complete metrics, that is, metrics whose geodesics (parametrized by arclength) can be extended infinitely from both sides. The Hilbert metrics are examples of geodesically complete metrics satisfying Hilbert’s problem requirement. In what follows, I shall present some ideas and make comments, several of them having a historical character, about Hilbert’s Problem IV and its ramifications, and I will concentrate on the ideas that arose from the work of Busemann. In fact, I generally followed the rule of not stating in the form Theorem: etc. anything that is not found in Busemann’s work. I have included in the footnotes some biographical notes on some of the mathematicians that appear in this paper, but not all of them. There are several available biographies of Hilbert and of Klein. For the latter, we refer the reader to [84]. The story of Hilbert’s Problem IV, of its various reformulations and of all the work done around it is an illustration of how mathematical problems can grow, transform and lead to new problems. And it also shows that mathematicians are persistent.

2 Precursors Hilbert’s problems did not come out of nowhere, and several of them have a history. Problem IV, like most of the others, arose as a natural question at that time. In the French version of the statement (see the appendix at the end of the present chapter), Hilbert mentions related work of Darboux. Darboux, in the work that Hilbert refers to,

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mentions Beltrami.2 We shall review some of Beltrami’s work related to the subject, but before that, we report on some works of Cayley and Klein. In a 1859 paper called A sixth Memoir upon Quantics [55], Cayley3 gave an interesting definition of a projective metric on the plane, that is, a metric on the projective plane such that the projective lines are geodesics for that metric. More specifically, Cayley started with a conic (which he called the absolute) in the projective plane, and he associated to it a metric d such that for any three points P1 , P2 , P3 in that order on a projective line, we have d.P1 ; P2 / C d.P2 ; P3 / D d.P1 ; P3 /. A few years later, Klein, in his two papers [86] and [87], re-interpreted Cayley’s ideas using the notion of cross-ratio, and he showed that by choosing the absolute to be a real conic in the projective plane and using Cayley’s construction, the interior of that conic is a model of hyperbolic geometry. Beltrami, in his paper [21] had already noticed this model, namely, he showed that the unit disc in the plane, with the Euclidean chords taken as 2 Eugenio Beltrami (1835–1900) was one of the major figures in differential geometry who continued the work of Riemann. He was born in a family of artists. He spent his childhood in a period of political turbulence: the Italian revolutions and the independence wars which led eventually to the unification of the country. He studied mathematics at the University of Pavia between 1853 and 1856, where he followed the lectures of Francesco Brioschi, but because of lack of money, his studies had to be interrupted. (Loria reports that Beltrami was expelled from the university because he was accused of promoting disorders against the rector [89], see also [72]). Beltrami worked during the next four years as a secretary of the director of the railway company of the Lombardo-Veneto in Verona. He undertook all over again his mathematical studies in Milan, between 1860 and 1862, where he followed the courses of Brioschi and of Cremona. Both men had a decisive influence on the future of Beltrami. Brioschi, who had moved to Milano, was very much involved in politics, and he played a major role in the formation of the new state of Italy. Beltrami published a mathematical paper in 1862, and he obtained in the same year a position at the University of Bologna. After that, he moved between several universities, partly because of the political situation in Italy. He spent his last years at the university of Rome. A stay in Pisa, between 1863 and 1866, where he was appointed professor of geodesy, was probably decisive for his mathematical future research. He was called there by Betti, and he also met there Riemann (who was in Italy for health reasons). Beltrami’s Saggio di Interpretazione della geometria non-Euclidea [21] and his Teoria fondamentale degli spazi di curvatura costante [22] which are mentioned in this survey were written during a second stay at the University of Bologna, where he held the chair of rational mechanics. The name of Beltrami is attached to the Beltrami differential, a basic notion in the theory of quasiconformal mappings, and with the Laplace–Beltrami operator. Besides mathematics, he cultivated physics, in particular thermodynamics, fluid dynamics, electricity and magnetism. He translated into Italian the work of Gauss on conformal representations. 3Arthur Cayley (1821–1895) grew up in a family of English merchants settled in Saint-Petersburg, and he lived in Russia until the age eight, when his family returned to England. Cayley is one of the main inventors of the theory of invariants. These include invariants of algebraic forms (the determinant being an example), but also algebraic invariants of geometric structures and the relations they satisfy (“syzygies”). He studied both law and mathematics at Trinity College, Cambridge, from where he graduated in 1842. He was talented in both subjects and as an undergraduate he wrote several papers, three of which were published in the Cambridge Mathematical Journal. The subject of these papers included determinants, which became later on one of his favorite subjects. After a four-year position at Cambridge university, during which he wrote 28 papers for the Cambridge journal, Cayley did not find any job to continue in academics. Looking for a new profession, law was naturally his second choice. He worked as a lawyer for 14 years, but he remained interested in mathematics, and he wrote about 250 mathematical papers. Cayley was appointed professor of mathematics at Cambridge in 1863. His list of publications includes about 900 items, on all the fields of mathematics of his epoch. The first definition of abstract group is attributed to him, cf. his paper [54]. Cayley proved that every finite group is a subgroup of some symmetric group. He is also one of the first discoverers of geometry in n dimensions. In his review of Cayley’s Collected Mathematical Papers edition in 13 volumes, G. B. Halsted writes: “‘Cayley not only made additions to every important subject of pure mathematics, but whole new subjects, now of the most importance, owe their existence to him. It is said that he is actually now the author most frequently quoted in the living world of mathematicians” [71]. The reader is referred to the interesting biography by Crilly [57].

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geodesics, is a model of the Lobachevsky plane with its non-Euclidean geodesics, but he did not have the formula for the distance in terms of the cross ratio. Klein’s model, with the cross ratio formula, constitutes what is called today the Klein model, or the Beltrami–Klein model, or also the Beltrami–Cayley–Klein model of hyperbolic space. It is worth noting that Klein also found models of the elliptic and of the Euclidean plane within the projective plane that use the cross ratio and a conic at infinity. In the case of the Euclidean plane, the conic is imaginary and the construction involves complex numbers. We refer the reader to Klein’s paper [86], and also to the commentary [1]. It may also be worth recalling here that Cayley considered the use of the cross ratio by Klein as a sort of circular reasoning, because he thought that the definition of the cross ratio is dependent upon some underlying Euclidean geometry, whereas Klein, in his paper [86], was aiming for a definition of the hyperbolic metric (and also, Euclidean and spherical metrics) based only on projective geometry. We can quote here Cayley from his comments on his paper [55] in Volume II of his Collected mathematical papers edition [62] (p. 605): I may refer also to the memoir, Sir R. S. Ball “On the theory of content,” Trans. R. Irish Acad. vol. xxix (1889), pp. 123–182, where the same difficulty is discussed. The opening sentences are – “In that theory [Non-Euclidian geometry] it seems as if we try to replace our ordinary notion of distance between two points by the logarithm of a certain unharmonic ratio. But this ratio itself involves the notion of distance measured in the ordinary way. How then can we supersede the old notion of distance by the non-Euclidian notion, inasmuch as the very definition of the latter involves the former?”

This is just to show that some of the most prominent mathematicians were missing an important idea, namely that the cross ratio does not depend on the underlying Euclidean geometry. In the same line of thought we quote Genocchi, an influential and well-established Italian mathematician of the second half of the nineteenth century4 who wrote ([67], p. 385): From the geometric point of view, the spirit may be chocked by certain definitions adopted by Mr. Klein: the notions of distance and angle, which are so simple, are replaced by complicated definitions [...] The statements are extravagant.”

The second paper of Klein [87] contains an elaboration on some of the ideas contained in the first one. In the meanwhile Beltrami found two models of hyperbolic geometry, and Klein noticed that his model coincides with one of the models that Beltrami found. This is the origin of the so-called Klein–Beltrami model of hyperbolic space. Klein’s aim in the two papers that we mentioned was to include the three 4Angelo

Genocchi (1817–1889) was in Italian mathematician who was also a very active politician. Like Cayley, he worked for several years as a lawyer. He taught law at the University of Piacenza, the city where he was born, and at the same time he was cultivating mathematics with passion. In 1859, Genocchi was appointed professor of mathematics at the University of Torino, and he remained there until 1886. During the academic year 1881–82, Guiseppe Peano served as his assistant, and he helped him later on with his teaching, when Genocchi became disabled after an accident. During several years, Genocchi was the main specialist in number theory in Italy.

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constant-curvature geometries in the setting of projective geometry (Cayley had done this only for Euclidean geometry), and although Klein did not formulate it in these words, he obtained models for these three geometries in which lines in projective space are geodesics. This fact was highlighted by Beltrami, who obtained in [21] the discmodel of hyperbolic geometry, where the Euclidean straight lines are the geodesics, but Beltrami did not have Klein’s formula involving the cross-ratio. One may add here that two years before Klein’s work on the subject, Beltrami wrote the following to Houël5 (letter dated July 29, 1869 [23], p. 96–97): The second thing [I plan to do in addition to the memoir I sent you]6 will be the most important, if I succeed to give it a concrete form, because up to now it only exists in my head in the state of a vague conception, although without any doubt it is based on the truth. This is the conjecture of a close analogy, and may be of an identity, between pseudo-spherical geometry7 and the theory of Mr Cayley on the analytical origin of metric ratios, using an absolute conic (or quadric). I almost did not know anything about that theory, when I was taken by the identity of certain forms. However, since the theory of invariants plays there a rather significant role and because I lost sight of it since a few years now, I want to do it again after some preliminary studies, before I address this comparison.

In another letter to Hoüel, written on July 5, 1872, Beltrami regrets the fact that he let Klein outstrip him ([23], p. 165): The principle which has directed my analysis8 is precisely that which Mr Klein has just developed in his recent memoir9 on non-Euclidean geometry, for 2-dimensional spaces. In other words, from the analytic point of view, the geometry of spaces of constant curvature is nothing else than Cayley’s doctrine of the absolute. I regret very much to have let Mr Klein supersede me on that point, on which I had already assembled some material, and it was my mistake for not giving to this matter enough importance.

There are comments on this work of Klein in the paper [1]. 5 Guillaume Jules Hoüel (1823–1886) is a major figure in the history of non-Euclidean geometry. There is a very interesting correspondence between him and Beltrami, and sixty-five of these letters were edited by Boi, Giacardi and Tazzioli in 1998 [23]. We refer to that correspondence several times in the present chapter. Hoüel translated into French and published in French and Italian journals works by several authors on non-Euclidean geometry, including Bolyai, Beltrami, Helmholtz, Riemann and Battaglini. In his first and memorable letter to Hoüel, written on November 18, 1868, Beltrami (who was seven years younger than Hoüel), starts as follows: “Professor, I take the liberty of sending you, in a file, two memoirs, the object of the first one is a real construction of non-Euclidean geometry, and the other, much less recent, contains the proof of some analytic results upon which this construction relies, but which, apart from that, have no immediate relation with that question”. The two memoirs are the Saggio di Interpretazione della geometria non-Euclidea [21] and the Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette [20]. It seems that Hoüel was the first to notice that Beltrami’s work on the pseudo-sphere implies that the parallel postulate is not a consequence of the other Euclidean postulates. Beltrami, in a letter dated January 2, 1870 ([23], p. 114), writes: “Your conclusion on the impossibility of proving in the plane the eleventh axiom, as far as it follows from my work, is also mine”. See the discussion in [95], §1.3 of the Introduction. 6 Beltrami, in this letter, gave a favorable response to Hoüel who had proposed to translate his two memoirs Saggio di Interpretazione della geometria non-Euclidea [21] and Teoria fondamentale degli spazi di curvatura costante [22] into French and to publish the translations in the Annales de l’École Normale Supérieure. 7 This is the term used by Beltrami to denote hyperbolic geometry. 8 Beltrami refers here to a note [18] which he had just published in the Annali di Matematica. 9 Beltrami refers to Klein’s paper [86].

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We next talk about Beltrami’s work. In 1865, Beltrami wrote a paper [20] whose title is Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette (A solution of the problem: Transfer the points of a surface over a plane in such a way that the geodesic lines are represented by straight lines). From the title, it is clear that the problem bears a direct relation to Hilbert’s problem. Indeed, if such a transfer map exists, then, by pushing forward the metric of the given surface to a metric on the Euclidean plane, we get a metric on that plane for which the Euclidean straight lines are geodesics. Beltrami worked on this subject in the framework of the differential geometry of surfaces, in the style developed by Gauss. In the introduction to his paper [20], Beltrami says that most of the research done before him on similar questions was concerned with the question of conservation of angles or of area, and that even though these two properties are regarded as the simplest and most important properties of such transfer maps, there are other properties that one might want to preserve. Indeed, the kind of question in which Beltrami was interested was motivated by the theory of geographical maps. A geographical map is a representation in the Euclidean plane of a part of the globe, considered as a sphere. It was clear since the beginning of the art of geographical maps – which dates back, as a science, to Greek antiquity – that there is no such a representation which is faithful (that is, where the proportions are respected) regarding distances and angles at the same time. Beltrami declares in this paper that since the projection maps that are used in this science are primarily concerned with the measure of distances, one would like to exclude projection maps where the images of distance-minimizing curves are too remote from straight lines. He mentions in passing that the central projection of the sphere is the only map that transforms geodesics into straight lines. As a matter of fact, this is the question that motivated Beltrami’s research on the problem referred to in the title of his essay, “Transfer the points of a surface over a plane etc.” Beltrami then writes that beyond its applications to geographic map drawing, the solution of the problem may lead to “a new method of geodesic calculus, in which the questions concerning geodesic triangles on surfaces can all be reduced to simple questions of plane trigonometry”. He finally acknowledges that his investigations on this question did not lead him to any general solution, and that the case of the central projection of the sphere is the only one he found where the required condition is realized. Beltrami proved though an important local result in that paper, viz. he showed that the only surfaces that may be (locally) mapped to the Euclidean plane in such a way that the geodesics are sent to Euclidean lines must have constant Gaussian curvature. In fact, Beltrami starts with a surface whose line element is given, in the tradition of Gauss, by p ds D Edu2 C 2F dudv C Gdv 2 : The line element represents the distance between the two infinitesimally nearby points .u; v/ and .uCdu; vCdv/ on the surface. After several pages of calculations Beltrami

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ends up with the following values for the functions E, F and G: 8 R2 .v 2 Ca2 / ˆ ˆ ˆ E D .u2 Cv2 Ca2 /2 ; < R2 uv F D .u2 Cv 2 Ca2 /2 ; ˆ ˆ 2 2 ˆ .u Ca2 / : G D .uR2 Cv 2 Ca2 /2 :

397

(2.1)

He then uses the following formula for the curvature, which he derived in his Ricerche [19], art. XXIV: ! !! @E @E @G F 2 @F CF 1 @ @ @E 1 @u E @v @v @u E @u D p C p p R1 R2 @v 2 EG F 2 @u EG F 2 EG F 2 After a computation, Beltrami finds that the curvature is constant and equal to 1 1 D 2: R1 R2 R He concludes from this: “Therefore our surfaces are those of constant curvature. In particular, if the quantity R is real, the formulae (2.1) are responsible for all surfaces that are applicable on the sphere of radius R”. Beltrami summarizes his result in the following statement: Theorem 2.1. The only surfaces that can be represented over a plane, in such a way that to every point corresponds a point and to every geodesic line a straight line are those whose curvature is everywhere constant (positive, negative or zero). When this constant curvature is zero, the correspondence does not differ from the ordinary homography. When it is nonzero, this correspondence is reducible to the central projection over the sphere and to its homographic transformations. Beltrami adds: Since among all the surfaces of constant curvature the only one that can have applications in the theory of geographic maps and in geodesy is probably the spherical surface, in this way from the point of view of these applications, what we asserted confirms that the only solution to the problem is obtained essentially by the central projection.

In conclusion to his paper, Beltrami states the following: In order for the points of a surface to be transported over a surface of constant curvature in such a way that the geodesic lines of the former be represented by geodesic lines of the latter, it is necessary that the first surface be of constant curvature.

Thus, in his conclusion, Beltrami considered only the case where the curvature is positive, and therefore the only surface he found that has the required property (apart from the plane itself) is the sphere (of a certain radius). We refer the reader to the chapter [120] in this Handbook for an exposition with a detailed proof of this result of Beltrami. Two years later, Beltrami wrote another paper, Saggio di Interpretazione della geometria non-Euclidea [21], which soon became famous, in which he worked on

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surfaces of constant negative curvature, a context in which he gave an interpretation of Lobachevsky’s geometry. In that paper, Beltrami defined a family of metrics on the disc of radius a > 0 centered at the origin of the Euclidean plane, using the formula ds 2 D R2

.a2 v 2 /du2 C 2ududv C .a2 u2 /dv 2 ; .a2 u2 v 2 /2

(2.2)

where ds 2 denotes as usual the square of the infinitesimal length element. He showed that the Gaussian curvature of that surface is constant and equal to 1=R2 , and that its geodesics are the Euclidean straight lines. He also stated the following: It follows from what precedes that the geodesics of the surface are represented in their total (real) development by the chords of the limit circle, whereas the prolongation of these chords in the exterior of the same circle are devoid of any (real) representation.10

Figure 1. Extracted from Beltrami’s paper [21]. The figure represents the unit disc model of the hyperbolic plane with a triangle pqr having two vertices at infinity.

Beltrami gave in the paper [21] the first Euclidean model of hyperbolic plane (the socalled “Beltrami” model, or “Klein” model, or “Klein–Beltrami” model or “CayleyKlein–Beltrami” model11 ), and at the same time he gives a new example of a metric on the plane satisfying Hilbert’s problem. We mention by the way that this research by Beltrami was motivated by his reading of the works of Lobachevsky. In a letter to Hoüel, written on November 18, 1868, Beltrami writes: “That writing [The Saggio 10 The 11 The

English translation in [114] of this sentence from [21] is not very faithful. four names are reasonable, see the comments in [1].

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etc.] was written during last year’s fall, after thoughts that abounded in me at the epoch of the publication of your translation of Lobatschewsky. I think that the idea of constructing non-Euclidean geometry on a perfectly real surface was entirely new” ([28]), p. 65–66). In Note I at the end of the paper [21], Beltrami recalls the metric of constant positive curvature 1=R2 that he defined in his 1865 paper [20], whose square of the infinitesimal line element is given by the formula ds 2 D R2

.a2 C v 2 /du2 2ududv C .a2 C u2 /dv 2 .a2 C u2 C v 2 /2

(2.3)

and he makes an interesting remark on the relation between the two metrics (2.2) and (2.3), namely, that the formula for one pmetric canpbe obtained from the other one by replacing the constants R and a by R 1 and a 1, thus confirming the statement that hyperbolic geometry is in some sense spherical geometry worked out on a “sphere of imaginary radius”. A modern proof of some of Beltrami’s results are contained in Busemann’s book [40], Chapter II, §15. Busemann introduces this result as follows: Although the methods are quite foreign to the rest of the book, we prove this fundamental Theorem of Beltrami in Section 15, because it is by far the most striking example of a Riemannian theorem without a simple analogue in more general spaces.

Busemann states Beltrami’s result as follows (see [40], p. 85, where Busemann refers to Blaschke [24]): Theorem 2.2. Let C be a connected open set of the projective plane P 2 equipped with a Riemannian metric whose geodesics lie on projective lines. Then the Gauss curvature is constant. For dimension 3, Busemann refers to Cartan [52]. There exists an earlier proof by Schur [109] and there are several modern proofs, see e.g. [90] and [60]. Remark 2.3.12 It is not sufficiently known that Beltrami’s theorem holds with very weak assumptions on the regularity on the metric. This was first noticed by Hartman and Wintner in an interesting paper [76], in which they prove that if a Riemannian metric is continuous and projective, then it is real analytic. Pogorelov rediscovered the result of Hartman and Wintner several years later. Note that by Brickell [32], in dimension 3, a continuous mapping between two smooth affine spaces which sends geodesics to geodesics (possibly without preserving parameterization) is automatically smooth. Darboux, who is quoted by Hilbert in his formulation of his fourth problem, in his Leçons sur la théorie générale des surfaces (Troisième partie), following Beltrami’s work, considers, in the setting of the calculus of variations, a similar problem. He 12 I

owe this remark to J. C. Álvarez Paiva.

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writes: “Proposons-nous d’abord le problème de M. Beltrami: Trouver toutes les surfaces qui peuvent être représentées géodésiquement sur le plan.” ([58], p. 59). That is to say, Darboux considers Beltrami’s problem of sending a surface into the plane by a map that sends the surface geodesics to the plane straight lines. According to Busemann ([44], p. 56), Darboux proposed several questions that are related to Hilbert’s problem, some of them being more general than the one in hand.13 In fact, one can relate the above-mentioned “problem of geographical maps” to problems that were studied by Lambert, Euler, Lagrange, Gauss, Darboux, Liouville and Bonnet. In his paper [61], Euler studied the problem of mapping a sphere on a plane such that “figures are similar” at the infinitesimal level. Gauss formulated again the problem of “representing a surface on the plane so that the representation is similar, at the very small level, to the original surface”. See [66]. This means that we ask for the similarity at the infinitesimal level for lines and angles between them, see Gauss’s quotes and the comments in the paper [94]. Using a more modern language, this is the question of finding conformal coordinates. In 1816, H. C. Schumacher, a famous German–Danish astronomer at Copenhagen who was a friend of Gauss and had been his student in Göttingen, announced a prize to be awarded by the Royal Society of Copenhagen for an answer to the problem of obtaining a general method for mapping conformally a surface onto another one. Gauss won the prize for a solution he submitted in 1822. This solution is contained in his paper [66].

3 Hilbert’s Problem IV The best way in which one can have an idea on Hilbert’s Problem IV is to read Hilbert’s text, and this is why we reproduce it in the appendix at the end of this chapter. It is also good to remember and to meditate on Hilbert’s conclusion to his Mathematical Problems.14 : The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection become ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. 13 Darboux’s “solution” of Hilbert’s fourth problem (before this problem was stated) misses however one of the most important features of the problem: by considering the variational problem for integrands of the form f .x; y; y 0 / instead of integrands of the form L.x; y; x 0 ; y 0 / that are homogeneous of degree one in the velocities x 0 and y 0 , Darboux gives many variational problems whose extremals are straight lines, but which cannot define metrics because they are not even defined on the whole tangent bundle (minus the origin). Darboux’s approach can be modified and be made to yield yet another solution to Hilbert’s fourth problem in dimension two in the smooth asymmetric case. Berck and Álvarez-Paiva did this, and later found out that it is also implicit in Hamel’s paper [75]. I owe this information to Juan Carlos Álvarez Paiva. 14 The translation is from the article in the 1902 Bulletin of the AMS [80].

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For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself more clearly. But we ask, with the extension of mathematical knowledge, will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science. The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of the exact knowledge of natural phenomena. That it may completely fulfill his high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples.

We now make some comments on Hilbert’s Problem IV. The formulation by Hilbert and the scope of his fourth problem are very broad compared to other problems in his list,15 and this problem admits several interpretations and generalizations. In fact, the problem was considered several times in the past as being solved.16 We shall recall in later sections how several mathematicians who worked on this problem interpreted it in their own manner and contributed to a solution. We mention right away the names of G. Hamel and P. Funk who worked on it in the early years of the 20th century, the fundamental work of H. Busemann over a span of fifty years (1930–1980), and the works of A. V. Pogorelov, Z. I. Szabó, R. Ambartzumian, R. Alexander, I. M. Gelfand, M. Smirnov and J.-C. Álvarez Paiva. We shall elaborate on some of these works below. There are at least five natural settings in which one can approach Hilbert’s Problem IV: (1) The setting of Riemannian geometry. In this setting, the problem was settled by Beltrami. 15As examples of Hilbert problems that are more precisely stated, we mention Hilbert’s Problem III, asking whether two 3-dimensional Euclidean polyhedra of the same volume are scissors-equivalent. This problem was solved in the negative by M. Dehn soon after Hilbert formulated it. We shall elaborate on this in Footnote 21 below. Another well-known example of a precise problem in Hilbert’s list is Part 2 of Problem VII, which asks whether given any algebraic number a 6D 0; 1 and given any irrational algebraic number b, ab is transcendental. This question was solved by A. O. Gelfand in 1934 and the proof was refined by T. Schneider in 1935. The result is called now the Gelfand–Schneider Theorem. But there are also problems in Hilbert’s list which are vaguely stated (and some of them are so because of their nature). For instance, Problem VI asks for a “mathematical treatment of the axioms of physics”. 16 One can search for papers titled “A solution to Hilbert’s Problem IV” and similar titles; there are several such papers.

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(2) The setting of the calculus of variations, which was the one of Darboux in the work we mentioned. In this setting, Hilbert’s problem is often described as an “inverse problem”; that is, one looks for a Lagrangian (defining the metric) which is associated to a given set of extremals (which are the geodesic lines). As is well known, the methods of the calculus of variations started in the works of Johann and Jakob Bernoulli, of Euler and of Lagrange. (3) The setting of metric geometry, where one does not assume any differentiability. This setting is at basis of the methods of Busemann and of the large amount of interesting works that arose from them. (4) The setting of Finsler geometry, where the metric is associated to a norm on each tangent space of the manifold. A minimum of differentiability is required in this setting, at least to give a meaning to tangent vectors. This setting can be considered as lying at the border of metric and of differential geometry. (5) The setting of the foundations of geometry. This was probably the most important setting for Hilbert, and we shall elaborate on it below. In fact, this setting is also inherent in one way or another in all the different settings mentioned above. There are also other points of view and relations with other questions. For instance, R. Alexander established relations between Hilbert’s Problem IV and zonoid theory (see [3] and the paper [115] by Z. Szabó), S. Tabachnikov made relations with the theory of magnetic flows (see [116]). I. M. Gelfand, M. Smirnov and J.-C. Álvarez Paiva developed relations with symplectic geometry (see [9], [7] and [8]). The result in [7] can be stated as follows: There exists a (twistorial) correspondence between smooth, symmetric solutions of Hilbert’s fourth problem in 3-dimensional projective space and anti-self-dual symplectic forms in S 2 S 2 . In any case, Hilbert’s Problem IV and the various approaches to solve it had an important impact on the development of several fields of mathematics, including all the ones we mentioned, to which we can add convex geometry and integral geometry. The developments that arose from that problem brought together major ideas originating from the various fields mentioned. Let us return to Hilbert’s statement, “the problem of the straight line as the shortest distance between two points”. Today, when we talk about straight lines being the shortest distance, we think of a subset of a Euclidean space Rn equipped with a metric to which the term “shortest distance” refers, and where the term “shortest line” refers to a Euclidean line, or a Euclidean segment. One should remember however that the axioms for a metric space, as we intend them today, were formulated (by M. Fréchet) only in 1906, that is, seven years after Hilbert stated his problem. From Hilbert’s own elaboration on his problem [80], it appears that he was thinking, rather than of a metric space, of a geometrical system, like Euclid’s axiomatic system (or like his own axiomatic system); that is, a system consisting of undefined objects (points, lines, congruence, etc.) and axioms that make connections between these objects. In fact, Hilbert’s problem can be included in a perspective that was dear to him on

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the axioms of geometry, and, more generally, on the foundations of mathematics.17 Indeed, in his comments about the problem, Hilbert gave the following formulation of the question18 : In analogy with the way Lobachevsky (hyperbolic) geometry sits next to Euclidean geometry, by denying the axiom of parallels and retaining all the other axioms of Euclidean geometry, explore other suggestive viewpoints from where geometries may be devised which stand, from such a point of view, next to Euclidean geometry.

This sentence clearly shows that Hilbert suggested to work out a theory in which one negates one of the axioms (other than the axiom of parallels) of Euclidean geometry, while keeping the other axioms untouched. Such an interpretation of Hilbert’s Problem IV led to interesting developments which we shall mention in the next section, but of course no geometry whose importance is comparable to the Lobachevsky geometry was discovered by keeping, among the axioms of Euclidean geometry, the parallel axiom untouched and denying another one. In fact, Hilbert gave a more precise formulation of that problem in the axiomatic framework, by adding that he asks for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms, except the one of the congruence of triangles and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.19

Hilbert’s problem, in this setting, can be formulated as follows: Consider the axiom system of Euclidean geometry, drop from it all the congruence axioms that contain the notion of angle20 and replace them by an axiom saying that in any triangle the sums of the lengths of any two sides is not smaller than the length of the third side (that is, the triangle inequality). Then: (1) characterize the geometries satisfying these conditions; (2) study individually such geometries. We shall elaborate more on the axiomatic point of view in the next section. 17 We can recall here Hilbert’s own words on a system of axioms for a geometry: “It must be possible to replace in all geometric statements the words point, line, plane by table, chair, beer, mug”, cf. Hilbert’s On the axiomatic method in mathematics (recalled in [106], p. 57). One may also recall here that others, before Hilbert (and probably since Euclid’s epoch), had the same point of view on the axiomatic method, even if they did not express it with the same words. 18 The translation is in [80]. 19 We note that the word “congruence”, in Hilbert’s point of view, designates a relation between primary objects like segments, angles, triangles, etc. satisfying certain properties (some of which are axioms), and that this word does not necessarily refer to an equivalence relation in a metric sense, that is, an isometry. In fact, as in Euclid’s axioms, there is no distance or length function involved in Hilbert’s axioms. Congruence can be thought of as an undefined notion. 20 For instance, in Hilbert’s axioms for Euclidean geometry, the following axiom is part of the set of congruence axioms for angles:

b 1 b

Given an angle ABC and given a point B 0 and a ray B 0 C 0 starting at B 0 , and given a choice of a side on the line B 0 C 0 , there exists a ray B 0 A0 starting at B 0 , with A0 being on the chosen side of B 0 C 0 , such that A0 B 0 C 0 ABC . Note that in Euclid’s Elements, this is a theorem (Proposition 23 of Book 1).

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4 Some early works on Hilbert’s Problem IV Among the major mathematicians who worked on Hilbert’s Problem IV, there are Hilbert himself and his students Dehn21 , Hamel22 and Funk23 . The title of Dehn’s dissertation is Die Legendre’schen Sätze über die Winkelsumme im Dreieck (1900) [59] (Legendre’s theorems on the sum of the angles in a triangle), the title of the one of Hamel is Über die Geometrien in denen die Geraden die kürzesten sind (1901) (On the geometries where the straight lines are the shortest), and the title of the one of Funk is Über Flächen mit lauter geschlossenen geodätischen Linien (1911) (On surfaces with none but closed geodesic lines). All these works are related to Problem IV. Among the other early works that are related to Hilbert’s problem, one also has to mention those of Finsler24 and of Berwald.25 We shall report on some of their works. 21 Max Dehn (1878-1952) is one of the main founders of combinatorial topology and combinatorial group theory. His thesis, defended in 1900, concerns questions on the foundations of geometry, and it is related to Hilbert’s Problem IV, in an extended form discussed by Hilbert. Dehn constructed a geometry in which there are infinitely many lines passing through a point and disjoint from another line and in which Legendre’s theorem stating that the sum of the angles in any triangle is at most fails. (Therefore this geometry is different from hyperbolic geometry.) Dehn’s geometry is non-Archimedean, and in this work Dehn constructed a nonArchimedean Pythagorean field. The same year, Dehn, who was only 22 years old, solved Hilbert’s Problem III which we already mentioned in Footnote 15. The problem asks whether two 3-dimensional (Euclidean) polyhedra that have the same volume are scissors-equivalent, that is, whether they can be obtained from each other by cutting along planes and re-assembling. The corresponding result in dimension 2 is true and is known as the Bolyai–Gerwien Theorem. Dehn showed that in dimension 3 the answer is no, which was Hilbert’s guess. In his work on that problem, Dehn introduced a scissors-equivalence invariant for 3-dimensional polyhedra, which is now called the Dehn invariant, and he showed that this invariant does not give the same value to the tetrahedron and the cube. In 1965, J.-P. Sydler – an amateur mathematician – completed Dehn’s theory by showing that two polyhedra are scissors-equivalent if and only if they have the same volume and the same Dehn invariant. 22 Georg Hamel (1877–1954) entered Göttingen University in 1900. In 1901 he was awarded his doctorate, under the supervision of Hilbert. Hamel is probably best known for his contribution on the so-called Hamel basis, a work published in 1905, in which he gave a construction of a basis of the real numbers as a vector space over the rationals. This construction is important because it involves one of the earliest applications of the axiom of choice. Hamel later made a name for himself in the field of mechanics. 23 Paul Funk (1886–1969) is mostly known for his work on the calculus of variations and for his introduction of the Funk transform (also called the Minkowski-Funk transform), an integral transform obtained by integrating a function on great circles of the spheres. The subject of his dissertation work (1911), surfaces with only closed geodesics, which he wrote under Hilbert, was further developed by Carathéodory. We owe him the Funk metric, an asymmetric metric defined on convex subsets of Rn which is a non-symmetric version of the Hilbert metric satisfying the requirements of Hilbert’s Problem IV. This metric is studied in detail in Chapter 2 of this Handbook [100]. 24 Paul Finsler (1894–1970) wrote his doctoral dissertation in Göttingen in 1919 under the supervision of Carathéodory. The title of the thesis is Über Kurven and Flächen in allgemeinen Räumen (On Curves and surfaces in general spaces). According to Busemann [38], Finsler was not the first to study Finsler spaces. These spaces had been discovered by Riemann, who mentioned them in his famous lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypothesis which lie at the foundations of geometry) (1854) [107]. Busemann recalls in this respect that the goal that Riemann set for himself was the definition and discussion of the most general finite-dimensional space in which every curve has a length derived from an infinitesimal length or line element. The name Finsler space is due to É. Cartan, after his paper Les espaces de Finsler [53] published in 1934. In their paper [50], Busemann and Phadke write the following: “Finsler was the first who investigated non-Riemannian spaces (under strong differentiability hypotheses) guided by Carathéodory, whose methods in the calculus of variations form the basis of Finsler’s thesis, which has no relation to our G-spaces. The misnomer was caused by the fact that Finsler’s thesis (1918) was inaccessible until 1951, when Birkhäuser reissued it unchanged”.

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Let us now say a few words on the work of Hilbert on the foundations on mathematics, in order to explain the axiomatic approach that he alludes to in the comments on his fourth problem. In the various revised editions of his Grundlagen der Geometrie26 (Foundations of geometry) [81], Hilbert worked out several sorts of geometries which he termed as “non-Euclidean”,27 a word he used in a very broad sense, in the formulation of his fourth problem. This includes first the well-known non-Archimedean geometry,28 in which all the axioms of Euclidean geometry are kept untouched except the Archimedean axiom. We already mentioned that a non-Archimedean geometry was also the subject of the doctoral thesis of Dehn (1902). Then, Hilbert constructed a nonArguesian geometry, which is a plane geometry in which the theorem of Desargues29 of projective geometry fails,30 and a non-Pascalian geometry, in which the theorem of Pascal fails.31 To define these geometries, one has to construct new number fields, in particular a field of non-Archimedean numbers and a field of non-Pascalian num25 Ludwig Berwald (1883–1942) obtained his doctorate at the Royal Ludwig-Maximilians-University of Munich under the guidance of Auriel Voss. The title of his dissertation was: Über die Krümmungseigenschaften der Brennflächen eines geradlinigen Strahlsystems und der in ihm enthaltenen Regelflächen (On the curvature properties of the caustics of a system of rays and the ruled surfaces contained in it). Berwald is considered as one of the founders of Finsler geometry. 26 The first edition appeared in 1899, that is, one year before Hilbert formulated his Problems. 27 The term non-Euclidean is used here in a broad sense, and not only in its classical sense which usually includes only hyperbolic and spherical geometry. 28 Guiseppe Veronese (1854–1917) constructed, in 1889 (that is, a few years before Hilbert did), a nonArchimedean geometry. Hilbert mentions the work of Veronese in the statement of his problem [80]. 29 Since this theorem plays a central role in the work of Hilbert and in the work of Busemann on axioms, we recall its statement. Consider in the projective plane two triangles abc and ABC . We say that they are in axial perspectivity if the three intersection points ab \ AB, ac \ AC , bc \ BC are on a common line. We say that the three triangles are in central perspectivity if the three lines Aa; Bb; C c meet in a common point. Desargues’ theorem says that for any two triangles, being in axial perspectivity is equivalent to being in central perspectivity. This theorem was published for the first time by A. Bosse, in his Manière universelle de M. Desargues pour manier la perspective par petit pied comme le géométral (Paris, 1648, p. 340). Bosse’s memoir is reproduced in Desargues’ Œuvres (ed. N. G. Poudra, Paris 1884, p. 413–415). Desargues’ proof of the theorem uses Menelaus’ Theorem. Von Staudt, in his Geometrie der Lage (Nuremberg, 1847) gave a proof of this theorem that uses only projective geometry notions, namely, notions of incidence between points and lines, etc., but in three-space, and not only in the plane. Klein, in his famous paper [87], noted that this theorem cannot be proved using only twodimensional projective geometry. Hilbert, in his Grundlagen der Geometrie (2nd. ed., §23) exhibited a plane geometry in which all the axioms of projective geometry are satisfied, but where the theorem of Desargues does not hold. Thus, the theorem of Desargues follows from the axioms of three-dimensional projective geometry, but not of those of the two-dimensional. It can be proved in two-dimensional projective geometry if one introduces metric notions. Conversely, Hilbert proved that for three-dimensional projective geometry, it is sufficient to have the axioms of two-dimensional geometry together with the plane theorem of Desargues. 30A model of non-Arguesian geometry, described by Hilbert in the first editions of his Grundlagen, is obtained by taking an ellipse in the Euclidean plane and replacing the Euclidean segments intersecting it by arcs of circles that pass by a common fixed point P . Thus, in this geometry, a line is taken in the usual sense if it does not intersect the (interior of the) ellipse, and in the case where it intersects it, it is made out of three pieces: two half-lines, outside the ellipse, connected by an arc of a circle inside the ellipse. The group of Euclidean axioms which Hilbert calls the “projective axioms” are satisfied by such a geometry provided the point P is not too close of the ellipse. In later editions, Hilbert replaced this model by a model found by F. R. Moulton, cf. [93]. 31 Pascal’s theorem in projective geometry says that given a hexagon in the projective plane, there is an equivalence between the following two statements: (1) the hexagon is inscribed in a conic; (2) The intersections of the lines containing pairs of opposite sides are aligned.

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bers. A non-Pascalian geometry is also non-Archimedean. Hilbert also considered a geometry he called non-Legendrian, in which it can be drawn from a point infinitely many parallels to a line that does not contain it, and where, like in spherical geometry, the angle sum in a triangle is greater than two right angles. He then considered a geometry he called semi-Euclidean, in which one can draw from a point infinitely many parallels to a line not containing it, and where the angle sum in a triangle is equal to two right angles. In fact, both non-Legendrian and semi-Euclidean geometries are considered by Dehn in his 1900 dissertation [59], as two distinct non-Archimedean geometries. Hilbert also considered a non-Pythagorean geometry. In his paper Über eine neue Begründung der Bolyai–Lobatschefskyschen Geometrie [82], he constructed a non-Archimedean Lobachevsky geometry. Another “non-Euclidean” geometry that Hilbert mentioned is the Minkowski geometry, which he described as a geometry in which all the axioms of Euclidean geometry are satisfied except a “triangle congruence” axiom. In this geometry, the theorem stating that the angles at the basis of an isosceles triangle are equal is not satisfied and the theorem stating that in any triangle the sum of the lengths of two sides is less than the length of the third side is taken as an axiom. In his comments on his fourth problem, Hilbert mentioned a geometry he described in his paper [78] where the parallel axiom is not satisfied but all the other axioms of a Minkowski geometry are satisfied. In this paraphernalia of geometries, only a few satisfy the requirements of Problem IV, but it is clear from the statement of the problem and the work that Hilbert did at the same period that he was thinking about new geometries in which a variety of other properties could replace the fact that “shortest distances are realized by straight line”. Poincaré commented on these geometries and on the number fields to which they give rise in his review of Hilbert’s Grundlagen der Geometrie, [105]. He writes (p. 2): Many contemporary geometers [...] in recognizing the claims of the two new geometries [the hyperbolic and the spherical, which Poincaré calls the Lobachevsky and the Riemann geometries] feel doubtless that they have gone to the extreme limit of possible concessions. It is for this reason that they have conceived what they call general geometry, which includes as special cases the three systems of Euclid, Lobachevsky, and Riemann, and does not include any other. And this term, general indicates clearly that, in their minds, so other geometry is conceivable. They will loose this illusion if they read the work of Professor Hilbert. In it they will find the barriers behind which they have wished to confine us broken down at every point.

In the conclusion to his comments on Problem IV, Hilbert declares that it would be desirable to make a complete and systematic study of all geometries in which shortest distances are realized by straight lines. The conclusion of the problem in its French version is different from that of the English one: in the former, Hilbert includes the problem in the setting of the calculus of variations, and he refers to the work of Darboux that we already alluded to. The reader can find an English translation of the conclusion in the Appendix to the present chapter. It may be worth mentioning that in order to make the relation between the axiomatic abstract setting and the metric setting, one may refer to Euclid. We already recalled

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that there is no distance function involved in Euclid’s Elements, but if we introduce this metric language, it is a consequence of Euclid’s axioms (in particular Axiom 2) that on a line, distance is additive, and it follows from Proposition 20 of Book I that if between three points distance is additive then the three points are on a line (see [77]). This implies that the Euclidean lines in the sense of the axioms of geometry are the (geodesic) lines in the metric sense. We now say a few words on other early works on Problem IV. Hamel considered in [75] metric spaces .; d / where is an open subset of a projective space P n for some n > 1 and where d satisfies the following strong version of Hilbert’s requirements (stated here in modern terms, following Busemann [44]): (1) .; d / is a geodesic metric space and the projective lines are geodesics; (2) the closed balls are compact; (3) if x; y; z do not lie on a projective line, then d.x; y/ C d.y; z/ > d.x; z/. Note that Conditions (1) and (3) imply that the metric d is uniquely geodesic. Under these hypotheses, Hamel proved that the set is necessarily convex. More precisely, he proved the following: Theorem 4.1. If the metric space .; d / satisfies (1) to (3), then is of one of the following two types: (1) can be mapped homeomorphically onto the projective space P n in such a way that the maximal geodesics of are mapped to great circles and all great circles have equal lengths; (2) can be mapped homeomorphically onto an open convex subset of affine space An in such a way that the maximal geodesics of have infinite length and are mapped to the intersection of with affine lines in An . In particular, closed geodesics and maximal open geodesics cannot coexist in .; d / . Let us note that the two geometries of constant curvature +1 and -1 are examples of metric spaces described in Theorem 4.1, namely, spherical geometry is an example of type (1), and hyperbolic geometry, through the Klein–Beltrami model, is an example of type (2). Hamel considered only smooth metrics, and his methods are variational. In his thesis [74] (1901), he considered non-necessarily symmetric metrics. In his later paper [75] (1903) he also considered non-symmetric metrics although he was not very explicit about that. He refers to symmetric metrics by the terms “Starke Monodromieaxiome” and to non-symmetric metrics by “Schwache Monodromieaxiome”.32 The main 32 The meaning is “strong monodromy hypothesis” and “weak monodromy hypothesis” respectively. The reason for this terminology comes from dimension two, where Hamel (and Funk after him) use polar coordinates. In this case, the Lagrangian depends on r and . The periodicity of the angle variable imposes that its value is invariant by adding integer multiples of 2 (this is the weak monodromy hypothesis). In the case of a symmetric metric, the value of must be invariant by adding integer multiple of (this is the strong monodromy hypothesis). I owe this remark to Marc Troyanov.

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result he obtained in that paper is a characterization of the differential of arclength, for a general smooth (symmetric and non-symmetric) metric satisfying Hilbert’s requirement, in dimensions 2 and 3. Hamel’s result is presented in Chapter 3 [120] in this Handbook. At its publication, Hamel’s paper [75] was considered as a solution to Hilbert’s problem, under the given restrictive assumptions. Concerning Hamel’s method, in his 1974 report [46] on Hilbert’s Problem IV, Busemann writes (p. 137): We do not reproduce it here, since it is not based on general ideas like integral geometry and the expression given by Hamel is not illuminating. Also, inevitably in view of the time it was written, the discussion of completeness does not satisfy our modern requirements; the necessary concepts did not yet exist. However, the immense number of possibilities becomes quite clear from [75].

Busemann, in [40], gave a proof of the same result, putting it in a modern setting and using completely different methods, which are purely metrical. He gave a nonsymmetric version which he attributed to Hamel. A proof of Hamel’s result is contained in Busemann [44], p. 37, which Busemann states as follows in the setting of nonsymmetric metrics: Theorem 4.2. Under the hypotheses of Theorem 4.1 but without the assumption that the metric is symmetric and where Condition (2) is replaced by the requirement that the right closed balls are compact,33 the space .; d / is of one of the following two types: • D P n , and each segment of a projective line in P n , traversed in either direction, is a geodesic. Furthermore, all such geodesic are projective lines which, traversed in either direction, have the same length. • is a convex set and contained in some affine n-dimensional subspace An of P n . Finally, we mention a result of Berwald related to Hilbert’s Problem IV. Berwald proved in [27] (with a simpler proof given by Funk in [64]) that 2-dimensional Finsler spaces of constant flag curvature34 that satisfy the requirements of Hilbert’s Problem IV are characterized by the property that any isometry of a geodesic onto another (or onto itself) is the restriction of a projectivity. This result generalizes Beltrami’s result which he obtained in the setting of Riemannian geometry. A proof of this result is given in Chapter 3 [120] of this Handbook. The works of Hamel, Funk and Berwald had a great influence on Busemann, who made a new definite breakthrough on the problem, which we describe in the next section. a non-symmetric metric d , one has to distinguish between the right open ball of center x and radius r, D fy j d.x; y < rg and the left open ball of center x and radius r, B C .x; r/ D fy j d.y; x/ < rg. The right and left closed balls are defined by replacing the strict inequalities by large inequalities. 34 There are several notions of curvature in the setting of Finsler spaces, and an important one is the notion of flag curvature, which is a kind of Finsler analogue of the sectional curvature of Riemannian geometry. We refer the reader to §41 of Busemann’s Geometry of geodesics [40] for a discussion of curvature in Finsler spaces and to Chapter 3 of this Handbook [120]. 33 For

B C .x; r/

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5 Busemann’s approach and Pogorelov’s solution We start with a reformulation of the problem. We already mentioned that, in its original formulation, Hilbert’s Problem IV is sometimes considered as too broad, and several mathematicians in their work on it made it more precise. Busemann, who spent a large part of his intellectual activity thinking and working on that problem reformulated it as follows (see [46]): The fourth problem concerns the geometries in which the ordinary lines, i.e. lines of an n-dimensional (real) projective space P n or pieces of them, are the shortest curves or geodesics. Specifically, Hilbert asks for the construction of these metrics and the study of the individual geometries. It is clear from Hilbert’s comments that he was not aware of the immense number of these metrics, so that the second part of the problem is not at all well posed and has inevitably been replaced by the investigation of special, or special classes of, interesting geometries.

The second part of the problem is indeed very wide, and because of that it is likely to remain open forever. Busemann writes in [47]: The discovery of the great variety of solutions showed that Part (2) of Problem 4 is not feasible. It is therefore no longer considered as part of the problem. But many interesting special cases have been studied since 1929.

Part (2) of the problem is nevertheless fascinating, in many cases of the individual geometries. There are various examples where new phenomena can be found, even in the following classes of classical metric spaces: (1) Minkowski metrics (that is, translation-invariant metrics on Rn defined by norms, which may be non-symmetric);35 (2) Funk metrics;36 (3) Hilbert metrics. Each of the geometries for which the Euclidean straight lines are geodesics defines a new world, and one may try to answer in that world numerous questions regarding triangles, boundary structure, trigonometry, parallelism, perpendicularity, horocycles, compass and straightedge constructions, and there are many others. The two- and three-dimensional cases are particularly worth investigating in detail. According to Busemann [47], at the time where Hilbert proposed his problem, the only classes of metrics (“besides the elementary ones”) that were known to satisfy the requirements of that problem were the Minkowski metrics, introduced around 1890 by H. Minkowski for their use in number theory, and the Hilbert metrics. The latter were discovered by Hilbert in 1894 [78], as a generalization of the Klein–Beltrami model of hyperbolic geometry. We note that both classes of metrics are Finsler and almost 35 The Minkowski metrics are the translation-invariant metrics associated to finite-dimensional normed spaces, except that the underlying norm function (and therefore the distance function) is not necessarily symmetric. See Chapter 1 [99] in this Handbook. 36 For a survey on Funk geometries, see Chapter 2 [100] in this Handbook.

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never Riemannian.37 Indeed, it is a consequence of Beltrami’s work that we already mentioned, that the only Riemannian metrics that satisfy Hilbert’s requirements are the metrics of constant curvature. The Minkowski and the Hilbert metrics are mentioned in Hilbert’s comments on Problem IV. Busemann notes furthermore that Minkowski geometries satisfy the parallel axiom and that Hilbert geometries generalize the hyperbolic one.38 Busemann adds that probably Hilbert did not think of “mixed situations”, which Busemann himself considered in his various books and papers and which we recall now. Busemann introduced new classes of metrics which satisfy Hilbert’s requirements and which are combinations of Minkowski and Hilbert metrics. In his book [40] (p. 111 ff.), he first makes the simple remark that the metric L.p; q/Ce.p; q/ where L is the Hilbert metric and e is the Euclidean metric has the required properties, and then he defines several variations on this metric. In particular, he considers a combination of the Euclidean metric with the Hilbert metric restricted to the intersection of the convex set with a flat subspace. He then defines (in dimension 2) several classes of combinations which depend on parameters and which satisfy interesting parallelism properties (in the sense of hyperbolic geometry). In [46], returning to these examples, he considers the explicit metric e.x; y/ C je xn e yn j on affine space with affine coordinates x D .x1 ; : : : xn /, and he notes that there are variations on this example where instead of the exponential function one can take any monotone function. He then adds the following: Many such functions can be added, and watching convergence even infinitely many. Also, the addition can be replaced by an integration over a continuous family. Of course, it is very easy to generate nonsymmetric distances by this method, for example by modifying nonsymmetric Minkowski distances. It seems clear that Hilbert was not aware of these simple possibilities.

For arbitrary open convex subsets of An , Busemann also exhibits “infinitely many essentially different Desarguesian metrics”, namely, he takes the metric e.x; y/ C j tan xn tan yn j defined on the strip =2 < xn < =2, and again, he notes that the only property used is that tan t increases monotonically from 1 to C1 in .=2; =2/. One of the major achievements of Busemann was to put Hilbert’s Problem IV in an adequate metric setting. In particular, he introduced the terminology of a G-space and of a Desarguesian space which we recall now, and over several decades he did extensive work on the geometry of these spaces. At several places, he gave slightly different but overall equivalent definitions of G-spaces and of Desarguesian. In [36] and [37], Busemann gave the following definition, see also Busemann’s book The geometry of geodesics [40]. 37 The exceptions in both cases are represented by ellipsoids: the case where the convex set on which the Hilbert metric is defined is an ellipsoid, and the case where the unit ball of the Minkowski metric is an ellipsoid centered at the origin. 38 Recall that the Hilbert metric of the disc is the hyperbolic metric.

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Definition 5.1 (G-space). A G-space39 is a metric space .X; d / satisfying the following properties: (1) (Finite compactness) Every bounded infinite set has an accumulation point. (2) (Menger convexity) For any x and z in X , there exists a point y which is between x and z, that is, y different from x and z and satisfies d.x; y/Cd.y; z/ D d.x; z/. (3) (Local extendability) For every point p of X there exists an open ball B centered at p such that for every x and y in B there exists a point z in B such that y is between x and z. (4) (Uniqueness of extension) If y is between x and z1 and between x and z2 and if d.y; z1 / D d.y; z2 / then z1 D z2 . In particular, in a G-space, geodesics are extendable, and the extension is unique. Busemann gave then the following definition of a Desarguesian space (see [40] and [47]): Definition 5.2 (Desarguesian space). A Desarguesian space is a metric space .R; d / satisfying the following properties: (1) R is an open nonempty subset of projective space P n equipped with a metric .x; y/ whose associated topology is the one induced from its inclusion in P n . (2) The closed balls in R are compact. (3) Any two points in R can be joined by a geodesic segment, that is, a curve of length .x; y/. (Here, the length of a curve is defined as usual as the supremum over all subdivisions of the image set of a segment of the set of lengths of polygonal curves joining the vertices of the subdivision; in modern terms, a metric where any two points can be joined by a geodesic segment is said to be geodesic). (4) For any three points x, y, z in P n that are not collinear, the strict triangle inequality .x; y/ C .y; z/ < .x; z/ holds between them; that is, the three points do not belong to a geodesic. Remark 5.3 (Non-symmetric metrics). In the later generalizations of G-spaces and of Desarguesian space to non-symmetric metrics that Busemann considered (see e.g. [44]), the condition that the closed balls are compact is replaced by the condition that the right closed balls are compact. 39 The expression G-space stands for “geodesic space”. The concept was introduced quite early by Busemann. In their paper [50], p. 181, Busemann and Petty recall that “The G-spaces were first introduced without this name by Busemann in his thesis in 1931.” This notion turned out to be of paramount importance, and a lot of work has been done, by Busemann and others, especially around the conjecture of Busemann saying that every finite-dimensional G-space is a topological manifold. Busemann himself proved the conjecture in dimensions 1 and 2 ([40] §9 and §10), Krakus [88] proved it in dimension 3 using a theorem of Borsuk [30] and P. Thurston [119] in dimension 4. See the survey [73] by Halverson and Repovš. It may be useful to recall in this respect that G-spaces are metric and separable, and therefore the usual concept of dimension in the sense of Meyer-Urysohn applies to them, cf. [83].

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Thus, whereas the definition of a G-space concerns general metric spaces, the one of a Desarguesian space concerns metrics on subsets of projective space. A Desarguesian space is a particular case of a G-space and it is also a particular case of a space satisfying the requirement of Hilbert’s Problem IV. In a Desarguesian space, the geodesic joining two points is unique and is necessarily the affine or projective line. The terminology chosen by Busemann comes from the fact that in such a space, Desargues’ theorem40 holds. Beltrami’s theorem, which we already quoted, implies that in the Riemannian case, the only Desarguesian spaces are the spaces of constant curvature, that is, the Euclidean, hyperbolic or elliptic spaces. The terminology used today for spaces satisfying Hilbert’s problem requirements, especially in the context of Finsler geometry, is that of a projective metric, or projectively flat metric (cf. also Busemann in [39]). Let us note that Busemann was suspicious about the Finsler geometry viewpoint, in the sense in which this word was used in the early 1950s. His feeling about it is expressed in his survey paper [38] which starts as follows: The term “Finsler space” evokes in most mathematicians the picture of an impenetrable forest whose entire vegetation consists of tensors. The purpose of the present lecture is to show that the association of tensors (or differential forms) with Finsler spaces is due to an historical accident, and that, at least at the present time, the fruitful and relevant problems lie in a different direction.

The paper ends with the following words: This confirms that in spite of all the work on Finsler spaces we are now at a stage which corresponds to the very beginning in the development of ordinary differential geometry. Therefore the mathematician who likes special problems has the field. After sufficiently many special results have been accumulated someone will create the appropriate tools. At the present time it is difficult to guess what they will be beyond a vague feeling that some theory of integro-differential invariants will be essential.

Busemann interpreted Hamel’s work in his setting of Desarguesian spaces; let us quote him from [40] (p. 66): Whereas Hamel [75] has given a method for constructing all Desarguesian, sufficiently differentiable G-spaces, no entirely satisfactory infinitesimal characterization of these spaces, in terms of analogues to curvature tensors say, has ever been given. The freedom in the choice of a metric with given geodesics is for non-Riemannian metrics so great, that it may be doubted, whether there really exists a convincing characterization of all Desarguesian spaces.

Busemann also considered the axiomatic viewpoint. In §13 of his book The geometry of geodesics [40], he made relations between the metrical point of view on Problem IV and the foundations of geometry point of view. The Section starts with the following introduction: This section is closely related to the classical results of the foundations of geometry, and the methods of this field are partly used here. We outline briefly the analogies as well as the differences of that work with the present. 40 See

Footnote 29.

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A. V. Pogorelov,41 in his monograph [104] that accounts for his solution of Hilbert’s Problem IV, notes that the Minkowski geometries (respectively the Hilbert geometries) realize the system of axioms of Euclidean geometry (respectively of Lobachevsky geometry) after we drop all the axioms of congruence involving the concept of angle and adding to them the triangle inequality axiom (which is one of Hilbert’s requirements at the level of axioms). He adds (without further elaboration) that Hilbert’s problem can be formulated in a similar way for elliptic geometries. Álvarez Paiva in [5] makes further comments on the elliptic geometry case. The idea of Pogorelov’s proof of Hilbert’s Problem IV came from Busemann, who, at the Moscow ICM (1966), gave a construction of a Desarguesian space, which introduced ideas from integral geometry in the approach to Hilbert’s problem.42 The method provided a metric on any bounded open convex set B in a projective space P n . We already noted that (by a result of Hamel) a solution of Hilbert’s problem is necessarily a metric defined either on P n or on an open convex subset of affine space (sitting as the complement of a hyperplane in P n ). We briefly recall Busemann’s definition in case where B D P n . Let H be the set of hyperplanes in P n . For any subset X of P n , we let X be the set of hyperplanes that intersect X. On the set H , we define a nonnegative measure m satisfying the following properties: (1) For any point p in P n , m.fpg/ D 0. (2) For every nonempty open subset X of P n , m.X / > 0. (3) m.P n / D 2k < 1. Then we notice that for every line L in P n we have L D P n , which shows that m.L/ D 2k. Now for every pair of distinct points x and y in P n , consider the line L that contains them. The two points divide L into two arcs, A and B, and m.A/ C m.B/ D 41Aleksei Vasil’evich Pogorelov (1919–2001) spent most of his career in Kharkiv. He belonged to the famous Russian mathematical school, and his two advisors wereAlexandrov and Efimov. He first studied at the Zhukowski Air ForceAcademy in Moscow, and he worked as an engineer before he started his graduate studies in mathematics in Moscow. Before his work on Hilbert’s Problem IV, Pogorelov had published several important papers, including solutions of some famous problems. In 1949, he gave a substantial generalization of Cauchy’s rigidity theorem from convex polyhedra to convex surfaces. We recall that Cauchy’s theorem (1813) states that any two convex polytopes in R3 which have congruent faces are congruent. Pogorelov’s theorem states that any two closed isometric convex surfaces in R3 are congruent. The proof used a famous result of Alexandrov on gluing convex surfaces, which was used by the latter to prove that any surface homeomorphic to a sphere and equipped with a metric of positive curvature is isometric to a convex surface, answering a famous problem which was asked by Weyl. The history of Cauchy’s theorem is interesting. A mistake in Cauchy’s proof was found by E. Steinitz in 1920, and corrected by him in 1928. Other improvements and extensions of the theorem are due to several people, among them Dehn (1916) and Alexandrov (1950). In 1952, Pogorelov published a solution of the socalled multi-dimensional regularity problem of the Minkowski problem, stating that any convex surface whose Gaussian curvature is positive and is a C m function of the outer normal, for m 3, is C mC1 -regular. (This problem has several later developments and generalizations, by Nirenberg, Cheng–Yau and others.) 42 There is no text by Busemann in the proceedings of this ICM, and his name is not mentioned in the list of participants. But there is a paper by him in Russian, in the Russian journal Uspehi of the same year, [43], in which he presents Hilbert’s problem and his results related to the solution of that problem.

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m.P n / D 2k. Therefore, one of the two values m.A/ or m.B/, say m.A/, is k. Setting .x; y/ D m.A/ defines a Desarguesian metric on P n . Explicit formulae can be obtained for such measures since the set of hyperplanes in projective space can easily be parametrized. One must remember here that Busemann, at the time of the Moscow 1966 ICM, was working in relative isolation, and that the subject of metric geometry was not fashionable in the West. On the contrary, in the Soviet Union, the subject, which flourished independently, was considered as important, and it was represented by a host of very good and influential mathematicians including A. D. Alexandrov, V. A. Zalgaller, V. A. Toponogov, N. V. Efimov, Yu. G. Rechetnjak, and A. V. Pogorelov himself. The interaction between the two schools, before Busemann’s visit to Moscow, was poor. One should also note that Busemann, during his college years, followed the lectures of the topologist P. S. Alexandrov and of other Russian mathematicians. In her biography of Courant (who was Busemann’s advisor in Göttingen), Constance Reid writes ([106], p. 106): Since 1923 Alexandrov had returned each year, either alone or accompanied by countrymen. From 1926 through 1930, Courant always arranged for him to give courses in topology, each for a quite different audience of mathematicians. The summer that Courant was trying to keep up some semblance of attendance at Wiener’s lectures, Alexandrov’s were crowded. “Alexandroff and the other Russian visitors were very important, very influential,” I was told by Herbert Busemann, who came in Göttingen in 1925 to study mathematics after “wasting” – as he said – several years of his life in business to please his father, one of the directors of Krupp. “The Russians filled a gap because they were familiar with certain more abstract tendencies which were not well represented in Göttingen. Courant, as probably many have told you, was rather reactionary in his mathematical outlook. He didn’t see the importance of many of the modern things”.

let us also recall that in 1985, Busemann received the Lobachevsky prize, which is the most prestigious reward in geometry that was given the Soviet Union (and, before that, in Russia).43 In last paper [50], written in collaboration with Phadke and published in 1993, Busemann recalls his beginning in metric geometry. He writes the following (p. 181): Busemann has read the beginning of Minkowski’s Geometrie der Zahlen in 1926 which convinced him of the importance of non-Riemannian metrics. At the same time he heard a course on point set topology and learned Fréchet’s concept of metric spaces. The older generation ridiculed the idea of using these spaces as a way to obtain results of higher differential geometry. But it turned out that a few simple axioms on distance suffice to obtain many non-trivial results of Riemannian geometry and, in addition, many which are quite inaccessible to the classical methods.

In the concluding remarks of the same paper, Busemann writes: 43 The prize was given to Busemann “for his innovative book The geometry of geodesics” which he had written 30 years before. We also recall that the first recipients of this prize were Sophus Lie in 1897, Wilhelm Killing in 1900 and David Hilbert in 1903. The prize was awarded to A. D. Alexandrov in 1951.

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The acceptance of our theory by others was slow in coming, but A. D. Alexandrov and A. M. Gleason were among the first who evinced interest and appreciated the results.

Let us now quote Pogorelov, from the introduction to his monograph [104]: The occasion for the present investigation is a remarkable idea due to Herbert Busemann, which I learned from his report to the International Congress of Mathematicians at Moscow in 1966. Busemann gave an extremely simple and very general method of constructing Desarguesian metrics by using a nonnegative completely additive set function on the set of planes and defining the length of a segment as the value of the function of the set of planes intersecting the segment. I suspected that all continuous Desarguesian metrics could be obtained by this method. The proof of this in the 2-dimensional case strengthened my belief in this conjecture and I announced a general theorem in [103]. However it turned out later, on making a detailed investigation of the three-dimensional case, that the completely additive set function figuring in Busemann’s construction may not satisfy the non-negativity condition. Therefore, the result given here, while preserving its original form, assumes that other conditions are satisfied.

Busemann’s construction mentioned by Pogorelov is based on the integral formula for distances defined as a measure on the set of planes that we mentioned above. The formula is contained in earlier work of Blaschke [25], based on an integral formula which is due to Crofton44 in the setting of geometric probability. We shall say a few more words on this formula below, since it was at the basis of most of the developments that followed. As already mentioned, Pogorelov’s solution of Hilbert’s Problem IV consisted in showing that every Desarguesian metric is given by Busemann’s construction, but with a slight modification. Pogorelov obtained the result, first in dimension n D 2 for a general continuous metric, and then for n D 3, with a smoothness assumption. He also proved that in dimension 3, every metric that is a solution of Hilbert’s Problem IV is a limit (in the sense of uniform convergence on compact sets) of a sequence of such metrics that are of class C 1 . It turned out that the proof for n D 3 needs more work than the one for n D 2; in particular, a new definition of the measure m, allowing it to take negative values (but with still positive values on every set of the form T , where T is a segment) is needed. In fact, Pogorelov’s solution (even in dimension 2) is variational and it uses the smoothness condition. For dimension 2, Pogorelov obtained the result for continuous metrics by using an approximation argument (which does not apply in dimensions > 2). One should also note that the fact that in dimension > 2 the measure m can take negative values is also mentioned in Busemann’s earlier paper [41] in which he showed that all Minkowski metrics which are smooth enough can be obtained by this construction. 44A formula for a metric on a subset of Euclidean space that uses a measure on the set of Euclidean hyperplanes, where the distance between two points x and y is equal to the measure of the set of Euclidean hyperplanes that meet the segment Œx; y, is sometimes referred to as a Crofton (or sometimes Cauchy–Crofton) formula. The name is after Morgan Crofton (1826–1915) who proved a result of integral geometry relating the length of a curve to the expected number of times a random line intersects it, see [108], p. 12–13.

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Pogorelov’s exposition in the book [104] is elementary and pleasant. Busemann wrote a review on that book in [47] in which he confirmed that Pogorelov’s method of proof for n D 3 works, with more technicalities, for n > 3. It is fair to recall here that Hilbert, in the spirit of his time, restricted himself to dimensions 2 and 3. In their paper [48], Busemann and Phadke work out a general and global45 version of the 1865 Beltrami Theorem [20] in the setting of Desarguesian spaces. They define a general class of metric spaces which they call chord spaces. These are metric spaces possessing a distinguished class of extremals (that is, locally isometric maps from an interval into the metric space), called chords, which behave locally like the affine lines, but which are not the unique extremals of the metric. Chord spaces are the subject of the monograph [49]. Busemann and Phadke obtain in [48] the following results, which they “believe to be the most general meaningful version of that theorem”: Theorem 5.4. A locally Desarguesian simply connected chord space is either defined in all of S n or is an arbitrary open convex set of an open hemisphere of S n (considered as the projective space An ). Theorem 5.5. A simply connected locally Desarguesian and locally symmetric Gspace is Minkowskian, hyperbolic or spherical. At the end of their paper [48], Busemann and Phadke write: Surprisingly, one can clearly deduce from a paragraph (which is too long to be produced here in total) in Beltrami’s paper [21] that he would have welcomed our generalization of his theorem. He says that a theorem which holds under weaker hypotheses than stated has not been fully understood and that the proper generalization may involve the disappearance of some of the original concepts (in our case the Riemannian metric).

6 Other works and other solutions More general versions of Pogorelov’s solution in the 2-dimensional case were later on obtained by completely different methods by Ambartzumian [16] and by Alexander [2]. Ambartzumian’s method can be used only in dimension 2 and it gives the result of Pogorelov directly for continuous metrics, without passing by smooth metrics and without using any approximation argument. The method has a combinatorial character, and it uses the scissors-congruence of triangles which is the subject of Hilbert’s Problem III which we already mentioned in Footnote 15. The fact that Ambartzumian’s method does not work in dimension > 2 can be compared to the fact that the solution to Hilbert’s Problem III says that in dimension > 2, two polyhedra with the same volume are not necessarily scissors-equivalent. 45 Let

us recall that Beltrami’s problem is indeed local.

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Szabó, in a paper [115] containing several new ideas on the subject, gave two new proofs of Pogorelov’s theorem in a Finslerian setting, valid in all dimensions, one of them being elementary and based on ideas originating in Ambartzumian’s work. Szabó’s result is given in terms of a partial differential equation satisfied by the indicatrix of the Finsler structure, which in dimension n is assumed to be of class C nC2 . The proof uses Fourier transform theory, and precisely, a relation between the problem of characterizing (symmetric) projective metrics of low regularity and that of characterizing the (distributional) Fourier transform of norms on finite-dimensional vector spaces. For the relation between the Fourier transform and integral geometry on normed and projective Finsler spaces, see the paper [6] by Álvarez Paiva and Fernandes. A new point of view on Hilbert’s problem, based on an integral formula in dimension 2, is given in the paper [2] by Alexander, whose result can be considered as a general solution which does not even assume the continuity of the metric. In that paper, the author considers an indefinite metric (that is, a metric which does not necessarily separates points) on a 2-dimensional space homeomorphic to a Euclidean plane, equipped with a set of “lines” (in an abstract sense), which are homeomorphic to the real line. He shows that under very mild conditions (the metric is not assumed to be continuous and the set of lines are only assumed to satisfy Desargues theorem) there exists a unique Borel measure on the set of lines such that the distance between two points is given by integrating this measure. The methods are based on the author’s combinatorial interpretation of the Crofton formula from integral geometry. This work includes Busemann’s integral construction in the abstract setting of the axioms of geometry which was one of Hilbert’s favorite settings. The relation of Hilbert’s Problem IV with the foundations of geometry is highlighted again in the paper [3] by Alexander in which this author extends his previous result to higher dimensions, using the notion of zonoids from convexity theory.46 Alexander works in the settings of hypermetrics, that is, metrics d such that for any points n P1 ; : : : ; P Pm in R and for any integers N1 ; : : : ; Nm satisfying N1 C C Nm D 1, one has i <j d.Pi ; Pj /Ni Nj 0 (see [85]). This is a strong version of the triangle inequality (and for m D 3, the property boils down to the triangle inequality). Alexander notes in [3] that in dimension 2, all solutions of Hilbert’s problem are hypermetrics, and the main result of this paper is that there is a natural linear isomorphism between the cone of hypermetrics on Rn that are additive along lines (that is, hypermetrics that satisfy the requirement of Hilbert’s Problem IV) and the cone of positive Borel measures m on the set of hypersurfaces of Rn satisfying m.P / D 0 and 0 < m.ŒP; Q/ D 0 < 1 for any distinct P and Q in Rn . Here, as before, for X Rn , X denotes the set of hyperplanes that intersect X . The isomorphism is given by a generalized Crofton-type formula d.P; Q/ D m.ŒP; Q/. Alexander’s 46 One first defines a zonotope as a finite Minkowski sum of line segments. Equivalently, a zonotope is the affine projection of a high-dimensional cube. This can be seen as a generalization of the notion of polytope. A zonoid is then a limit of zonotopes in the sense of Blaschke–Hausdorff convergence. Equivalently, a zonoid is the limit of sums of segments. We refer the reader to the survey [29] by E. D. Bolker.

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methods using zonoids were further generalized by Schneider (see [112] and [113]), with a result valid for a higher-dimensional analogue of Hilbert’s problem involving area, volume, etc. and not only length. In the last section of this chapter, we recall Busemann’s formulation of this higherdimensional analogue. In the paper [15], Ambartzumian and Oganian give (Theorem 1) a necessary and sufficient condition for a Finsler metric in dimension 2 to be a solution of Hilbert’s problem. In the same paper, they study a parametric version of the problem, where the convex domain which is the support of the metric (which is assumed to be centrally symmetric) depends continuously on parameters. They also consider the question of when the family of Finsler metrics obtained is a Minkowski metric. They obtain a complete solution in the case of a dependence on a single (1-dimensional) parameter. The smoothness condition is essential in their work. Among the other recent works on the Finsler smooth case, we mention the works of Z. Shen and of R. L. Bryant. We recall that Funk, in his papers [63] and [64] gave a classification of Finsler metrics on convex domains in the plane which have constant flag curvature. In his paper [111], Shen described a class of smooth projective Finsler metrics of constant flag curvature using algebraic equations. In the paper [65], Funk proved that the standard metric of the sphere S 2 is the only flat projective Finsler metric, satisfying some additional conditions. Bryant, in [31] and [34] generalized the result of Funk; he described a 2-parameter family of projective Finsler metrics on S 2 and he showed that this family gives all such metrics. In his work, Bryant made use of the symplectic geometry of the space of geodesics (the quotient of the unit tangent bundle by the geodesic flow). Finally, we mention that Álvarez Paiva gave in [8] another point of view on Hilbert’s Problem IV in the framework of symplectic and contact topology. He obtained a characterization of Finsler metrics which are solutions of Hilbert’s problem in terms of a class of symplectic forms on the space of lines in Rn . At the same time, the result of Álvarez Paiva makes a relation between Hilbert’s problem and the “black and white” theory of Guillemin and Sternberg [70]. The results of Álvarez Paiva in his paper [8] were first announced in the paper [9] by himself, Gelfand and Smirnov, The paper [56] by M. Crampin contains reformulations and new versions of the works of Hamel and of Álvarez Paiva.

7 Further developments and perspectives A problem like Hilbert’s Problem IV never dies, and as we tried to show, it admits several generalizations and formulations in various settings. We mention that there are classical works on problems that are more general than Hilbert’s problem, namely, problems where the metrics on subsets of the plane are to be found, with prescribed geodesics which are curves that are not necessarily straight lines, and which satisfy

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certain conditions, e.g. they foliate the plane and each of them is homeomorphic to a real line that separates the plane into two components. Blaschke and Bol consider such problems in [26]. Before that, Darboux had already considered such a general problem, in the setting of the calculus of variations, see [58] and he obtained local existence results using theorems on partial differential equations. Busemann and Salzmann also considered such general problems in [51]. The case of nonsymmetric metrics is still wide open although, talking about this case, Busemann mentions in [44] (p. 36): “most contributors to Hilbert’s Problem IV have treated this general case”. In fact, Hilbert, when he formulated his problem, certainly had this case in mind, since he knew about the interesting examples of non-symmetric metrics discovered by Minkowski, and he was aware of the fact that extremal metrics obtained by the methods of the calculus of variation may lead to non-symmetric length functions. We already mentioned that Hamel, who proposed in his thesis [74] the first proof of Hilbert’s problem for dimensions 2 and 3, considered the case of nonsymmetric metrics. Szabó, in his paper [115], considered only the symmetric case and in the final section of that paper he outlined a sequel to that work which would treat the non-symmetric case; the sequel never appeared. Alexander, in the paper [2] that we mentioned above discusses the nonsymmetric case but without being able to settle it. Busemann also asked for the generalization of Hilbert’s problem (see [41] and [42]) to the study of k-dimensional area in n-dimensional affine spaces for which the affine k-flats minimize area. We quote Busemann, from the introduction to [42]: In his Problem IV, Hilbert [Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 1900, 253–297] suggested studying all geometries in which the straight lines are the shortest connections or minimize length. Although the problem in this form proved too general to be fruitful, it has given rise to many investigations dealing either with special geometries or with special questions. The higher-dimensional analogue to Problem IV concerns a-dimensional metrics in subsets of the n-dimensional projective space in which the a-flats minimize area. Such geometries apparently have never been treated, at least in the large. The present paper represents a first step in this direction.

In the same paper, Busemann asks for the study of k-dimensional areas that are given by a Crofton-type formula, similar to the one he gave for distances. We already mentioned advances on this problem, cf. the papers [6] byÁlvarez Paiva and Fernandes, in which the authors prove the Crofton formulas for smooth projective metrics, and [112] by Schneider, in which the author establishes again the Crofton-type formula for the Holmes–Thompson area for Finsler projective hypermetrics. Besides the two questions in Hilbert’s Problem, viz. the characterization of the metrics with the required properties and the study of such metrics individually, there is a third question, that lies in between, which is the question of finding general properties of such metrics. Busemann considered this question in various papers and books, and we mention as an example the following embedding result (part of it is obtained in [40], p. 81, and it was completed in [44], p. 32):

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Theorem 7.1. For any n-dimensional Desarguesian G-space R there exists an .n C 1/-dimensional Desarguesian space R such that R is a hyperplane in R and the restriction of R to R is the given metric on R. Another general result on Desarguesian spaces, also due to Busemann, is the following characterization of Hilbert and Minkowski geometries among all Desarguesian spaces ([44], p. 38): Theorem 7.2. Let .; d / be a non-compact Desarguesian space whose distance function d is non-necessarily symmetric. Assume that any isometry between two lines in is a projectivity. Then .; d / is of one of the following two sorts: • is a convex subset of projective space and d is its Hilbert metric; • is affine space and d is a Minkowski metric. This is a special case of a theorem that Berwald obtained under an additional differentiability hypothesis and which we mentioned at the end of Section 4. The following result of Busemann gives a characterization of Funk and Minkowski geometries among Desarguesian spaces [45]: Theorem 7.3. Any Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space. We also mention a result by Busemann which is more in the axiomatic spirit of Hilbert. Busemann develops in [40], §23 and 24, a theory of parallelism between geodesics in metric spaces which generalizes the Euclidean theory. He introduces in particular a parallel axiom (see [40], p. 141) and he proves the following (Theorem 24.1 of [40]): Theorem 7.4. A Desarguesian space in which the parallel axiom holds and where the spheres are convex is Minkowskian. Several of Busemann’s students and collaborators worked on general properties of Desarguesian spaces. We mention in particular the following result of E. M. Zaustinsky (one of Busemann’s students) [125]. Theorem 7.5. A Desarguesian metric defined on affine space and whose right and left open balls are all compact is necessarily a Minkowski metric. B. B. Phadke [102] (who was also a student of Busemann), proved the following. Theorem 7.6. A Desarguesian space in which the equidistant loci to hyperplanes are hyperplanes is necessarily a Minkowski space. We mention the following problems:

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Problem 7.7. Modify Hilbert’s fourth problem so that geometric circles replace lines, in spaces of constant curvature and on more general metric spaces. Problem 7.8. Generalize Hilbert’s Problem IV to the setting of metric spaces or convex subsets of metric spaces other than Euclidean spaces; that is, study the metrics on subspaces such that the geodesics of the ambient metric are geodesics for the new metric. (We recall that Busemann developed a theory of convexity in general metric spaces, see [40] and the exposition in [101].) Problem 7.9. Study in particular the preceding problem on Teichmüller space equipped with its various metrics. Problem 7.10. Study a complex analogue of Hilbert’s problem. Problem 7.11. In the paper [12], the authors used the great variety of explicit constructions of projective metrics to show that totally geodesic submanifolds (in that case, planes) are not minimal surfaces for the Hausdorff measure. A nice question mentioned in that paper is to see whether the following holds: Suppose that for a given projective metric, planes (or hyperplanes) are minimal for the Hausdorff measure; is the metric then either flat or Riemannian? Problem 7.12. The set of projective pseudo-metrics (pseudo distance functions) on Rn is a closed convex cone. Determine the extreme rays of this cone, for n 3. Problem 7.13. (This is an open version of Beltrami’s theorem): Consider a Riemannian metric defined on the real projective space such that all projective planes are minimal surfaces. Is the metric of constant curvature? Let us make a few comments on these problems. Problem 7.7 was solved by Álvarez Paiva and Berck in [13], for Finsler metrics and in the case of circles on the two-sphere. The 2-dimensional smooth version was solved by Segre [110]. Regarding Problem 7.8, in the case where the metric is Riemannian or semiRiemannian, this problem was extensively studied in the literature, see for example [91]. See also Chapter 12 of [35] for a list of open problems related to this topic. We also mention the work in [97] and the survey in Chapter 13 of this Handbook [98], which are related to a version of Hilbert’s Problem IV in the setting of constant curvature geometries. We study there two metrics (namely a Funk-type metric and a Hilbert-type metric) on open subsets of the sphere and of hyperbolic space such that the geodesics for the restriction of the ambient metric are geodesics for our two metrics. We can also mention the so-called Apollonian weak metric, which is a metric defined on open subsets of Rn . In the case where the convex set is the open unit disc or

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the upper half-plane (considered as the ambient spaces of the Poincaré models of hyperbolic geometry), the non-Euclidean geodesics are also geodesics of the Apollonian metric; see [96], and the recent paper [123]. The reason for the mention of Teichmüller space in Problem 7.9 is the amount of interest and of profound work that was done around it in the last six or seven decades. We note that there is a so-called Weil–Petersson Funk metric metric on Teichmüller space, which was first introduced by Yamada in [122], defined in analogy with the Funk metric on convex subsets of Rn (see Chapter 12 [92] in this Handbook and see also the survey [121]). There are several ways to formulate precisely Problem 7.10, and we propose the use of complex geodesics (holomorphic maps from the unit disc to a complex manifold). The Kobayashi metric could play the role played by the Euclidean metric in the real case. Problem 7.12 is due to Álvarez Paiva. It is solved for n D 2 in [14]. Problem 7.13 is due to Bryant, and it was communicated to me by Álvarez Paiva who also noted that there are examples of such metrics with non-constant curvature in R3 , but there are no known examples defined on the whole projective space. Let us also mention that Álvarez Paiva studied the problem of extending Hilbert’s problem to rank one symmetric spaces. In a recent preprint, [10], he showed that if M is a manifold diffeomorphic to a compact rank-one symmetric space equipped with a reversible (i.e. symmetric) Finsler metric F all of whose geodesics are closed and of the same length and if L is a Finsler metric on M whose geodesics coincide as unparametrized geodesics with those of F , then L is the sum of a reversible Finsler metric and an exact 1-form. Finally, Álvarez Paiva and Berck obtained recently a complete characterization of projective Finsler (not necessarily symmetric) metrics. The preprint [11] contains the details of the result for metrics defined on the whole projective n-space.

Appendix I: Hilbert’s Problem IV (The translation is from the article in the 1902 Bulletin of the AMS.) Problem 4: Problem of the straight line as the shortest distance between two points. Another problem relating to the foundations of geometry is this: If from among the axioms necessary to establish ordinary Euclidean geometry, we exclude the axiom of parallels, or assume it as not satisfied, but retain all the other axioms, we obtain, as is well known, the geometry of Lobachevsky (hyperbolic geometry). We may therefore say that this is a geometry standing next to Euclidean geometry. If we require further that that axiom be not satisfied whereby, of three points on a straight line, one and only one lies between the other two, we obtain Riemann’s (elliptic) geometry, so that this geometry appears to be the next after Lobachevsky’s. If we

Chapter 15. Hilbert’s fourth problem wish to carry out a similar investigation with respect to the axiom of Archimedes, we must look upon this as not satisfied, and we arrive thereby at the non-Archimedean geometries which have been investigated by Veronese and myself. The more general question now arises: Whether from other suggestive standpoints geometries may not be devised which, with equal right, stand next to Euclidean geometry. Here I should like to direct your attention to a theorem which has, indeed, been employed by many authors as a definition of a straight line, viz., that the straight line is the shortest distance between two points. The essential content of this statement reduces to the theorem of Euclid that in a triangle the sum of two sides is always greater than the third side – a theorem which, as is easily seen, deals solely with elementary concepts, i.e., with such as are derived directly from the axioms, and is therefore more accessible to logical investigation. Euclid proved this theorem, with the help of the theorem of the exterior angle, on the basis of the congruence theorems. Now it is readily shown that this theorem of Euclid cannot be proved solely on the basis of those congruence theorems which relate to the application of segments and angles, but that one of the theorems on the congruence of triangles is necessary. We are asking then, for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom. One finds that such a geometry really exists and is no other than that which Minkowski constructed in his book, Geometrie der Zahlen,47 and made the basis of his arithmetical investigations. Minkowski’s is therefore also a geometry standing next to the ordinary Euclidean geometry; it is essentially characterized by the following stipulations: 1. The points which are at equal distances from a fixed point O lie on a convex closed surface of the ordinary Euclidean space with O as a center. 2. Two segments are said to be equal when one can be carried to the other by a translation of the ordinary Euclidean space. In Minkowski’s geometry the axiom of parallels also holds. By studying the theorem of the straight line as the shortest distance between two points, I arrived48 at a geometry in which the parallel axiom does not hold, while all other axioms of Minkowski’s geometry are satisfied. The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e.g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seems to me desirable.

47 Leipzig, 48 Math.

1896. Annalen, vol. 46, p. 91.

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Appendix II: Extract from the French version Sur les problèmes futurs des mathématiques, 1902 The conclusion of the French version is different from the English one: In the plane, and assuming the continuity axiom, the problem in hand leads to a question treated by M. Darboux49 : To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines; this question seems to me worth of and interesting generalizations.50 [Dans le plan et en admettant l’axiome de la continuité, le problème dont il s’agit conduit à la question traitée par M. Darboux : Déterminer tous les problèmes du calcul des variations dans le plan où les solutions sont toutes des droites du plan ; questions qui me semble digne de généralisations fécondes et intéressantes.]

Acknowledgements. I would like to thank Juan Carlos Álvarez Paiva, Charalampos Charitos, Jeremy Gray, François Laudenbach, Darshana Prapajati, Vladimir Matveev, Marc Troyanov and Sumio Yamada for comments on a preliminary version of this chapter. This work was supported by the French ANR project FINSLER. The paper was completed during a stay at the Erwin Schrödinger International Institute for Mathematical Physics (Vienna).

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49 Leçons

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50 Compare

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Open problems

We present nineteen open problems on Hilbert geometry. Other open problems are contained in various chapters of this Handbook. In particular Chapter 8 and Chapter 15 contain several problems.

Problem 1 (B. Lemmens) Let (X, dX ) be a Hilbert geometry and let f : X → X be a nonexpansive map on X, that is, a map satisfying dX (f (x), f (y)) ≤ dX (x, y) for all x, y ∈ X. Beardon proved in [3] that if X is a strictly convex domain and f has no fixed point in X, then there exists η ∈ ∂X such that limk→∞ f k (x) = η for all x ∈ X, and the convergence is uniform on compact subsets of X. What about general Hilbert geometries? It was conjectured independently by Anders Karlsson and Roger Nussbaum, see [26], that if (X, dX ) is a general Hilbert geometry and f is a fixed point free nonexpansive map on X, then there exists a convex subset ⊂ ∂X such that for all x ∈ X, each limit point of the orbit Ox = {f k (x) : k = 0, 1, 2, . . .} lies inside . Besides strictly convex Hilbert geometries, the conjecture has only been confirmed for polyhedral domains, see the paper [22] by Brian Lins. Further details can be found in the articles by Karlsson and Lemmens and Nussbaum in this volume [15], [20].

Problem 2 (B. Lemmens) It is well known that the Hilbert geometry of an n-simplex is isometric to (Rn , · H ), where xH = (max xi ) ∨ 0 − (min xi ) ∧ 0 i

i

is a polyhedral norm. In general, if f : → is a nonexpansive map under a norm with polyhedral unit ball, and f has a fixed point, then every orbit of f converges to a periodic orbit of f , i.e., for each x there exists an integer p such that f kp (x) → ξx as k → ∞, where ξx is a periodic point of f with period p. Moreover, there exists an a priori upper bound for the maximum period of periodic points of f in terms of the number of facets of the unit ball of the norm. Thus, if f is a Hilbert metric nonexpansive map on the n-simplex and f has a fixed point, then every orbit converges to a periodic orbit of f , and there exists an a priori upper bound for the maximum period of its periodic points in terms of n. What is the optimal upper bound? For the 2-simplex, it was shown by Lemmens in [19] that 6 is the optimal upper bound. The Rn

Rn

434

Open problems

current best upper bound is:

m max 2 , k k k

where m = n(n + 1)/2. It is believed, however, that there exists c > 2 such that the periods of the periodic points do not exceed cn , but no proof is known.

Problem 3 (B. Lemmens) The following conjecture was posed by Bletz-Siebert and Foertsch in [9]: It is impossible to isometrically embed a Euclidean plane into any Hilbert geometry (X, dX ), or, as they say the Euclidean rank of any Hilbert geometry is 1. They proved the conjecture for all strictly convex domains X, for all domains whose boundary is C 1 , and for all 3-dimensional domains. The conjecture is also known to be true for polyhedral domains X, as it is impossible to have a nonexpansive map f : S → S, with S ⊆ X, which has a fixed point and whose orbits do not converge to periodic orbits. (For further details see the contribution of Lemmens and Nussbaum in this volume [20].) A variant of this problem was posed by Cormac Walsh. Walsh defined the Minkowski rank of a Hilbert geometry (X, dX ) as the largest dimension of a finitedimensional normed space that can be isometrically embedded into (X, dX ). As the Hilbert metric on the n-simplex is isometric to an n-dimensional normed space, its Minkowski rank is n. It also easy to show that the Minkowski rank of any strictly convex Hilbert geometry is 1. Walsh conjectured that in general the Minkowski rank of a Hilbert geometry (X, dX ) is the maximum dimension of a affine subspace H such that H ∩ X is a simplex.

Problem 4 (B. Lemmens) In a recent paper [35], Cormac Walsh showed that the collineation group of each finite-dimensional Hilbert geometry is a subgroup of index at most 2 in the group of isometries. Moreover, it has index 2 if and only if the domain is not the projection of a symmetric cone that is not a Lorentz cone. Is there an extension to infinite-dimensional Hilbert geometries of these results? Some results in this direction have been obtained by Lemmens, Roelands and Wortel in [21].

Problem 5 (A. Papadopoulos) The Hilbert metric is a generalization of Klein’s model of hyperbolic space, where the underlying set is the interior of an n-dimensional ellipsoid and where the distance between two distinct points x and y is the logarithm of the cross ratio of the quadruple

Open problems

435

consisting of x and y and the two intersection points of the Euclidean line that joins x and y with the boundary of the ellipsoid, taken in the natural order. A fact which is not so well known is that Klein also obtained models for spherical and Euclidean geometries using the cross ratio and taking instead of the ellipsoid other kinds of conic sections. (In the case of Euclidean geometry, the conic is degenerate.) The distance again is defined as a multiple of a (complex) logarithm of a cross ratio. The methods and the formulae that define the three geometries of constant curvature are formally very similar, and the constructions of the three geometries are hereby done in a unified way in the realm of projective geometry. This is the main theme of Klein’s two papers On the so-called non-Euclidean geometry, I and II ([17], [18]). An exposition together with a mathematical commentary of these two papers are contained in [1]. We propose the following problems: (1) Develop generalizations of the two other geometries that are parallel to the generalization of hyperbolic geometry to Hilbert geometry. (2) Are there non-symmetric versions of such metrics similar to the Funk metric? (3) Under some regularity conditions to be specified, are such metrics Finslerian with constant flag curvature? In this case the possible geometries would be limited by the results stated in Appendix A of Chapter 3 of this volume [30] (in particular Theorem A3).

Problem 6 (A. Papadopoulos) Study families of Hilbert and of Funk metrics parametrized by convex sets.

Problem 7 (I. Kim) In a recent paper [16], Kim and Zhang showed that the space of convex real projective structures admits a natural L2 Kähler metric, where the Weil–Petersson metric on Teichmüller space is totally geodesic. Is this Kähler metric non-positively curved?

Problem 8 (I. Kim) In the preprint [13], Foulon and Kim exhibited an example of a sequence of Hilbert metrics on a surface where the entropy of the geodesic flow goes to zero with respect to the Sinai–Ruelle–Bowen measure, yet remains bounded below with respect to the Bowen–Margulis measure. Is the entropy with respect to the Sinai–Ruelle–Bowen measure zero if and only if the Hilbert area of the surface is infinite?

436

Open problems

Problem 9 (M. Troyanov) Cheng and Yau proved the existence of a natural Riemannian metric on a sharp convex domain of Rn that is invariant under the group of projective transformation. This metric is obtained by solving the Monge–Ampère equation. It is related to the theory of affine spheres and is sometimes called the affine metric on the convex domain, see Theorem 3.2 in Chapter 11 of this Handbook or the book [25] by Nomizu and Sasaki. Benoist and Hulin recently proved that the affine metric is bi-Lipschitz equivalent to the Hilbert metric. A general construction of a Riemannian metric from a Finsler one is given by the Binet–Legendre metric, see [23]. In the case of the Hilbert metric, this metric is also known to be bi-Lipschitz equivalent to the Hilbert metric (see [24]). We ask the following: (1) What are the relations between the Binet–Legendre and the affine metric in a sharp convex domain? (2) Study these metrics from a Riemannian viewpoint: their curvature, etc. These metrics are obtained by rather indirect constructions, and computing some explicit examples should be a useful (and non trivial) first step. Similar questions could be asked about the metric constructed by Vinberg as the Hessian of the characteristic function of the cone over a convex domain, see Section 4.3 in Chapter 5 of this Handbook.

Problem 10 (S. Yamada) In a sense, modern differential geometry started with the Gauss equation; namely, given a codimension-1 submanifold in Rn , the canonical affine connection that is both torsion-free and metric compatible is singled out from the canonical Euclidean connection and the nearest point projection operators on the tangent spaces of the submanifold. Although there have been some studies on connections and parallel transport in the framework of Finsler geometry, little has been investigated in terms of the theory of submanifolds. Hence, we propose to study the issue of canonical connection(s) defined on a submanifold within an ambient space where some Hilbert/Funk geometry is defined. In particular, the case of the finite-dimensional simplex as the ambient space is already interesting, where the Hilbert metric is known to be isometric to a normed space, hence the space is equipped with a Berwald connection. On the other hand, infinite-dimensional simplices naturally appear as ambient spaces for Teichmüller spaces as submanifolds [27], [36] and in that framework, the Teichmüller metric, the Thurston asymmetric metrics and the length spectrum metrics are manifested.

437

Open problems

Problem 11 (L. Marquis) In [5], Benoist exhibits an example of a strongly irreducible subgroup of SL(4, R) and a properly convex open set of RP4 preserved by such that / is compact, but the convex set is not strictly convex nor with C 1 -boundary. Benoist suggests in the same paper that this construction can be easily generalized to dimensions 4, 5, 6, 7. Is it possible to find such a pair (, ) in any dimension? We recall that Benzécri shows in [6] that one cannot find such a pair in dimension 2.

Problem 12 (L. Marquis) Kapovich in [14] and Benoist in [4] gave examples of subgroups of SL(4, R) and of strictly convex open sets of RP4 preserved by such that / is compact but equipped with its Hilbert metric is not quasi-isometric to the real hyperbolic space H4 of dimension 4. Can one find infinitely many pairs (, ) with / compact and strictly convex up to quasi-isometry? In such a pair, the group has to be Gromov-hyperbolic. A nice invariant of quasiisometry of a Gromov-hyperbolic group is the conformal dimension of the boundary ∂ of . Can one compute the conformal dimension of ∂ in Kapovich’s or Benoist’s examples using Hilbert geometry?

Problem 13 (C. Vernicos) Using the Raleigh quotient, and regardless of any definition of a Laplacian, it is possible to define the bottom of the spectrum, λ1 , of a Hilbert geometry (see [11], [10], [31]). I proved in [31] that for any n-dimensional Hilbert geometry one has λ1 ≤

(n − 1)2 . 4

(∗)

Colbois conjectured that the equality should characterize hyperbolic geometry. In the case of a C 1 divisible convex set, the conjecture is true. To see this, we can use a classical method in Riemannian geometry which relates λ1 to the volume entropy h. The method consists in using the so-called test functions in Rayleigh quotients. For instance, for any h > h, one can consider the functions

fr (t) = e−h t − e−h r for t ≤ r and 0 for t > r. Then, for a given Hilbert geometry and ω ∈ , we compute the limit of the Rayleigh quotients of the functions fr d (ω, ·) as r → +∞.

438

Open problems

From that we can deduce the inequality λ1 ≤ (h )2 /4 which finally gives λ1 ≤

h2 . 4

In the case of the Hilbert geometry of a C 1 divisible convex set, Crampon [12] proved that h ≤ n−1 with equality if and only if the Hilbert geometry is the hyperbolic one. Hence the equality in (∗) implies the equality h = n − 1 and by Crampon’s theorem we easily conclude.

Problem 14 (C. Vernicos) A definition of amenability is possible in the realm of Hilbert geometry and was introduced in [31]. One possible definition is that a Hilbert geometry is amenable if and only if the bottom of its spectrum λ1 is equal to zero. We would like to have a necessary and sufficient condition in terms of properties of the convex set for its Hilbert geometry to be amenable. For instance, one can consider the action of the group PGL(n, R) of projective transformations on the set of open convex sets in Rn endowed with the Hausdorff distance. The Hilbert geometries arising from elements in the orbit of a given convex body under this action are all isometric. Again in [31], it was proved that if the closure of that orbit with respect to the Hausdorff distance contains an open convex polytope, then the initial Hilbert geometry is amenable. We conjecture that the Hilbert geometry of a convex set is amenable if and only if the closure of its orbit PGL(n, R) · contains an open convex polytope.

Problem 15 (C. Vernicos) I proved in [33] that there exists a universal constant an such that the volume of any ball of radius r in any n-dimensional Hilbert geometry is ≥ an r n . We ask: Is this constant an equal to the volume of the unit ball of the n-dimensional simplex? Another way of looking at this problem is the following: One can associate to any pointed open convex set (, ω), a function V (ω, ·) : R+ → R+ which maps r ∈ R+ to the Holmes–Thompson (or Busemann) volume of the ball of radius r centred at ω, with respect to the Hilbert metric of . Let us call this function the volume function. Is it true that the volume function is bounded from below by the volume function of the n-simplexes Sn and from above by the volume function of the hyperbolic space Hn ? That is, do we have, for all r ≥ 0, VSn (p, r) ≤ V (ω, r) ≤ VHn (q, r) ?

Open problems

439

As the group of projective transformations fixing a simplex (resp. an ellipsoid) acts transitively, the volume function of the simplex (resp. of hyperbolic geometry) does not depend on the centre p (resp. q) of the balls. Such an equality would prove the upper entropy bound conjecture which can be stated as follows: log V (ω, r) ≤ n − 1. h() := lim inf r→+∞ r

Problem 16 (C. Vernicos) The volume function can be seen as a function from the set of pointed open convex sets into the set of real functions (see Problem 15). Is this function 1-to-1?

Problem 17 (C. Vernicos) The pseudo-Gaussian curvature of a convex set was defined by G. Berck, A. Bernig and C. Vernicos in [7] as a function on the boundary ∂ of a convex body as follows. Fix a Euclidean metric in Rn . Following A. D. Alexandrov [2] at almost every point p of ∂ there is a well-defined normal. Such points are called smooth. For such a point p ∈ ∂, let n(p) be the outward normal of ∂ at p. For each unit vector e ∈ Tp ∂, let He (p) be the affine plane containing p and directed by the vectors e and n(p). We then define Re as the radius of the biggest disc containing p inside ¯ e := ∩ He (p). The pseudo-Gaussian curvature k(p) of ∂ at p is the minimum of the numbers n−1 Rei (p)−1 , i=1

where e1 , . . . , en−1 range over all orthonormal bases of Tp ∂. The fact that the square of this function is in L1 for any plane convex body is a key ingredient in the proof of the upper entropy bound conjecture in the plane. We ask: Is the square root of the pseudo-Gaussian curvature always in L1 ? A positive answer would imply, by Theorem 3.1 in [7], the entropy upper bound conjecture and an asymptotic of the ratio of the volume function by the function r → sinh(r)n−1 , introduced as the centro-projective area in [7]. We also ask: Is there a Bishop volume comparison type theorem in Hilbert geometry, i.e., for any and ω ∈ , is there a ρ0 > 0 such that for any R > r > ρ0 , VHn (o, R) V (ω, R) ≤ ? V (ω, r) VHn (o, r)

440

Open problems

In other words, is it true that the function V (ω, r)/VHn (o, r) is a non-increasing function for r > ρ0 . Note that it suffices to prove this statement for a dense subset of the set of convex sets, and that this would also yield the entropy upper bound conjecture.

Problem 18 (C. Vernicos) The approximability of a convex body, a concept introduced by Schneider and Wieacker [28], is a constant which somehow measures how well a convex set can be approximated by polytopes. More precisely, let N(ε, ) be the smallest number of vertices of a polytope whose Hausdorff distance to the convex body is less than ε > 0. Then the approximability of is a() := lim inf ε→0

log N(ε, ) . − log ε

We proved in [34] that in dimensions 2 and 3 the volume entropy and twice the approximability coincide, and that 2a() ≤ h() in all dimension. Does the equality between the entropy and twice the approximability always occur? Again, this would prove the entropy upper bound conjecture. What metric spaces can be embedded, in general, into a Hilbert geometry and, in particular, in a Hilbert geometry of the same dimension? For example, we believe that it is not possible to embed the three-dimensional Heisenberg group in a three-dimensional Hilbert geometry.

Problem 19 (C. Vernicos) Consider in a Hilbert geometry (with C 2 boundary to start with) a point p, and for any R > 0 a polytope PR with minimal vertices which contains the ball of radius R − 1 and is contained in the ball of radius R + 1, both centred at p. Compute the difference (resp. the ratio) between the Busemann volume of the ball B(p, r) and the Busemann volume of PR . Study the behaviour as R → +∞. Does this yield a functional related to the centro-projective area? (Note that this function has to be a projective invariant because of the projective invariance of Hilbert geometries.)

Open problems

441

References [1]

N. A’Campo and A. Papadopoulos, On Klein’s So-called Non-Euclidean geometry. In Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics, (L. Ji and A. Papadopoulos, eds.), Eurpoean Mathematical Society, Zürich, to appear.

[2]

A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 (1939), 3–35.

[3]

A. F. Beardon, The dynamics of contractions. Ergodic Theory Dynam. Systems 17 (1997), no. 6, 1257–1266.

[4]

Y. Benoist, Convexes hyperboliques et quasiisométries. Geom. Dedicata 122 (2006), 109–134.

[5]

Y. Benoist, Convexes divisibles. IV. Structure du bord en dimension 3. Invent. Math. 164 (2006), no. 2, 249–278.

[6]

J.-P. Benzécri, Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88 (1960), 229–332.

[7]

G. Berck, A. Bernig and C. Vernicos, Volume entropy of Hilbert geometries. Pacific J. Math. 245 (2010), no. 2, 201–225.

[8]

A. Bernig, Hilbert geometry of polytopes. Archiv Math. 92 (2009), 314–324.

[9]

O. Bletz-Siebert and T. Foertsch, The Euclidean rank of Hilbert geometries. Pacific J. Math. 231 (2007), no. 2, 257–278.

[10] B. Colbois and C. Vernicos, Les géométries de Hilbert sont à géométrie locale bornée. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 4, 1359–1375. [11] B. Colbois and C. Vernicos, Bas du spectre et delta hyperbolicité en géométrie de Hilbert plane Bull. Soc. Math. France 134 (2006), no. 3, 357–381. [12] M. Crampon, Entropies of compact strictly convex projective manifolds. J. Mod. Dyn. 3 (2009), no. 4, 511–547. [13] P. Foulon and I. Kim, Entropy and deformation of real projective structures on surface. In preparation. [14] M. Kapovich, Convex projective structures on Gromov–Thurston manifolds. Geom. Topol. 11 (2007), 1777–1830. [15] A. Karlsson, Dynamics of Hilbert nonexpansive maps. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 263–273. [16] I. Kim and G. Zhang, Kähler metric on Hitchin component. Preprint, arXiv:1312.1965. [17] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie (erster Aufsatz). Math. Ann. IV (1871), 573–625; English translation by J. Stillwell, [29], 69–111. [18] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie (zweiter Aufsatz). Math. Ann. VI (1873), 112–145. [19] B. Lemmens, Nonexpansive mappings on Hilbert’s metric spaces. Topol. Methods Nonlinear Anal. 38 (2011), no. 1, 45–58.

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Open problems

[20] B. Lemmens and R. Nussbaum, Birkhoff’s version of Hilbert’s metric and its applications in analysisIn Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 275–303. [21] B. Lemmens, M. Roelands, and M. Wortel, Isometries of infinite dimensional Hilbert geometries. Preprint, arXiv:1405.4147. [22] B. Lins, A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on a polyhedral domain. Math. Proc. Cambridge. Philos. Soc., 143 ( 2007) no. 1, 157–164. [23] V. Matveev and M. Troyanov, The Binet–Legendre metric in Finsler geometry. Geom. Topol. 16 (2012), 2135–2170. [24] V. Matveev and M. Troyanov, Completeness and incompleteness of the Binet–Legendre metric. Preprint, arXiv:1408.6401. [25] K. Nomizu and T. Sasaki, Affine differential geometry. Cambridge Tracts in Math. 111, Cambridge University Press, Cambridge 1994. [26] R. D. Nussbaum, Fixed point theorems and Denjoy–Wolff theorems for Hilbert’s projective metric in infinite dimensions. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 199–250. [27] K. Ohshika, H. Miyachi and S.Yamada, Weil–Petersson Funk metric on Teichmüller space. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 339–352. [28] R. Schneider and J. A. Wieacker, Approximation of convex bodies by polytopes. Bull. London Math. Soc. 13 (1981) 149–156. [29] J. Stillwell, Sources of hyperbolic geometry. History of Mathematics Series 10, Amer. Math. Soc., Providence, RI; London Math. Soc., London 1996. [30] M. Troyanov, Funk and Hilbert geometries from the Finslerian viewpoint. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 69–110. [31] C. Vernicos, Spectral Radius and amenability in Hilbert geometries. Houston J. Math. 35 (2009), no. 4, 1143–1169. [32] C. Vernicos, Lipschitz caracterisation of polytopal Hilbert geometries to appear in Osaka J. Math., arXiv :0812.1032v1. [33] C. Vernicos, Asymptotic volume in Hilbert geometries. Indiana Univ. Math. J. 62 (2013), no. 5, 1431–1441. [34] C. Vernicos, Approximability of convex bodies and volume entropy of Hilbert geometries. Preprint, arXiv:1207.1342. [35] C. Walsh, Gauge-reversing maps on cones, and Hilbert and Thompson isometries. Preprint, arXiv:1312.7871. [36] S. Yamada, Local and global aspects of Weil–Petersson geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zürich 2014, 43–112.

List of Contributors

Mickaël Crampon, Universidad de Santiago de Chile, Av. El Belloto 3580, Estación Central, Santiago, Chile email: [email protected] Ren Guo, Department of Mathematics, Oregon State University, Corvallis, OR 973314605, U.S.A. email: [email protected] Anders Karlsson, Section de mathématiques, Université de Genève, 2-4 Rue du Lièvre, Case Postale 64, 211 Genève 4, Switzerland email: [email protected] Inkang Kim, School of Mathematics, KIAS, Heogiro 85, Dongdaemun-gu Seoul, 130-722, Republic of Korea email: [email protected] Bas Lemmens, School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, UK email: [email protected] Ludovic Marquis, Institut de Recherche Mathématique de Rennes, Rue Sine 190, 31062 Toulouse Cedex 4, France email: [email protected] Hideki Miyachi, Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan email: [email protected] Roger Nussbaum, Department of Mathematics, Rutgers, The State University Of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, U.S.A. email: [email protected] Ken’ichi Ohshika, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, 560-0043, Osaka, Japan email: [email protected] Athanase Papadopoulos, Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France email: [email protected] Marc Troyanov, Section de mathématiques, École Polytechnique Fédérale de Lausanne, SMA–Station 8, 1015 Lausanne, Switzerland email: [email protected]

444

List of Contributors

Constantin Vernicos, Institut de mathématique et de modélisation de Montpellier, Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34395 Montpellier Cedex, France email: [email protected] Cormac Walsh, INRIA Saclay & Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France e-mail: [email protected] Sumio Yamada, Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima, Tokyo 171-8588, Japan email: [email protected]

Index absolutely continuous measure, 193 area centro-projective, 439 affine chart, 209 affine metric, 316, 318 affine midpoint, 21 affine normal, 316 affine patch, 58, 310 affine ratio, 37, 61 affine sphere, 316 affinely convex, 52 Alexandrov’s theorem, 202 algebraic horosphere, 216 almost-geodesic, 130 ˛-conformal density, 332 amenable Hilbert metric, 438 Anosov decomposition, 189 Anosov flow, 189, 327 approximability convex body, 440 arithmetic symmetrization metric, 34 asymptotic convex hypersurface, 231 asymptotically harmonic, 195 attractor, 298 automorphism group, 166 axioms continuity, 385 Hilbert, 385 incidence, 385 order, 385

forward, 43 left, 43 right, 43 Beltrami Theorem, 99 Beltrami, Eugenio, 393 Beltrami–Cayley–Klein model, 59,308, 310, 394 Benzécri, Jean-Paul, 308, 312 Berwald metric, 74 Berwald, Ludwig, 405 ˇ-convexity, 186 bi-, 213 Binet–Legendre metric, 436 biproximal, 213 Birkhoff theorem, 282 Birkhoff version of Hilbert’s metric, 276 Blaschke connection, 316 bottom of the spectrum, 437 boundary horofunction, 130 Bowen–Margulis measure, 194, 332 bulging deformation, 324 Busemann G-space, 27 Busemann cocycle, 330 Busemann function, 127, 178, 215, 266, 330 Busemann point, 131 Busemann volume, 210, 311, 438 Busemann zero curvature, 28 Busemann, Herbert, 392, 401, 408, 409 Busemann–Hausdorff volume, 166

backward complete Finsler manifold, 73 backward complete metric, 47 backward open ball, 43 backward proper metric, 47 backward sphere, 43 ball backward, 43

Cayley, Arthur, 393 centered John ellipsoid, 20 centro-projective area, 439 character variety, 321 characteristic function, 143 closed orbit, 170 collineation, 115, 140

446 cometric, 172 compactification horofunction, 130 complete Finsler manifold, 73 cone, 277, 317 above , 212 ice-cream, 143 Lorentz, 143 normal, 277 symmetric, 143 total, 277 cone linear mapping, 283 conformal density, 332 conformal dimension, 437 conical face, 113 conical flag, 114 connection, 173 convex, 209 with C 1 boundary, 211 affinely, 52 ˇ-convex, 246 class C 1C" , 246 cone, 222 divisible, 208 geodesically, 52 Menger, 411 projective structure, 167, 198 properly, 163, 209 quasi-divisible, 208 round, 211 strictly, 211 strongly, 246 convex body, 209 approximability, 440 convex cone, 222, 317 decomposable, 222 indecomposable, 222 sharp, 217 convex domain proper, 35 sharp, 60 convex hull, 218 convex hypersurface, 231

Index

asymptotic, 231 convex polytope, 42 convex projective structure, 308 deformation space, 321 Goldman parameters, 321 marked, 313 convex real projective manifold, 313 convex set divisible, 313 hyperbolic space, 358 projective automorphism group, 310 projective space, 310 properly, 310 sphere, 358 symmetric, 223 convex structure dual, 326 convex structures deformation space, 313 moduli space, 313 C 1C regularity, 186 Crofton formula, 115 cross ratio Euclidean, 364 hyperbolic, 364 spherical, 365 current, 329 curvature of a Finsler metric, 175 of Hilbert geometry, 177 of the boundary, 202 pseudo-Gaussian, 439 curvature zero (Busemann), 28 decomposable convex cone, 222 Dehn invariant, 404 Dehn, Max, 404 Denjoy–Wolff theorem, 298 density function, 154 Desargues’ property, 27 Desarguesian space, 27, 411 detour cost, 131 detour metric, 132

Index

developing map, 198 divisible, 167, 226, 308 convex set, 313 divisible convex, 208 division ratio, 37, 61 dominate, 277 dual, 217 dual convex structure, 326 dual wedge, 277 dyadic number, 21 earthquake deformation, 324 Ehresmann, Charles, 308, 311 eigenvalues, 293 ellipsoid, 220 centered John, 20 John, 20 elliptic, 214 energy of a curve, 84 entropy measure-theoretical, 193 topological, 193, 198, 328 volume, 197, 331 epigraph topology, 134 equiaffine metric, 316 ergodic measure, 192 Euclidean Jordan algebra, 288 Euler lemma, 25 Euler–Lagrange equations, 85 exposed face, 49 exposed point, 49 extreme set, 135 face, 113, 214, 278 dimension, 214 open, 214 support, 214 face (of a convex set), 48 Fenchel–Nielsen parameters, 320 Finsler cometric, 172 Finsler metric, 163, 310 for Hilbert geometry, 165 regular, 164 Ricci curvature, 96

447

Finsler structure, 72, 151 Euclidean tautological, 362 hyperbolic tautological, 362 spherical tautological, 363 tautological, 34 Finsler, Paul, 404 fixed point, 296 fixed point theorem, 264 flag curvature, 96 Hilbert metric, 100 flip map, 198 flow Anosov, 327 topologically mixing, 328 foot, 53 formal Christoffel symbols, 86 forward bounded metric, 47 forward boundedly compact metric, 47 forward Cauchy sequence, 47 forward open ball, 43 forward proper metric, 47 forward sphere, 43 functional Minkowski, 19 fundamental tensor (of a Finsler metric), 83 Funk metric, 36, 76, 128, 280, 340 hyperbolic, 359 relative, 58 reverse, 41 spherical, 360 Funk, Paul, 404 gauge, 136 gauge-preserving map, 144 gauge-reversing map, 142 Genocchi, Angelo, 394 geodesic, 85 geodesic current, 329

448

Index

geodesic flow, 327 for regular Finsler metrics, 164 for Hilbert geometries, 169 geodesic metric space, 210, 328 Gromov-hyperbolic, 237 geodesically convex, 51, 52 geodesics, 164 in Hilbert geometry, 166 geometry Hilbert, 148 non-Archimedean, 405 non-Arguesian, 405 non-Euclidean, 405 non-Legendrian, 406 non-Pascalian, 405 semi-Euclidean, 406 Goldman parameters, 321 Gromov product, 269, 292 Gromov-hyperbolic, 237 group, 168 space, 168 G-space, 27, 410

Hilbert, David, 383 Hitchin component, 308, 325 Hoüel, Guillaume Jules, 395 holonomy map, 198 homogeneous Hilbert geometry, 166 homogeneous tangent bundle, 165 homography, 115 horoball, 129, 132, 178, 266 horofunction, 127, 130, 266 horofunction boundary, 130 horofunction compactification, 130 horosphere, 178, 215 algebraic, 216 hyperbolic, 214 isometry, 168 orbit, 186 space, 165 hyperbolic metric, 307 hyperbolic metric space, 328 hypermetric, 417 hyperplane supporting, 211

Hölder cocycle, 331 Hölder regularity, 186 Hamel potential, 93 Hamel, Georg, 404 harmonic measure, 195 harmonic symmetrization, 81 Hessian, 164, 202 Hilbert 1-form, 171 Hilbert form, 92 Hilbert measure, 155 Hilbert metric, 58, 128, 147, 148, 163, 209, 276, 310 amenable, 438 Birkhoff version, 278 Finsler norm, 310 flag curvature, 100 hyperbolic, 366 spherical, 367 Hilbert Problem IV, 422 Hilbert volume, 311

ice-cream cone, 143 ideal triangle, 155 idempotents, 293 complete system of orthogonal, 293 primitive, 293 indecomposable, 222 indecomposable convex cone, 222 indicatrix, 17, 73 invariant measure, 192 isometry group, 166 Jacobi field, 175 Hilbert geometry, 177 Jacobi operator, 175 Hilbert geometry, 177 John ellipsoid, 20, 218, 247 Klein model, 59, 81, 394 Krein–Milman theorem, 288 Laguerre, Edmond, 384

Index

metric left open ball, 43 affine, 316, 318 Legendre transform, 172 arithmetic symmetrization, 34 on HM , 172 backward complete, 47 Legendre–Clebsch condition, 83 backward proper, 47 Legendre–Fenchel transform, 137 Binet–Legendre, 436 Lemma Blaschke, 316 Euler, 25 detour, 132 Schur, 97 equiaffine, 316 Selberg, 239 forward bounded, 47 Zassenhaus–Kazhdan–Margulis, 251 forward boundedly compact, 47 length forward proper, 47 translation, 330 Funk, 36, 128, 340 length spectrum, 326 hyperbolic, 359 length spectrum metric, 61 spherical, 360 limit set, 296 geodesically convex, 51 Liouville measure, 171, 193 Gromov hyperbolic, 328 locally finite fundamental domain, 233 Hilbert, 128, 147, 209 locally Ptolemaic, 150 hyperbolic, 366 Lorentz cone, 143, 287 spherical, 367 Lyapunov decomposition, 188 hyperbolic, 307 Lyapunov exponents, 188 length spectrum, 61 of a periodic orbit, 188 max-symmetrization, 61 marked convex projective structure, 242, projective, 36, 70, 249 313 projectively flat, 13 marked length spectrum, 326 relative Funk, 58 marked projective structure, 242 reverse Funk, 41, 61 convex, 242, 313 reverse-Funk, 129 equivalent, 242 Sasaki, 176 marking, 313 Thurston, 33, 349 max-symmetrization variational formulation of metric, 61 Teichmüller, 348 Mazur–Ulam Theorem, 20 weak, 12, 33 measure weak Minkowski, 13 Bowen–Margulis, 332 Weil–Petersson, 341 Hilbert, 155 completion, 342 Patterson–Sullivan, 331 Weil–Petersson Funk, 60, 344 Sinai–Ruelle–Bowen, 435 metric space measure of maximal entropy, 194 geodesic, 210 measure-theoretical entropy, 193 proper, 210, 266 Menelaus, 372 uniquely geodesic, 210 Menelaus Theorem, 373 midpoint property, 21 Menger convexity, 411 minimal displacement, 265

449

450

Index

Minkowski functional, 19, 38 Funk metric Euclidean, 362 hyperbolic, 362 spherical, 363 Minkowski norm strictly convex, 25 strongly convex, 25 Minkowski rank, 434 mixing measure, 192 topologically, 190 natural, 249 nearest point, 53 non-Archimedian field, 404, 405 non-Arguesian geometry, 405 non-Euclidean geometry, 405 non-Legendrian geometry, 406 non-Pascalian field, 406 non-Pascalian geometry, 405, 406 nonexpansive map, 265, 433 orbit, 170 ordered geometry, 385 orthogonal representation, 226 osculating Riemannian metric, 94 Painlevé–Kuratowski topology, 132 pair of pants, 314 decomposition geodesic, 315 topological, 315 parabolic, 214 parallel transport, 174 parts, 278 parts of boundary, 132 Pasch, Moritz, 385 patch affine, 310 Patterson–Sullivan geodesic current, 333 Patterson–Sullivan measure, 331 period, 296 cocycle, 331

periodic point, 296 perpendicular, 55 Perron–Frobenius operator, 284 Perron–Frobenius theory, 264 perspectivity, 148 physical measure, 200 planar automorphism, 219 Pogorelov, Alexei Vasil’evich, 399, 409, 413 polytope, 113 positive hyperbolic transformation, 322 positively biproximal, 213 positively proximal, 213 positively semi-proximal, 213 power, 213 primitive element, 170 Problem Hilbert fourth, 422 product, 222 projective metric, 249 natural, 249 volume, 249 projective center, 152 projective diameter, 282 projective line, 309 projective manifold, 242, 311 convex, 313 strictly, 313 isomorphism, 311 projective metric, 36, 70, 249 natural, 249 projective structure, 311 convex, 308, 312 flat, 242 strictly convex, 308 projective volume, 249 natural, 249 projective weak metric, 13 projectively equivalent, 242 projectively flat metric, 13

Index

projectively flat Finsler manifold, 88 proper convex, 310 proper convex domain, 35 proper metric space, 210 properly convex, 163, 209 indecomposable, 222 product, 222 properly convex set, 310 proximal, 213, 225, 233 action, 233 bi, 213 group, 233 positively, 213 semi, 213 pseudo-Gaussian curvature, 439 Ptolemaic, 150 Ptolemaic inequality, 150 quadratic representation, 294 quasi-divisible, 226, 208 quasi-hyperbolic, 214 rank Hilbert geometry, 434 Randers metric, 73 Rayleigh quotient, 437 reductive, 234 regular orbit, 188 regular Finsler metric, 164 relative Funk metric, 58 representation equal mod 2, 226 highest restricted weight, 225 highest weight, 225 lattice of restricted weights, 225 orthogonal, 226 proximal, 225 restricted weight, 225 symplectic, 226 weight, 225 reverse Funk metric, 41, 61, 129 reversible norm, 310 reversible tautological

451

Finsler structure, 34 Ricci curvature of a Finsler metric, 96 right open ball, 43 round, 211 Ruelle’s inequality, 198, 201 Sasaki metric, 176 Schur Lemma, 97 Schwarzian derivative, 106 Selberg lemma, 239 semi-proximal, 213 semi-simple k-semi-simple, 213 S1 -semi-simple, 213 semicontraction, 265 sharp convex cone, 217 sharp convex domain, 60 simplicial cone, 289 Sinai–Ruelle–Bowen measures, 200 space G-, 410 Busemann G-, 27 Desarguesian, 27, 411 spectral radius, 213 spectrum, 293 sphere backward, 43 forward, 43 spherical metric projective model, 88 spray, 87 stable set, 179 standard positive cone, 287 -map, 143 strictly convex, 211 strictly convex Minkowski norm, 25 strictly convex projective structure, 308 strictly convex set, 49 strongly convex, 246 strongly convex Finsler metric, 83 strongly convex Minkowski norm, 25 subhomogeneous map, 282 support hyperplane, 38, 211

452 support of a face, 214 supporting functional, 38 symmetric cone, 143 symmetric convex set, 223 symmetric Hilbert geometry, 166 symplectic representation, 226 tautological Finsler structure, 34, 56 tautological Finsler structure Euclidean, 362 hyperbolic, 362 spherical, 363 Teichmüller component, 308 Teichmüller space, 341 tensor, 83 Theorem Beltrami, 99 Birkhoff, 282 Ceva, 63 Denjoy–Wolff, 298 Desargues, 405 Funk–Berwald, 70 Krein–Milman, 49, 288 Mazur–Ulam, 20 Menelaus, 61, 373 Pascal, 405 sector figure, 372 Thurston metric, 33, 349 topological entropy, 193, 198, 328 topologically mixing, 190, 328 transitive, 190 topology epigraph, 134 Painlevé–Kuratowski, 132 transitive flow, 190 translation distance, 168 translation length, 214, 265,

Index

330 translation number, 265 triangle ı-thin, 328 unipotent, 213 uniquely geodesic metric space, 210 unstable set, 179 upper asymptotic volume, 116 variation norm, 289 variational formulation of Teichmüller metric, 348 variational principle, 194 Veronese, Guiseppe, 405 vertex, 113 Vinberg’s -map, 143 volume, 165 Busemann, 210, 438 Busemann–Hausdorff, 166 projective, 249 volume entropy, 197, 331 von Staudt, Karl Georg Christian, 383 weak Finsler structure, 72 weak metric, 12, 33 geodesic, 51 projective, 13 weak Minkowski metric, 13 wedge, 277 Weil–Petersson Funk metric, 60, 344 Weil–Petersson metric, 341 Zariski-dense, 168 Zermelo transform, 75 zero curvature Busemann, 28 zonoid, 417 zonotope, 417

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 6 Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature 7 Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) 8 AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) 9 Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) 10 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)

Handbook of Hilbert Geometry Athanase Papadopoulos Marc Troyanov Editors

Editors: Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 Rue René Descartes 67084 Strasbourg Cedex France

Marc Troyanov Section de mathématiques École Polytechnique Fédérale de Lausanne SMA-Station 8 1015 Lausanne Switzerland

2010 Mathematics Subject Classification: 01A55, 01-99, 35Q53, 37D25, 37D20, 37D40, 47H09, 51-00, 51-02, 51-03, 51A05, 51B20, 51F99, 51K05, 51K10, 51K99, 51M10, 52A07, 52A20, 52A99, 53A20, 53A35, 53B40, 53C22, 53C24, 53C60, 53C70, 53B40, 54H20, 57S25, 58-00, 58-02, 58-03, 58B20, 58D05, 58F07. Key words: Hilbert metric, Funk metric, non-symmetric metric, Finsler geometry, Minkowski space, Minkowski functional, convexity, Cayley-Klein-Beltrami model, projective manifold, projective volume, Busemann curvature, Busemann volume, horofunction, geodesic flow, Teichmüller space, Hilbert fourth problem, entropy, geodesic, Perron-Frobenius theory, geometric structure, holonomy homomorphism

ISBN 978-3-03719-147-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2014 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Foreword

The idea of collecting the surveys that constitute this Handbook came out of a desire to present in a single volume the foundations as well as the modern developments of Hilbert geometry. In the last two decades the subject has grown into a very active field of research. The Handbook will allow the student to learn this theory, to understand the questions and problems that it leads to, and to acquire the tools that can be used to approach them. It should also be useful to the confirmed researcher and to the specialist, for it contains an exposition and an update of the most recent developments. Thus, some chapters contain classical material, highlighting works of Beltrami, Klein, Hilbert, Berwald, Funk, Busemann, Benzécri and the other founders of the theory, and other chapters present recent developments. Hilbert geometry can be regarded from different points of view: the calculus of variations, Finsler geometry, projective geometry, dynamical systems, etc. At several places in this volume, the fruitful relations between Hilbert geometry and other subjects in mathematics are reported on. These subjects include Teichmüller spaces, convexity theory, Perron–Frobenius theory, representation theory, partial differential equations, coarse geometry, ergodic theory, algebraic groups, Coxeter groups, geometric group theory, Lie groups, and discrete group actions. All these important topics appear in one way or another in this book. We would like to take this opportunity to thank Gérard Besson who helped us at an early stage of this project. Our warm thanks go to Manfred Karbe from the European Mathematical Society who encouraged the project and to Irene Zimmermann for a very efficient collaboration and for the seriousness of her work. This work was supported in part by the French research program ANR FINSLER and by the Swiss National Science Foundation. Strasbourg and Lausanne, September 2014

Athanase Papadopoulos Marc Troyanov

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I. Minkowski, Hilbert and Funk geometries Chapter 1. Weak Minkowski spaces by Athanase Papadopoulos and Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2. From Funk to Hilbert geometry by Athanase Papadopoulos and Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter 3. Funk and Hilbert geometries from the Finslerian viewpoint by Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 4. On the Hilbert geometry of convex polytopes by Constantin Vernicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry by Cormac Walsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Chapter 6. Characterizations of hyperbolic geometry among Hilbert geometries by Ren Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Part II. Groups and dynamics in Hilbert geometry Chapter 7. The geodesic flow of Finsler and Hilbert geometries by Mickaël Crampon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Chapter 8. Around groups in Hilbert geometry by Ludovic Marquis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

viii

Contents

Chapter 9. Dynamics of Hilbert nonexpansive maps by Anders Karlsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Chapter 10. Birkhoff’s version of Hilbert’s metric and and its applications in analysis by Bas Lemmens and Roger Nussbaum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Part III. Developments and applications Chapter 11. Convex real projective structures and Hilbert metrics by Inkang Kim and Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Chapter 12. Weil–Petersson Funk metric on Teichmüller space by Hideki Miyachi, Ken’ichi Ohshika and Sumio Yamada . . . . . . . . . . . . . . . . . . . . . 339 Chapter 13. Funk and Hilbert geometries in spaces of constant curvature by Athanase Papadopoulos and Sumio Yamada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Part IV. History of the subject Chapter 14. On the origin of Hilbert geometry by Marc Troyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Chapter 15. Hilbert’s fourth problem by Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Introduction

The project of editing this Handbook arose from the observation that Hilbert geometry is today a very active field of research, and that no comprehensive reference exists for it, except for results which are spread in various papers and a few classical (and very inspiring) pages in books of Busemann. We hope that this Handbook will serve as an introduction and a reference for both beginners and experts in the field. Hilbert geometry is a natural geometry defined in an arbitrary convex subset of real affine space. The notion of convex set is certainly one of the most basic notions in mathematics, and convexity is a rich theory, offering a large supply of refined concepts and deep results. Besides being interesting in themselves, convex sets are ubiquitous; they are used in a number of areas of pure and applied mathematics, such as number theory, mathematical analysis, geometry, dynamical systems and optimization. In 1894 Hilbert discovered how to associate a length to each segment in a convex set by way of an elementary geometric construction and using the cross ratio. In fact, Hilbert defined a canonical metric in the relative interior of an arbitrary convex set. Hilbert geometry is the geometric study of this canonical metric. The special case where the convex set is a ball, or more generally an ellipsoid, gives the Beltrami–Klein model of hyperbolic geometry. In this sense, Hilbert geometry is a generalization of hyperbolic geometry. Hilbert geometry gives new insights into classical questions from convexity theory, and it also provides a rich class of examples of geometries that can be studied from the point of view of metric geometry or differential geometry (in particular Finsler geometry). Let us recall Hilbert’s construction. The line joining two distinct points x and y in a bounded convex domain intersects the boundary of that domain in two other points p and q. Assuming that y lies between x and p, the Hilbert distance from x to y is the logarithm of the cross ratio of these four points: p y x

d.x; y/ D

1 jx log 2 jy

pj jy pj jx

qj : qj

q

This distance was defined in a letter to Klein written by Hilbert in 1894. It is a distance in the usual sense, and the relative interior of the convex set is a complete metric space for this distance. The Hilbert metric is invariant under projective transformations and depends in a monotonic way on the domain: a larger domain induces a smaller Hilbert distance. The Hilbert metric is projective in the sense that the straight

2

Introduction

lines are geodesics. In other words, d.x; y/ D d.x; z/ C d.z; y/ whenever z 2 Œx; y. Furthermore, if the convex domain is strictly convex, then the affine segment is the unique geodesic joining two points. The fourth Hilbert problem asks for a description of all projective metrics in a convex region, that is, metrics for which the straight lines are geodesics. At the beginning of the twentieth century, Hamel, who was a student of Hilbert, worked on this problem; he discovered new examples of projective metrics and found some deep results using differential calculus and the calculus of variations. The subject of Finsler geometry gradually emerged as an independent topic, and at the end of the 1920s, Funk and Berwald gave a differential geometric characterization of Hilbert metrics among all Finsler metrics on a domain with a smooth and strongly convex boundary. All these facts and several others which we describe below are reported on in this Handbook. The deepest and most thorough studies in Hilbert geometry during the twentieth century are due to Busemann and his students and collaborators. During the period from the 1940s to the 1990s this school investigated Hilbert geometry from the viewpoint of metric geometry. A variety of questions regarding these metrics were studied, concerning their geodesics, their convexity theory (convexity of the distance function, of the spheres, etc.), their curvature, area, asymptotic geometry, limit cycles (horocycles) and limit spheres (horospheres), and several other features. For instance, these authors gave several characterizations of the ellipsoid in terms of its Hilbert geometry. They noticed (like Hilbert did before them) that the special case of the simplex is particularly interesting and they studied it in detail. They worked on the metrical properties of the Hilbert metric as well as on the axiomatic theory, making relations with the axioms and the basic notions of Euclidean and of non-Euclidean geometries, in particular the theory of parallels. They established relations between Hilbert geometry and other fields, including the foundations of mathematics, the calculus of variations, convex geometry, Minkowski geometry, geometric group theory and projective geometry. They also developed the basics of the closely related Funk metric. Busemann formulated and initiated the study of several problems of which he (and his collaborators) gave only partial solutions, and a large amount of the research on Hilbert geometry that was done after him is directly or indirectly inspired by his work. Let us briefly mention two further important directions in which the subject developed in the last century. In the late 1950s, Birkhoff found a new proof of the classical Perron–Frobenius theorem on eigenvectors of non-negative matrices based on the Hilbert metric in the positive cone. This new proof brought a new point of view on the subject and initiated a rich generalization of Perron–Frobenius theory. During the same period, Benzécri initiated the theory of divisible convex domains, that is, convex domains admitting a discrete cocompact group of projective transformations. The quotient manifold or orbifold naturally carries a Finsler structure whose universal cover is a Hilbert geometry. The reader will find more information on twentieth

3

Introduction

century developments in Chapters 3, 10 and 15 of this Handbook. During the last fifteen years, the subject grew rapidly and a number of these recent developments are discussed in the other chapters. We now describe the content of the book. The various chapters are written by different authors and are meant to be read independently from each other. Each chapter has its own flavor, due to the variety of tastes and viewpoints of the authors. Although we tried to merge the chapters into a coherent whole, we did not unify the different notation systems, nor did we try to avoid repetitions, hopefully to the benefit of the reader. The book is divided into four parts. Part I contains surveys on Minkowski, Funk and Hilbert geometries and on the relations between them. In Chapter 1, A. Papadopoulos and M. Troyanov treat weak Minkowski spaces. A weak metric on a set is a non-negative distance function ı that satisfies the triangle inequality, but is allowed to be non-symmetric (we may have d.x; y/ 6D d.y; x/) or degenerate (we may have d.x; y/ D 0 for some x 6D y). A weak Minkowski metric on a real vector space is a weak metric that is translation-invariant and projective. The authors define the fundamental concept of weak Minkowski space and they give several examples and counterexamples. The basic results of the theory are stated and proved. Minkowski geometry shares several properties with Hilbert geometry, one of them being that the Euclidean geodesics are geodesics for that geometry. An important observation is that the infinitesimal – or tangential – geometry of a Hilbert or a Funk geometry is of Minkowski type. One of the main results of this chapter is the following: A continuous weak metric ı on Rn is a weak Minkowski metric if and only if it satisfies the midpoint property, that is, ı.p; q/ D 2ı.p; m/ D 2ı.m; q/ for any points p, q where m is the affine midpoint. Other characterizations of weak Minkowski distances are given, providing various important aspects of this geometry. The relations with Busemann’s G-spaces and Desarguesian spaces and comparisons with the Funk and the Hilbert metrics are also highlighted. Chapter 2, From Funk to Hilbert geometry, by the same authors, is devoted to the study of the distance in a convex domain introduced by P. Funk in 1929. Using the notation of the figure on page 1, the Funk distance is defined by Â

Ã

jx pj : F .x; y/ D log jy pj Observe that this distance is a non-symmetric version of the Hilbert distance. Many properties of the Hilbert distance can be obtained as consequences of similar properties of the Funk distance. In this chapter, metric balls, the topology, convexity and orthogonality properties in Funk geometry are studied. A full description of Funk geodesics is also given. In the case of smooth curves, the property can be described as follows: A smooth curve .t / in a convex domain is geodesic for the Funk metric if and only if there is a face D in @ such that the velocity vector P .t / points toward D for all t .

4

Introduction

Chapter 3, by M. Troyanov, concerns the Funk and the Hilbert metrics from the point of view of Finsler geometry. This approach dates back to works done at the end of the 1920s, by Funk and by Berwald, who gave a characterization of Hilbert geometry from the Finslerian viewpoint. Funk and Berwald proved the following theorem: A smooth Finsler metric defined on a convex bounded domain of Rn is the Hilbert metric of that domain if and only if this geometry is complete (in an appropriate sense), if its geodesics are straight lines and if its flag curvature is equal to 1. The author explains these notions in detail and he gives a complete proof of this result. At the same time, the chapter constitutes an introduction to the Finsler nature of the Funk and Hilbert metrics, where the Funk and the Hilbert Finsler structures appear respectively as the tautological and the symmetric tautological Finsler structures on . Chapter 4, On the Hilbert geometry of convex polytopes, by C. Vernicos, concerns the Hilbert geometry of an open set Rn which is a polytope. A bounded convex domain is a polytope if and only if its Hilbert metric is bi-Lipschitz equivalent to a Euclidean space. An equivalent condition is that the domain is isometrically embeddable in a finite-dimensional normed vector space. Another characterization states that the volume growth is polynomial of order equal to the dimension of the convex domain. The author discusses several other aspects of the Hilbert geometry of polytopes. The main goal of Chapter 5, by C. Walsh, is to give an explicit description of the horofunction boundary of a Hilbert geometry. This notion is based on ideas that go back to Busemann but which were formally introduced by Gromov. The results in this chapter are mainly due to Walsh. Walsh gives a sketch of how this boundary may be used to study the isometry group of these geometries. The main result in the chapter is that the group of isometries of a bounded convex polyhedron which is not a simplex coincides with the group of projective transformations leaving the given domain invariant. Let us note that the horofunction boundaries of several other spaces have been described during the last few years (mostly by Walsh), in particular, Minkowski spaces and Teichmüller spaces equipped with the Thurston and with the Teichmüller metrics. These descriptions have also been applied to the characterization of the isometry groups of the corresponding metrics spaces. Chapter 6, by R. Guo, gives a number of characterizations of hyperbolic geometry (or, equivalently, the Hilbert geometry of an ellipsoid) among Hilbert geometries. All these geometric characterizations are formulated in simple geometric terms. Part II concerns the dynamical aspects of Hilbert geometry. It consists of four chapters: Chapters 7 to 10. Chapter 7, by M. Crampon, concerns the geodesic flow of a Hilbert geometry. The study is based on a comparison of this flow with the geodesic flow of a negatively curved Finsler or Riemannian manifold. The main interest is in Hilbert geometries that have some hyperbolicity properties. Such geometries correspond to convex sets with C 1 boundary. In this case, stable and unstable manifolds exist, and there is a relation between the asymptotic behaviour of these manifolds along an orbit of the flow and the shape of the boundary at the endpoint of that orbit. The author then

Introduction

5

studies the particular case of the geodesic flow associated to a compact quotient of a strictly convex Hilbert geometry, and he shows that such a flow satisfies an Anosov property. This property implies some regularity properties at the boundary of the convex set. The author then describes the ergodic properties of the geodesic flow, and he also surveys several notions of entropy that are associated to Hilbert geometry. In Chapter 8, L. Marquis studies Hilbert geometry in the general setting of projective geometry. More precisely, the author surveys the various groups of projective transformations that appear in Hilbert geometry. In the first part of the chapter, he describes the projective automorphism group of a convex set in terms of matrices and then from a dynamical point of view. He shows the existence of convex sets with large groups of symmetries. He exhibits relations with several areas in mathematics, in particular with the theory of spherical representations of semi-simple Lie groups and with Schottky groups. He then explains how Hilbert geometry involves geometric group theory in various contexts. For instance, the Gromov hyperbolicity of a socalled divisible Hilbert metric (that is, one that admits a compact quotient action by a discrete group of isometries) is equivalent to a smoothness property of the boundary of the convex set. Note that the fact that the convex set is divisible means in some sense that it has a large group of symmetries. The other aspects of Hilbert geometry in which group theory is involved include differential geometry, convex affine geometry, real algebraic group theory, hyperbolic geometry, the theories of moduli spaces, of symmetric spaces, of Hadamard manifolds, and the theory of geometric structures on manifolds. In Chapter 9, A. Karlsson considers the dynamical aspect of the theory of nonexpansive (or Lipschitz) maps in Hilbert geometry. He makes relations with works of Birkhoff and Samelson done in the 1950s on Perron–Frobenius theory, and with a more recent work by Nussbaum and Karlsson–Noskov. He explains in particular how the theory of Busemann functions, horofunctions and horospheres in Hilbert geometries appear in the study of nonexpansive maps of these spaces and in their asymptotic theory. This is another example of the fact that a theory which is quite developed in the setting of spaces of negative curvature can be generalized and used in an efficient way in Hilbert geometry, which is not negatively curved (except in the case where the convex set is an ellipsoid). In Chapter 10, B. Lemmens and R. Nussbaum give a thorough survey of the development of the ideas of Birkhoff and Samelson on the applications of Hilbert geometry to the contraction mapping principle and to the analysis of non-linear mappings on cones and in particular to the so-called non-linear Perron–Frobenius theory. The setting is infinite-dimensional. The authors also show how this theory leads to the result that the Hilbert metric of an n-simplex is isometric to a Minkowski space whose unit ball is a polytope having n.n C 1/ facets. Part III contains extensions and generalizations of Hilbert and Funk geometries to various contexts. It consists of three chapters: Chapters 11 to 13. Chapter 11, written by I. Kim and A. Papadopoulos, concerns the projective geometry setting of Hilbert geometry. A Hilbert metric is defined on any convex subset

6

Introduction

of projective space and descends to a metric on convex projective manifolds, which are quotients of convex sets by discrete groups of projective transformations. Thus, it is natural in this Handbook to have a chapter on convex projective manifolds and to study the relations between the Hilbert metric and the other properties of such manifolds. There are several parametrizations of the space of convex projective structures on surfaces; a classical one is due to Goldman and is an analogue of the Fenchel– Nielsen parametrization of hyperbolic structures. A more recent parametrization was developed by Labourie and Loftin in terms of hyperbolic structures on a Riemann surface together with cubic differentials, making use of the Cheng–Yau classification of complete hyperbolic affine spheres. Chapter 11 contains an introduction to these parametrizations and to some related matters. Teichmüller spaces appear naturally in this setting as they are important subspaces of the deformation spaces of convex projective structures of surfaces. The authors also discuss higher-dimensional analogues, covering in particular the work of Johnson–Millson and of Benoist and Kapovich on deformations of convex projective structures on higher-dimensional manifolds. Relations with geodesic currents and topological entropy are also treated. The Hilbert metrics which appear via their length spectra in the parametrization of the deformation spaces of convex projective structures also show up in the compactifications of these spaces. Some of the subjects mentioned are only touched on; this chapter is intended to open up new perspectives. In Chapter 12, S. Yamada, K. Ohshika and H. Miyachi report on a new weak metric which Yamada defined recently on Teichmüller space and which he calls the Weil–Petersson Funk metric. This metric shares several properties of the classical Funk metric. It is defined using a similar variational formula, involving projections on hyperplanes in an ambient space that play the role of support hyperplanes, namely, they are the codimension-one strata of the Weil–Petersson completion of the space. The ambient Euclidean space of the Funk metric is replaced here by a complex which Yamada introduced in a previous work, and which he calls the Teichmüller–Coxeter complex. It is interesting that the Euclidean setting of the classical Funk metric can be adapted to a much more complex situation. In Chapter 13, A. Papadopoulos and S. Yamada survey analogues of the Funk and Hilbert geometries on convex sets in hyperbolic and in spherical geometries. These theories are developed in a way parallel with the classical Funk theory. The existence of a Funk geometry in these non-linear spaces is based on some non-Euclidean trigonometric formulae, and the fact that the analogies can be carried over is somehow surprising because the study of the classical Funk metric on convex subsets of Euclidean space involves a lot of similarity properties and the use of parallels, which do not exist in the non-Euclidean settings. The geodesics of the non-Euclidean Funk and Hilbert metrics are studied, and variational definitions are given of these metrics. These metrics are shown to be Finsler. The Hilbert metric in each of the constant curvature convex sets is also a symmetrization of the Funk metric. The Hilbert metric of a convex subset in a space of constant curvature can also be defined using a notion

Introduction

7

of a cross ratio which is adapted to that space. A relation is made with a generalized form of Hilbert’s Problem IV. Part IV consists of a two chapters which have a historical character. Chapter 14, written by M. Troyanov, contains a brief description of the historical origin of Hilbert geometry. The author presents a summary with comments of a letter of Hilbert to Klein in which Hilbert announces the discovery of that metric. Chapter 15, written by A. Papadopoulos, is a report on Hilbert’s Fourth Problem, one of the famous twentythree problems that Hilbert presented at the second ICM held in Paris in 1900. The problem asks for a characterization and a study of metrics on subsets of projective space for which the projective lines are geodesics. The Hilbert metric is one of the finest examples of metrics that satisfy the requirements of this problem. The author also reports on the relations between this problem and works done before Hilbert on metrics satisfying this requirements, in particular, by Darboux in the setting of the calculus of variations and by Beltrami in the setting of differential geometry. Making historical comments and giving information on the origin of a problem bring into perspective the motivations behind the ideas. The comments that we include also make relations between the theory that is surveyed here and other mathematical subjects. We tried to pay tribute to the founders of the theory as we feel that historical comments usually make a theory more attractive. Several chapters contain questions, conjectures and open problems, and the book also contains a special section on open problems proposed by various authors.

Part I

Minkowski, Hilbert and Funk geometries

Chapter 1

Weak Minkowski spaces Athanase Papadopoulos and Marc Troyanov

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Weak metric spaces . . . . . . . . . . . . . . . . . . . . . . 3 Weak Minkowski norms . . . . . . . . . . . . . . . . . . . 4 The midpoint property . . . . . . . . . . . . . . . . . . . . 5 Strictly and strongly convex Minkowski norms . . . . . . . 6 The synthetic viewpoint . . . . . . . . . . . . . . . . . . . 7 Analogies between Minkowski, Funk and Hilbert geometries References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 12 14 21 25 26 29 30

1 Introduction In the last decade of the 19th century, Hermann Minkowski (1864–1909) initiated new geometric methods in number theory, which culminated with the celebrated Geometrie der Zahlen1 . Minkowski’s work is referred to several times by David Hilbert in his 1900 ICM lecture [15], in particular in the introduction Hilbert declares: The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski’s work, Die Geometrie der Zahlen.

Regarding the influence of this book on the birth of metric geometry, let us mention the following from the paper [10] by Busemann and Phadke, p. 181: 1 Minkowski’s first paper on the geometry of numbers was published in 1891. A first edition of his book Geometrie der Zahlen appeared in 1896, and a more complete edition was posthumously edited by Hilbert and Speiser and published in 1910 [23]. This edition was reprinted several times and translated, and it is a major piece of the mathematical literature of the early 20th century.

12

Athanase Papadopoulos and Marc Troyanov Busemann had read the beginning of Minkowski’s Geometrie der Zahlen in 1926 which convinced him of the importance of non-Riemannian metrics.

An early result of Minkowski in that theory, related to number theory, states that any convex domain in R2 which is symmetric around the origin and has area greater than four contains at least one non-zero point with integer coordinates. One step in Minkowski’s proof amounts to considering a metric on the plane for which the unit ball at any point in R2 (in fact in Z2 ) is a translate of the initial convex domain. Such a metric is not Euclidean, it is translation-invariant and the Euclidean lines are shortest paths. The geometric study of this type of metrics is called (since Hilbert’s writings) Minkowski geometry. We refer to [4], [12], [20], [21], [27] for general expositions of the subject. Minkowski formulated the basic principles of this geometry in his 1896 paper [22], and these principles are recalled in Hilbert’s lecture [15] (Problem IV). An express description of Minkowski geometry is the following: choose a convex set in Rn that contains the origin. For points p and q in Rn , define a number ı > 0 as follows. First dilate by the factor ı and then translate the set in such a way that 0 is sent to p and q lies on the boundary of the resulting set. In other words, ı is defined by the condition q 2 @.p C ı /: (1.1) We denote by ı.p; q/ the number defined in this way. The function ı W Rn Rn ! RC is what we call a weak metric. It satisfies the triangle inequality and ı.p; p/ D 0. It is not symmetric in general and it can be degenerate in the sense that ı.p; q/ D 0 does not imply p 6D q. On the other hand, the straight lines are geodesics for this metric and ı is translation-invariant. Minkowski geometry is the study of such weak metrics. It plays an important role in convexity theory and in Finsler geometry, where Minkowski spaces play the role played by flat spaces in Riemannian geometry. There is a vast literature on Minkowski metrics, and the goal of the present chapter is to provide the reader with some of the basic definitions and facts in the theory of weak Minkowski metrics, because of their relation to Hilbert geometry, and to give some examples. We give complete proofs of most of the stated results. We end this chapter with a discussion about the relations and analogies between Minkowski geometry and Funk and Hilbert geometries.

2 Weak metric spaces We begin with the definition of a weak metric space. Definition 2.1 (Weak metric). A weak metric on a set X is a map ı W X X ! Œ0; 1 satisfying the following two properties: a) ı.x; x/ D 0 for all x in X; b) ı.x; y/ C ı.y; z/ ı.x; z/ for all x, y and z in X .

Chapter 1. Weak Minkowski spaces

13

We often require that a weak metric satisfies some additional properties. In particular one says that the weak metric ı on X is c) separating if x ¤ y implies ı.x; y/ > 0, d) weakly separating if x ¤ y implies max fı.x; y/; ı.y; x/g > 0, e) finite if ı.x; y/ < 1, f) reversible (or symmetric) if ı.y; x/ D ı.x; y/, g) quasi-reversible if ı.y; x/ C ı.x; y/ for some constant C , for all x and y in X . One sometimes says that ı is strongly separating if condition b) holds, in order to stress the distinction with condition d). Observe that for reversible metrics both notions of separation coincide. A metric in the classical sense is a reversible, finite and separating weak metric. Thus, it satisfies 0 < ı.x; y/ D ı.y; x/ < 1 for all x ¤ y in X . Definition 2.2. Let U X be a convex subset of a real vector space X . A weak metric ı in U is said to be projective (or projectively flat) if satisfies the condition ı.x; y/ C ı.y; z/ D ı.x; z/

(2.1)

whenever the three points x, y and z in U are aligned and y 2 Œx; z, the affine segment from x to z (equivalently, if y D tx C .1 t /z for some 0 t 1). The weak metric is strictly projective if it is projective and ı.x; u/ C ı.u; z/ > ı.x; z/ whenever u 62 Œx; z. Definition 2.3 (Weak Minkowski metric). A weak Minkowski metric on a real vector space X is a weak metric ı on X that is translation-invariant and projective. Example 2.4. Let X be a real vector space and ' W X ! R a linear form. Define ı' .x; y/ D maxf0; '.y x/g. Then ı is a weak Minkowski metric. It is finite, but it is neither reversible nor weakly separating. We note that in functional analysis, given a real vector space X, the collection of sets B.';x;r/ D fy 2 X j ı' .x; y/ < rg X; where x 2 X is an arbitrary point, r > 0 and ' 2 X is an arbitrary linear form generates a topology which is called the weak topology on X . This observation is a possible justification for the name “weak metric” that we give to such functions. The terminology has its origin in the work of Ribeiro who was interested around 1943 [26]

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in some generalization of the Urysohn metrization theorem for the topology associated to ı. Example 2.5 (Counterexample). Let X be a real vector space and let k k W X ! R be a norm on X. Then ı.x; y/ D maxfky xk; 1g is a metric that is translation-invariant, but it is not a Minkowski metric because it is not projective. Indeed, suppose kzk D 1, then ı.0; 2z/ D 1 < ı.0; z/ C ı.z; 2z/ D 2: In this example, the metric is “projective for small distances”, in the sense that if kz xk 1 and y 2 Œx; z, then (2.1) holds. On the other hand, large closed balls are not compact; in fact any ball of radius 1 is equal to the whole space X . Example 2.6 (Counterexample). This is a variant of the previous example. Let again k k W X ! R be a norm on the real vector space X. Then ˛ .x; y/ D ky xk˛ is a metric if and only if 0 < ˛ 1. It is clearly translation-invariant, but it is not projective if ˛ < 1, and thus it is not a Minkowski metric. Unlike the previous metric ı, the metric ˛ is not projective for small distances (if ˛ < 1). On the other hand, every closed ball is compact.

3 Weak Minkowski norms Proposition 3.1. Let ı be a weak Minkowski metric on some real vector space X and set F .x/ D ı.0; x/. Then the function F W X ! Œ0; 1 satisfies the following properties. i) F .x1 C x2 / F .x1 / C F .x2 / for all x1 ; x2 2 X. ii) F .x/ D F .x/ for all x 2 X and for all 0. Proof. The first property is a consequence of the triangle inequality together with the fact that ı is translation-invariant: F .x C y/ D ı.0; x C y/ ı.0; x/ C ı.x; x C y/ D ı.0; x/ C ı.0; y/ D F .x/ C F .y/: To prove the second property, observe for any x 2 X and any ; 0 we have ı.0; x/ C ı.x; . C /x/ D ı.0; . C /x/; because x belongs to the segment Œ0; . C /x. Since we have ı.x; . C /x/ D ı.0; x/ D F .x/, the previous identity can be written as F .x/ C F .x/ D F .. C /x/

(3.1)

Chapter 1. Weak Minkowski spaces

15

and we conclude from the next lemma that F .x/ D F .x/ for all > 0. We also have F .0 x/ D 0 F .x/ D 0 since F .0/ D ı.0; 0/ D 0. Lemma 3.2. Let f W RC ! Œ0; 1 be a function such that f . C / D f ./ C f ./ for any ; 2 RC , then f ./ D f .1/ for every > 0. Proof. We first assume f .a/ < 1 for every a 2 RC . We have by hypothesis f .k a/ D f ...k 1/ C 1/ a/ D f ..k 1/ a/ C f .a/ for any k 2 N. We thus have by induction f .k a/ D k f .a/ for any k 2 N and any a 2 RC . Using the above identity with k; m 2 N, we have

k k Df m D f .k/ D k f .1/: mf m m Dividing both sides of the identity by m, we obtain f .˛/ D ˛f .1/ for any ˛ 2 QC . Consider now 2 RC arbitrary, and choose ˛1 ; ˛2 2 QC such that ˛1 < < ˛2 . Then f ./ D f .˛1 / C f . ˛1 / > f .˛1 / D ˛1 f .1/ and f ./ D f .˛2 / f .˛2 / < f .˛2 / D ˛2 f .1/: Since ˛2 ˛1 > 0 is arbitrarily small, we deduce that f ./ D f .1/ for any > 0. So far we assumed f .a/ < 1 for any a > 0. Assume now there exists a > 0 such that f .a/ D 1. Then f ./ D 1 for any > 0. Indeed choose an integer k such that k > a. Then kf ./ D f .k/ D f .k a/ C f .a/ f .a/ D 1: Therefore f ./ D f .1/ D 1. Definition 3.3. A function F W X ! Œ0; 1 defined on a real vector space X is a weak Minkowski norm if the following two conditions hold: i) F .x1 C x2 / F .x1 / C F .x2 / for all x1 ; x2 2 X; ii) F .x/ D F .x/ for all x 2 X and for all 0. Proposition 3.1 states that a weak Minkowski metric determines a weak Minkowski norm. Conversely, a weak Minkowski norm defines a weak Minkowski metric ıF by the formula (3.2) ıF .x; y/ D F .y x/: We then naturally define a weak Minkowski norm F to be

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• separating if x ¤ 0 implies F .x/ > 0; • weakly separating if x ¤ 0 implies maxfF .x/; F .x/g > 0; • finite if F .x/ < 1; • reversible (or symmetric) if F .x/ D F .x/; for any x 2 X. Example 3.4. The function F W RN ! Œ0; 1 defined by F ..xk // D sup maxfxk ; 0g k2N

is a weak Minkowski norm which is neither finite, nor separating, nor symmetric. It is however weakly separating. A weak Minkowski norm with similar properties also exists in finite dimensions: Example 3.5. The function F W R2 ! Œ0; 1 defined by F .x1 ; x2 / D maxfx1 ; 0g if x2 D 0 and F .x1 ; x2 / D 1 if x2 ¤ 0 is a weak Minkowski norm with the same properties: it is neither finite, nor separating, nor symmetric, but it is weakly separating. Observe that in both examples F is finite on some vector subspace of R2 . This is a general fact: Proposition 3.6. Let F W X ! Œ0; 1 be a weak Minkowski norm on the real vector space X and set DF D fx 2 X j F .x/ < 1g. Then DF is a vector subspace of X. Furthermore, the restriction of F to any finite-dimensional subspace E DF is continuous. Proof. If x; y 2 DF , then F .x/ and F .y/ are finite and therefore F .x C y/ F .x/ C F .y/ < 1 and F .x/ D F .x/ < 1. Therefore x C y 2 DF and x 2 DF , which proves the first assertion. To prove the second assertion, we consider a finite-dimensional subspace E DF and we choose a basis e1 ; e2 ; : : : ; em 2 E. Define the constant C D max .F .ej / C F .ej //: 1j m

For an arbitrary vector x D

Pm

j D1

F .x/

˛j ej 2 E, we then have

m X j D1

F .˛j ej / C

m X

j˛j j:

j D1

In particular, if x ! 0, then F .x/ ! 0. More generally, if a sequence x 2 E converges to some a 2 E, then lim sup F .x / D lim sup F .a C .x a// F .a/ C lim sup F ..x a// D F .a/: !1

!1

!1

Chapter 1. Weak Minkowski spaces

17

Since F .a/ F .x / C F .a x / we also have F .a/ lim inf .F .x / C F .a x // D lim inf F .x /: !1

!1

It follows that lim sup F .x / F .a/ lim inf F .x /; !1

!1

and the continuity on E follows. Corollary 3.7. Any weak Minkowski norm on a finite-dimensional vector space X is lower semi-continuous. Proof. We need to prove that F .a/ lim inf !1 F .x / for every sequence x 2 X converging to a. If F .a/ D 1, then a belongs to the open set X n DF . It follows that x 62 DF for large enough and therefore lim inf F .x / D 1 D F .a/: !1

If F .a/ < 1, then two cases may occur. If infinitely many x belong to DF , then by the previous proposition we have F .a/ D

lim

!1;x 2DF

F .x / D lim inf F .x /: !1

If on the other hand DF contains only finitely many elements of the sequence x , then lim inf F .x / D 1 > F .a/: !1

Definition 3.8. Given a weak Minkowski norm F on a vector space X, we define the open and closed unit balls at the origin as F D fx 2 X j F .x/ < 1g

and

x F D fx 2 X j F .x/ 1g:

The set F D fx 2 X j F .x/ D 1g is called the unit sphere or the indicatrix of F . Proposition 3.9. Let F be a weak Minkowski norm on a finite-dimensional vector space X . Then the following are equivalent: (1) F is finite; (2) F is continuous; (3) F is open; (4) 0 is an interior point of F . Proof. The implication (1) ) (2) is Proposition 3.6 and the implications (2) ) (3) ) (4) are obvious. To prove (4) ) (1), we suppose that F is not finite. Then there exists a 2 X such that F .a/ D 1. Thus, F .a/ D 1 for all > 0, in particular a 62 F for all > 0 and therefore 0 is not an interior point of F .

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Proposition 3.10. Let F be a weak Minkowski norm on Rn . Then the following are equivalent: (1) F is separating (i.e. F .x/ > 0 for all x ¤ 0); (2) F is bounded below on the Euclidean unit sphere S n1 U; x F is bounded. (3) Proof. (1) ) (2): Suppose that F is not bounded below on S n1 . Then there exists a sequence xj 2 S n1 such that F .xj / ! 0. Choosing a subsequence if necessary, we may assume, by compactness of the sphere, that F .xj / < 1 for all j , i.e. xj 2 DF \ S n1 , and that xj converges to some point x0 2 DF \ S n1 . Since F is continuous on DF , we have F .x0 / D limj !1 F .xj / D 0. Since x0 ¤ 0 (it is a point on the sphere), it follows that F is not separating. (2) ) (3): Condition (2) states that there exists > 0 such that F .x/ for all x 2 S n1 . Therefore F .y/ 1 implies kyk 1 . (3) ) (1): Suppose F is non-separating. Then there exists x ¤ 0 with F .x/ D 0. x F which is therefore Therefore F .x/ D 0 for any > 0. In particular RC x unbounded. Definition 3.11. A Minkowski norm is a weak Minkowski norm that is finite and separating. It is simply called a norm if it is furthermore reversible. To a finite and separating norm is associated a well-defined topology, viz. the topology associated to the symmetrization of the weak metric defined by Equation (3.2) (which is a genuine metric). For a deeper investigation of various topological questions we refer to the book [11] by S. Cobzas. Corollary 3.12. The topology defined by the distance (3.2) associated to a Minkowski norm on Rn coincides with the Euclidean topology. Proof. Proposition 3.6 implies that F is continuous. From the compactness of the Euclidean unit sphere S n1 we thus have a constant > 0 such that F .x/ 1 for all points x on S n1 . It follows that kxk F .x/

1 kxk

(3.3)

for all x 2 Rn and therefore F induces the same topology as the Euclidean norm. The next result shows how one can reconstruct the weak Minkowski norm from its unit ball. Proposition 3.13. Let Rn be a convex set containing the origin. Define a function F W Rn ! Œ0; 1 by F .x/ D infft 0 W x 2 t g:

(3.4)

19

Chapter 1. Weak Minkowski spaces

x F , that is, Then F is a weak Minkowski norm and the closure of coincides with x D fx 2 Rn j F .x/ 1g: Moreover, if is open, then D fx 2 Rn j F .x/ < 1g. The function F defined by (3.4) is called the Minkowski functional of . Proof. We need to verify the two conditions in Definition 3.3. For > 0, we have F .x/ D inffs 0 W x 2 s g s D inffs 0 W x 2 g D infft 0 W x 2 t g .sDt /

D F .x/: Now because is convex we have for s; t > 0 s xs C t y xCy x 2 and 2 H) D s s sCt sCt

y t

2 :

Therefore F .x/ < s and F .y/ < t H) F .x C y/ < s C t; which is equivalent to F .x C y/ F .x/ C F .y/. This proves the first part of the proposition. To prove the remaining assertions, observe that F .x/ 1 means that tx 2 for x This shows that any 0 < t < 1 and thus x 2 . x fx 2 Rn j F .x/ 1g : x fx 2 Rn j F .x/ 1g follows from the lower semiThe converse inclusion continuity of F (Corollary 3.7). Finally, if is open, then F is continuous (Proposition 3.9) and therefore D fx 2 Rn j F .x/ < 1g. As a result, we have established one-to-one correspondences between weak Minkowski metrics on Rn , weak Minkowski norms and closed convex sets containing the origin. The closed convex set associated to a weak Minkowski norm F is the set x F D fx 2 X j F .x/ 1g. The associated weak metric is separating if and only if the associated convex set is bounded and the metric is finite if and only if the origin is an interior point of the convex set. Remark 3.14. These concepts have some important consequences in convex geometry. For instance one can easily prove that every unbounded convex set in Rn must contain a ray. Indeed, let Rn be unbounded and convex. One may assume that contains the origin. Then, by Proposition 3.10, its weak Minkowski functional F is not separating, that is, there exists a ¤ 0 in Rn such that F .a/ D 0; but then F .a/ D 0 for every > 0 and therefore the ray contains the ray RC a. Let us conclude this section with two important results from Minkowski geometry. A Minkowski norm on Rn is said to be Euclidean if it is associated to a scalar product.

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Proposition 3.15. Let ı be a Minkowski metric on Rn . Then ı is a Euclidean metric if and only if the ball B.a;r/ D fx 2 X j ı.a; x/ < rg Rn (for some a 2 Rn and r > 0, or, equivalently, for any a 2 Rn and r > 0) is an ellipsoid centered at a. Notice that the above proposition is false if the ellipsoid is not centered at a. Proof. Recall that, by definition, an (open) ellipsoid is a convex set in Rn that is the affine image of the open Euclidean unit ball. If the weak metric ı is Euclidean, then it is obvious that every ball is an ellipsoid. Conversely, suppose that some ball of an arbitrary Minkowski metric ı is an ellipsoid. Then the ball with the same radius centered at the origin is also an ellipsoid since ı is translation-invariant, that is, B.0;r/ D fx 2 X j F .x/ D ı.0; x/ < rg is an ellipsoid. But then 1 B.0;1/ r is also an ellipsoid. Changing coordinates if necessary, one may assume that ˚ P D x 2 Rn j i xi2 < 1 ; D B.0;1/ D

which is the Euclidean unit ball. It follows that F .x/ D infft > 0 j x 2 t g D infft > 0 j kxk < t g D kxk where k k denotes the Euclidean norm. We have the following result on the isometries of a Minkowski metric. Theorem 3.16. Let ı be a Minkowski metric on Rn . Then every isometry of ı is an affine transformation of Rn , and the group Iso.Rn ; ı/ of isometries of ı is conjugate within the affine group to a subgroup of the group E.n/ of Euclidean isometries of Rn . Furthermore Iso.Rn ; ı/ is conjugate to the full group E.n/ if and only if ı is a Euclidean metric. Proof. The first assertion is the Mazur–Ulam Theorem, see [18]. To prove the second assertion, we recall that every bounded convex set in Rn with non-empty interior contains a unique ellipsoid J of maximal volume, called the John ellipsoid of , see [1]. Let us consider the unit ball D B.ı;0;1/ of our Minkowski metric and let us denote by J its John ellipsoid and by z 2 J its center. We call J D J z the centered John ellipsoid of . Consider now an arbitrary isometry g 2 Iso.Rn ; ı/. Set g.x/ Q D g.x/ b, where b D g.0/. Then gQ is an isometry for ı fixing the origin. By construction and uniqueness, the centered John ellipsoid is invariant: g.J Q / D J .

Chapter 1. Weak Minkowski spaces

21

There exists an element A 2 GL.n; R/ such that AJ D B is the Euclidean unit ball. Let us set f WD A B gQ B A1 . Then f .B/ D B: By the Mazur–Ulam theorem, gQ is a linear map, therefore f is a linear map preserving the Euclidean unit ball, which means that f 2 O.n/. We thus obtain g.x/ D A1 .f .x/ C Ab/A where A is linear and x 7! f .x/ C Ab is a Euclidean isometry. To prove the last assertion, one may assume, changing coordinates if necessary, that Iso.Rn ; ı/ D E.n/. Then the ı-unit ball is invariant under the orthogonal group O.n/ and it is therefore a round sphere. We now conclude from Proposition 3.15 that ı is Euclidean.

4 The midpoint property Definition 4.1. A weak metric ı on the real vector space X satisfies the midpoint property if for any p; q 2 X we have ı.p; m/ D ı.m; q/ D

1 ı.p; q/ 2

where m D 12 .p C q/ is the affine midpoint of p and q. To describe the main features of this property, we shall use the notion of dyadic numbers. Definition 4.2. A dyadic number is a rational number of the type D 2k m with m; k 2 Z. We denote the set of dyadic numbers by DD

1 [

2k Z;

kD0

and the subset of non-negative dyadic numbers by DC D: Proposition 4.3. Let ı be a weak metric on the real vector space X . Then ı satisfies the midpoint property if and only if for any pair of distinct points p; q 2 X and for any ; in D with we have ı../; .// D . / ı.p; q/;

(4.1)

where .t / D tp C .1 t /q. Proof. It is obvious that if (4.1) holds, then ı satisfies the midpoint property. The proof of the other direction requires several steps. Assume that ı satisfies the midpoint

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Athanase Papadopoulos and Marc Troyanov

property. Then we have 1 ı.p; q/ and 2 By an induction argument, we then have ı.p; . 12 // D

ı.p; .2/ D 2ı.p; q/:

ı.p; .2m // D 2m ı.p; q/

(4.2)

for any k 2 N. Because p is the midpoint of .2m / and .2m /, we deduce that ı..2m /; .2m // D 2mC1 ı.p; q/:

(4.3)

Now we have, for k 2 Z, ı..k 1/; .k// D ı..k/; .k C 1// since .k/ is the midpoint of .k 1/ and .k C1/. Because ı.p; q/ D ı..0/; .1//, we deduce that ı..k/; .k C 1// D ı.p; q/; and by the triangle inequality we have ı..i/; .j // .j i /ı.p; q/

(4.4)

for any i; j 2 Z with i < j . We show that this inequality is in fact an equality. Choose m 2 N with 2m max.jij; jj j//. Then we have from (4.3) and (4.4) 2mC1 ı.p; q/ D ı..2m /; .2m // ı..2m /; .i// C ı..i /; .j // C ı..j /; .2m //: Using now (4.4) we have ı..i/; .j // .j i /ı.p; q/, but also ı..2m /; .i// .i C 2m /ı.p; q/; and ı..j /; .2m // .2m j /ı.p; q/: Since .i C 2k / C .j i/ C .2m j / D 2mC1 ; all the above inequalities must be equalities. Thus, we have established that ı..i/; .j // D .j i /ı.p; q/ for any i; j 2 Z. Let us now fix k 2 N and set qk D .2k / and k .t/ D .t 2k / D tp C .1 t /qk : Applying (4.5) to k we have ı.k .i/; k .j // D .j i/ı.p; qk / D .j i /2k ı.p; q/:

(4.5)

Chapter 1. Weak Minkowski spaces

23

The last equality can be rewritten as ı.. 2ik /; . 2jk // D . 2jk

i /ı.p; q/ 2k

for any i; j 2 Z and k 2 N, which is equivalent to (4.1) for any dyadic numbers , with . The next result is a generalization to the case of weak metrics of a characterization of Minkowski geometry due to Busemann, see §17 in [4]. Theorem 4.4. A finite weak metric ı on Rn is a weak Minkowski metric if and only if it satisfies the midpoint property and if its restriction to every affine line is continuous. More precisely, the last condition means that if a and b are two points in Rn , then for any t0 2 R we have lim ı..t /; b// D ı..t0 /; b//

t!t0

and lim ı.a; .t // D ı.a; .t0 //;

t!t0

where .t / D t a C .1 t /b. Proof. If ı is a Minkowski metric, then it is projective and since ı is finite (by hypothesis), it follows from Propositions 3.1 and 3.6 that the distance is given by ı.x; y/ D F .y x/; where F is a weak Minkowski norm. The continuity of ı follows now from Proposition 3.6 and the midpoint property follows from property (ii) in Proposition 3.1. Conversely, let us assume that the weak metric ı satisfies the midpoint property and that it is continuous on every line. We need to show that ı is projective and translation-invariant. We first observe that if a; b 2 Rn are two distinct points with ı.a; b/ ¤ 0 and if x and y are two points aligned with a and b such that .y x/ is a non-negative multiple of .b a/, then jy xj ı.x; y/ D ; (4.6) ı.a; b/ jb aj where jq pj denotes the Euclidean distance between p and q in Rn . This follows from Proposition 4.3 together with the continuity of ı on lines and the density of D in R. This immediately implies that ı.p; z/ C ı.z; q/ D ı.p; q/ whenever z 2 Œp; q, meaning that the weak metric ı is projective. To prove the translation invariance, we consider four points p, q, p 0 , q 0 with 0 .q p 0 / D .q p/. If the four points are on a line, then (4.6) implies that ı.p 0 ; q 0 / D ı.p; q/. If the four points are not on a line, then pqq 0 p 0 is a nonC the degenerate parallelogram. Assume also that 0 < ı.p; q/ < 1 and denote by Lpq

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Athanase Papadopoulos and Marc Troyanov

ray with origin p through q and by Lqq 0 the line passing through q and q 0 . Choose a C such that jyj pj ! 1 and set xj D Lp0 yj \ LC sequence yj 2 Lpq qq 0 . yj

xj

q0

q

p

p0

We then have 1

ı.p; yj / ı.p; p 0 / ı.p 0 ; yj / ı.p; p 0 / D ı.p; yj / ı.p; yj / ı.p; yj / 0 ı.p ; p/ C ı.p; yj / ı.p; yj / 0 ı.p ; p/ C 1: D ı.p; yj /

Using (4.6), we have ı.p; yj / ! 1, therefore ı.p 0 ; xj / ı.p 0 ; yj / D lim D 1: j !1 ı.p; q/ j !1 ı.p; yj / lim

Because xj ! q 0 on the line Lqq 0 , we have by hypothesis lim ı.xj ; q 0 / D lim ı.q 0 ; xj / D 0;

j !1

j !1

and since ı.p 0 ; q 0 / ı.xj ; q 0 / ı.p 0 ; xj / ı.p 0 ; q 0 / C lim ı.q 0 ; xj /; j !1

0

0

0

we have ı.p ; xj / ! ı.p ; q /. Therefore ı.p 0 ; xj / ı.p 0 ; q 0 / D lim D 1: j !1 ı.p; q/ ı.p; q/ It follows that for a non-degenerate parallelogram pqq 0 p 0 , we have ı.p 0 ; q 0 / D ı.p; q/. Suppose now that ı.p; q/ D 0. Then we also have ı.p 0 ; q 0 / D 0 for otherwise, exchanging the roles of p, q and p 0 , q 0 in the previous argument, we get a contradiction. We thus established that in all cases ı.p 0 ; q 0 / D ı.p; q/ if q 0 p 0 D q p. In other words, ı is translation-invariant. Since it is projective, this completes the proof that it is a weak Minkowski metric.

Chapter 1. Weak Minkowski spaces

25

Example 4.5 (Counterexample). Let X a be real vector space and let h W X ! R be an injective Q-linear map. Then the function ı W X X ! R defined by ı.x; y/ D jh.x/ h.y/j is a metric which is translation-invariant and satisfies the midpoint property. Yet it is in general not projective (unless h is R-linear, and thus dimR .X / D 1).

5 Strictly and strongly convex Minkowski norms Definition 5.1. (i) Let F be a (finite and separating) Minkowski norm in Rn with unit ball F . Then F is said to be strictly convex if the indicatrix @F contains no non-trivial segment, that is, if for any p; q 2 @F , we have Œp; q @ H) p D q: (ii) The function F is said to be strongly convex if F is smooth on Rn n f0g and the hypersurface @F Rn has everywhere positive Gaussian curvature. Equivalently, the Hessian 1 @2 ˇˇ F 2 .y C u1 1 C u2 2 / (5.1) gy . 1 ; 2 / D ˇ 2 @u1 @u2 u1 Du2 D0 of F 2 .y/ is positive definite for any point y 2 Rn n f0g. There are several equivalent definitions of strict convexity in Minkowski spaces, see e.g. [12], [23]. It is clear that a strongly convex Minkowski norm is strictly convex. The converse does not hold: the Lp -norm kykp D

n X

jyj jp

1=p

j D1

is an example of a smooth strictly convex norm which is not strongly convex. Proposition 5.2. Let F be a strongly convex Minkowski norm on Rn . Then F can be recovered from its Hessian via the formula q F .y/ D gy .y; y/; (5.2) where gy is defined by (5.1). This result follows by applying twice the following lemma, which is sometimes called the Euler Lemma.

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Lemma 5.3. Let W R n 0 ! R be a positively homogeneous functions of degree r. @ If is of class C k for some k 1, then the partial derivatives @y i are positively homogenous functions of degree r 1 and r

.y/ D

n X

yi

iD1 @ In particular, y i @y i D 0 if

@ : @y i

is 0-homogenous.

Recall that a function W Rn n 0 ! R is said to be positively homogenous of degree r if .y/ D r .y/ for all y 2 Rn n 0 and all > 0. Proof. This is elementary: we just differentiate the function t 7! obtain @ .ty/ y i D rt r1 .y/; @y i

.ty/ D t r .y/ to

and we set t D 1. If F is a strongly convex Minkowski norm on Rn , then Formula (5.1) defines a Riemannian metric gy on Rn n f0g. Using Lemma 5.3, on gets that gy is invariant under homothety, that is, we have gy D gy for every > 0 and y 2 Rn n f0g. Furthermore, F is determined from this metric by Equation (5.2). We conclude from these remarks the following: Proposition 5.4. There is a natural bijection between strongly convex Minkowski norms on Rn and Riemannian metrics This observation can be used as a founding stone for Minkowski geometry, see e.g. [28], and it plays a central role in Finsler geometry.

6 The synthetic viewpoint Definition 2.3 of a weak Minkowski space is based on a real vector space X as a ground space. In fact, only the affine structure of that space plays a role and we could equivalently start with a given affine space instead of a vector space. The synthetic viewpoint is to start with an abstract metric space and to try to give a list of natural conditions implying that the given metric space is Minkowskian. This question, and similar questions for other geometries, has been a central and recurrent question in the work of Busemann, and it is implicit in Hilbert’s comments on his Fourth Problem [15]. Some answers are given in Busemann’s book The Geometry of Geodesics [4]. In that book Busemann introduces the notions of G-spaces and Desarguesian spaces. The goal of this section is to give a short account on this viewpoint. We restrict ourselves to the case of ordinary metric spaces.

Chapter 1. Weak Minkowski spaces

27

Definition 6.1 (Busemann G-space). A Busemann G-space is a metric space .X; d /, satisfying the following four conditions: (1) (Menger Convexity) Given distinct points x; y 2 X, there is a point z 2 X different from x and y such that d.x; z/ C d.z; y/ D d.x; y/. (2) (Finite Compactness) Every d -bounded infinite set has an accumulation point. (3) (Local Extendibility) For every point p 2 X, there exists rp > 0, such that for any pair of distinct points x; y 2 X in the open ball B.p; rp /, there is a point z 2 B.p; rp / n fx; yg such that d.x; y/ C d.y; z/ D d.x; z/. (4) (Uniqueness of Extension) Let x; y; z1 ; z2 be four points in X such that d.x; y/C d.y; z1 / D d.x; z1 / and d.x; y/Cd.y; z2 / D d.x; z2 /. Suppose that d.y; z1 / D d.y; z2 /, then z1 D z2 . A typical example of a Busemann G-space .X; d / is a strongly convex Finsler manifold of class C 2 . (In fact class C 1;1 suffices, by a result of Pogorelov.) It follows from the definition that any pair of points in a Busemann G-space .X; d / can be joined by a minimal geodesic and that geodesics are locally unique. It is also known that every G-space is topologically homogeneous and that it is a manifold if its dimension is at most 4. We refer to [2] for further results on the topology of G-spaces. Among G-spaces, Busemann introduced the class of Desarguesian spaces. Definition 6.2 (Desarguesian space). A Desarguesian space is a metric space .X; d / satisfying the following conditions: (1) .X; d / is a Busemann G-space. (2) .X; d / is uniquely geodesic, that is, every pair of points can be joined by a unique geodesic. (3) If the topological dimension2 of X equals 2, then Desargues’ theorem holds for the family of all geodesics. (4) If the topological dimension of X is greater than 2, then any triple of points lie in a plane, that is, a two-dimensional subspace of X which is itself a G-space. The reason for assuming Desargues’ property in the two-dimensional case as an axiom is due to the well-known fact from axiomatic geometry that it is possible to construct exotic two-dimensional planes in which the axioms of real projective or affine geometry are satisfied but which are not isomorphic to RP 2 or R2 (an example of such exotic object is the Moufang plane); these objects do not satisfy the Desargues property. Similar objects do not exist in higher dimensions and Desargues’ property is a theorem in all dimensions 3. In fact, Klein showed in his paper [17] that Desargues’ theorem in the plane, although a theorem of projective geometry, cannot be proved using only 2 On page 46 in [4], regarding this definition of Desarguesian space, Busemann states that he uses the Menger– Urysohn notion of dimension, but any reasonable notion of topological dimension is equivalent for a G-space.

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two-dimensional projective geometry. Hilbert, in his Grundlagen (2nd. ed., §23), constructed a plane geometry in which all the axioms of two-dimensional projective geometry hold but where the theorem of Desargues fails. This theorem follows from the axioms of three-dimensional projective geometry. Condition (3) in the above definition could be rephrased as follows: If X is two-dimensional, then it can be isometrically embedded in a three-dimensional Desarguesian space. We refer to [4] and to Chapter 15 of this volume [24] for further discussion of Desarguesian spaces. A deep result of Busemann states that a Desarguesian space can be mapped onto a real projective space or on a convex domain in a real affine space with a projective metric. More precisely he proved the following Theorem 6.3 (Theorems 13.1 and 14.1 in [4]). Given an n-dimensional Desarguesian space .X; d /, one of the following condition holds: (1) Either all the geodesics are topological circles and there is a homeomorphism ' W X ! RP n that maps every geodesic in X onto a projective line; (2) or there is a homeomorphism from X onto a convex domain C in Rn that maps every geodesic in X onto the intersection of a straight line with C . Using the notion of Desarguesian space and following Busemann, we now give two purely intrinsic characterizations of finite-dimensional Minkowski spaces among abstract metric spaces. Note that a Minkowski space .X; d / is a G-space if and only if its unit ball is strictly convex. The first result is a converse to that statement. Theorem 6.4 ([4], Theorem 24.1). A metric space .X:d / is isometric to a Minkowski space if and only if it is a Desarguesian space in which the parallel postulate holds and the spheres are strictly convex. We refer the reader to [4], p. 141, for a discussion of the parallel postulate in this context. Observe that in a Desarguesian space there are well defined notions of lines and planes and therefore Euclid’s parallel postulate can be formulated. Using Theorem 6.3 and the parallel postulate, we obtain that .X; d / is isometric to Rn with a projectively flat metric. To prove the theorem, Busemann uses the strict convexity of spheres to establish the midpoint property. The next result involves the notion of Busemann zero curvature. Recall that a geodesic metric space is said to have zero curvature in the sense of Busemann if the distance between the midpoints of two sides of an arbitrary triangle is equal to half the length of the remaining side. Busemann formulates the following characterization: Theorem 6.5 ([4], Theorem 39.12). A simply connected finite-dimensional G-space of zero curvature is isometric to a Minkowski space. Busemann came back several times to the problem of characterizing Minkowskian and locally Minkowskian spaces. In his paper with Phadke [9], written 25 years after

Chapter 1. Weak Minkowski spaces

29

[4], he gave sufficient conditions that are more technical but weaker than those of Theorem 6.5.

7 Analogies between Minkowski, Funk and Hilbert geometries Given a Minkowski metric ı in Rn whose unit ball at the origin is open and bounded, the distance between two points is obtained by setting ı.x; x/ D 0 for all x in Rn and, for x 6D y, jx yj ı.x; y/ D j0 aC j where j j denotes the Euclidean metric and the point aC is the intersection with @ of the ray starting at the origin 0 of Rn and parallel to the ray R.x; y/ from x to y. This formula is equivalent to (1.1) and it suggest an analogy with the formula for the Funk distance in the domain (see Definition 2.1 in Chapter 2 of this volume [25]). It is also in the spirit of the following definition of Busemann ([4], Definition 17.1): A metric d.x; y/ in Rn is Minkowskian if for the Euclidean metric e.x; y/ the distances d.x; y/ and e.x; y/ are proportional on each line. Minkowski metrics share several important properties of the Funk and the Hilbert metrics, and it is interesting to compare these three classes of metrics. Let us quickly review some of the analogies. We start by recalling that in the formulation of Hilbert’s fourth problem which asks for the construction and the study of metrics on subsets of Euclidean (or of projective) space for which the Euclidean segments are geodesics, the Minkowski and Hilbert metrics appear together as the two examples that Hilbert gives (see [15] and Chapter 15 in this volume [24]). A rather simple analogy between the Minkowski and the Funk geometries is that both metrics are uniquely geodesic if and only if their associated convex sets are strictly convex. (Here, the convex set associated to a Minkowski metric is the unit ball centered at the origin (Definition 3.8). The convex set associated to a Funk metric is the set on which this metric is defined.) Another analogy between Minkowski and Hilbert geometries is the well-known fact that a Minkowski weak metric on Rn is Riemannian if and only if the associated convex set is an ellipsoid, see Proposition 3.15. This fact is (at least formally) analogous to the fact that the Hilbert geometry of an open bounded convex subset of Rn is Riemannian if and only if this convex set is an ellipsoid (see [16], Proposition 19). As a further relation between Minkowski and Hilbert geometries, let us recall a result attributed to Nussbaum, de la Harpe, Foertsch and Karlsson. Nussbaum and de la Harpe proved (independently) in [19] and [14] that if Rn is the interior of the standard n-simplex and if H denotes the associated Hilbert metric, then the metric space .; H / is isometric to a Minkowski metric space. Foertsch and Karlsson proved the converse in [13], thus completing the result saying that a bounded open

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convex subset of Rn equipped with its Hilbert metric is isometric to a Minkowski space if and only if is the interior of a simplex. It should be noted that the result (in both directions) was already known to Busemann since 1967. In their paper [8], p. 313, Busemann and Phadke write the following, concerning the simplex: The case of general dimension n is most interesting. The (unique) Hilbert geometry possessing a transitive abelian group of motions where the affine segments are the chords (motion means that both distance and chords are preserved) is given by a simplex S, ([5], p. 35). If we realize [the interior of the simplex] as the first quadrant xi > 0 of an affine coordinate system, the group is given by xi0 D ˇi xi , ˇi > 0 [...] m is a Minkowski metric because it is invariant under the translations and we can take the affine segments as chords.

We finally mention the following common characterizations of Minkowski–Funk geometries and of Minkowski–Hilbert geometries: Theorem 7.1 (Busemann [6], p. 38). Among noncompact and nonnecessarily symmetric Desarguesian spaces in which all the right and left spheres of positive radius around any point are compact, the Hilbert and Minkowski geometries are characterized by the property that any isometry between two (distinct or not) geodesics is a projectivity. Theorem 7.2 (Busemann [7]). A Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space. Acknowledgement. The first author is partially supported by the French ANR project FINSLER.

References [1]

A. Barvinok, A course in convexity. Grad. Stud. in Math. 54, Amer. Math. Soc., Providence, RI, 2002.

[2]

V. N. Berestovski˘ı, D. M. Halverson and D. Repovš, Locally G-homogeneous Busemann G-spaces. Differential Geom. Appl. 29 (2011), no. 3, 299–318.

[3]

H. Busemann, The foundations of Minkowskian geometry. Comment. Math. Helv. 24 (1950), 156–187.

[4]

H. Busemann, The geometry of geodesics. Academic Press, New York 1955; reprinted by Dover in 2005.

[5]

H. Busemann, Timelike spaces. Dissertationes Math. (Rozprawy Mat.) 53, Warsaw 1967.

[6]

H. Busemann, Recent synthetic differential geometry. Ergeb. Math. Grenzgeb. 54, Springer-Verlag, Berlin 1970.

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[7]

H. Busemann, Spaces with homothetic spheres. J. Geometry 4 (1974), 175–186.

[8]

H. Busemann and B. B. Phadke, A general version of Beltrami’s theorem in the large. Pacific J. Math. 115 (1984), 299–315.

[9]

H. Busemann and B. B. Phadke, Minkowskian geometry, convexity conditions and the parallel axiom. J. Geometry 12 (1979), no. 1, 17–33.

[10] H. Busemann and B. B. Phadke, Novel results in the geometry of geodesics. Adv. in Math. 101 (1993), 180–219. [11] S. Cobzas, Functional analysis in asymmetric normed spaces. Front. Math., Birkhäuser/Springer Basel AG, Basel 2013. [12] M. M. Day, Normed linear spaces. 3rd ed., Ergeb. Math. Grenzgeb. 21, Springer-Verlag, Berlin 1973. [13] T. Foertsch, and A. Karlsson, Hilbert metrics and Minkowski norms. J. Geometry 83 (2005), no. 1–2, 22–31. [14] P. de la Harpe, On Hilbert’s metric for simplices. In Geometric group theory (Graham A. Niblo et al., eds.,) Vol. 1, London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge 1993, 97–119. [15] D. Hilbert, Mathematische Probleme. Göttinger Nachrichten 1900, 253–297, reprinted in Archiv der Mathematik und Physik, 3d. ser., vol. 1 (1901) 44–63 and 213–237; English version, “Mathematical problems”, reprinted also in Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436. [16] D. C. Kay, The Ptolemaic inequality in Hilbert geometries. Pacific J. Math. 21 (1967), 293–301. [17] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz). Math. Ann. VI (1873), 112–145. [18] B. Nica, The Mazur–Ulam theorem. Expo. Math. 30 (2012), no. 4, 397–398. [19] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 391 (1988). [20] H. Martini, K. Swanepoel and G. Weiss, The geometry of Minkowski spaces – a survey. I. Expo. Math. 19 (2001), no. 2, 97–142. [21] H. Martini and M. Spirova, Recent results in Minkowski geometry. East-West J. Math., Special Vol. (2007), 59–101. [22] H. Minkowski, Sur les propriétés des nombres entiers qui sont dérivées de l’intuition de l’espace. Nouvelles annales de mathématiques, 3e série, 15, 1896. [23] H. Minkowski, Geometrie der Zahlen. B. G. Teubner, Leipzig and Berlin, 1896 and 1910 (several editions and translations). [24] A. Papadopoulos, Hilbert’s fourth problem. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 391–431. [25] A. Papadopoulos and M. Troyanov, From Funk to Hilbert geometry. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 33–67. [26] H. Ribeiro, Sur les espaces à métrique faible. Portugaliae Math. 4 (1943), 21–40.

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[27] A. C. Thompson, Minkowski geometry. Encyclopedia Math. Appl. 63. Cambridge University Press, Cambridge 1996. [28] O. Varga, Zur Begründung der Minkowskischen Geometrie. Acta Univ. Szeged. Sect. Sci. Math. 10 (1943), 149–163.

Chapter 2

From Funk to Hilbert geometry Athanase Papadopoulos and Marc Troyanov

Contents 1 2 3 4 5 6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Funk metric . . . . . . . . . . . . . . . . . . . . . . . . . The reverse Funk metric . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The geometry of balls in the Funk metric . . . . . . . . . . . . On the topology of the Funk metric . . . . . . . . . . . . . . . The triangle inequality and geodesics . . . . . . . . . . . . . . 7.1 On the triangle inequality . . . . . . . . . . . . . . . . . 7.2 Geodesics and convexity in Funk geometry . . . . . . . . 8 Nearest points in Funk geometry . . . . . . . . . . . . . . . . . 9 The infinitesimal Funk distance . . . . . . . . . . . . . . . . . 10 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A projective viewpoint on Funk geometry . . . . . . . . . . . . 12 Hilbert geometry . . . . . . . . . . . . . . . . . . . . . . . . . 13 Related questions . . . . . . . . . . . . . . . . . . . . . . . . . A Menelaus’ Theorem . . . . . . . . . . . . . . . . . . . . . . . B The classical proof of the triangle inequality for the Funk metric References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

33 35 41 42 43 46 48 48 51 53 55 56 57 58 60 61 64 66

1 Introduction The Funk metric associated to an open convex subset of a Euclidean space is a weak metric in the sense that it does not satisfy all the axioms of a metric: it is not symmetric, and we shall also allow the distance between two points to be zero; see Chapter 1 in this volume [19], where such metrics are introduced. Weak metrics often occur in the calculus of variations and in Finsler geometry and the study of such metrics has been revived recently in low-dimensional topology and geometry by Thurston who introduced an asymmetric metric on Teichmüller space, which became the subject of intense research. In this chapter, we shall sometimes use the expression “metric” instead of “weak metric” in order to simplify.

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The Funk weak metric F .x; y/ is associated to an open convex subset of a Euclidean space Rn . It is an important metric for the subject treated in this Handbook, because the Hilbert metric H of the convex set is the arithmetic symmetrization of its Funk metric. More precisely, for any x and y in , we have H .x; y/ D

1 .F .x; y/ C F .y; x// : 2

Independently of its relation with the Hilbert metric, the Funk metric is a nice example of a weak metric, and there are many natural questions for a given convex subset of Rn that one can ask and solve for such a metric, regarding its geodesics, its balls, its isometries, its boundary structure, and so on. Of course, the answer depends on the shape of the convex set , and it is an interesting aspect of the theory to study the influence of the properties of the boundary @ (its degree of smoothness, the fact that it is a polyhedron, a strictly convex hypersurface, etc.) on the Funk geometry of this set. The same questions can be asked for the Hilbert geometry, and they are addressed in several chapters of this volume. The Funk geometry is studied in Busemann’s book The geometry of geodesics [2], although the name “Funk metric” is not used there, but it is used by Busemann in his later papers and books, see e.g. [3], and in the memoir [28] by Zaustinsky (who was a student of Busemann). We studied some aspects of this metric in [17], following Busemann’s ideas. But a systematic study of this metric is something which seems to be still missing in the literature, and the aim of this chapter is somehow to fill this gap. Euclidean segments in are geodesics for the Funk metric, and in the case where the domain is strictly convex (that is, if there is no nonempty open Euclidean segment in @), the Euclidean segments are the unique geodesic segments. For a metric d on a subset of Euclidean space, the property of having the Euclidean segments d geodesics is the subject of Hilbert’s Problem IV (see Chapter 15 in this volume [16]). One version of this problem asks for a characterization of non-symmetric metrics on subsets of Rn for which the Euclidean segments are geodesics. The Funk metric is also a basic example of a Finsler structure, perhaps one of the most basic. In the paper [17], we introduced the notions of tautological and of reversible tautological Finsler structures of the domain . The Funk metric F is the length metric induced by the tautological weak Finsler structure, and the Hilbert metric H is the length metric induced by the reversible tautological Finsler structure of the domain . This is in fact how Funk introduced his metric in 1929, see [9], [24]. The reversible Finsler structure is obtained by the process of harmonic symmetrization at the level of the convex sets in the tangent spaces that define these structures, cf. [18]. This gives another relation between the Hilbert and the Funk metrics. The Finsler geometry of the Funk metric is studied in some detail in Chapter 3 [24] of this volume. A useful variational description of the Funk metric is studied in [27]. There are also interesting non-Euclidean versions of Funk geometry, see Chapter 13 in this volume ([20]).

Chapter 2. From Funk to Hilbert geometry

35

In the present chapter, we start by recalling the definition of the Funk metric, and we give several basic properties of this metric, some of which are new, or at least formulated in a new way. In Section 3, we introduce what we call the reverse Funk metric, that is, the metric rF defined by rF .x; y/ D F .y; x/. This metric is much less studied than the Funk metric. In Section 4, we give the formula for the Funk metric for the two main examples, namely, the cases of convex polytopes and the Euclidean unit ball. In Section 5, we study the geometry of balls in the Funk metric. Since the metric is non-symmetric, one has to distinguish between forward and backward balls. We show that any forward ball is the image of by a Euclidean homothety. This gives a rigidity property, namely, that the local geometry of the convex set determines the convex set up to a scalar factor. In Section 6, we show that the topologies defined by the Funk and the reverse Funk metrics coincide with the Euclidean topology. In Section 7, we give a proof of the triangle inequality for the Funk metric and at the same time we study its geodesics and its convexity properties. In Section 8, we study the property of the nearest point projections of a point in on a convex subset of equipped with the Funk metric, and the perpendicularity properties. In the case where is strictly convex, the nearest point projection is unique. There is a formulation of perpendicularity to hyperplanes in in terms of properties of support hyperplanes for . In Section 9, we study the infinitesimal (Finsler) structure associated to a Funk metric. In Section 10, we study the isometries of a Funk metric. In the case where is bounded and strictly convex, its isometry group coincides with the subgroup of affine transformations of Rn that leave invariant. In Section 11, we propose a projective viewpoint and a generalization of the Funk metric and in Section 12, we present some basic properties of the Hilbert metric which are direct consequences of the fact that it is a symmetrization of the Funk metric. In the last section we give some new perspectives on the Funk metric, some of which are treated in other chapters of this volume. Appendix A contains two classical theorems of Euclidean geometry (the theorems of Menelaus and Ceva). Menelaus’ Theorem is used in Appendix B to give the classical proof of the triangle inequality for the Funk metric.

2 The Funk metric In this section, is a proper convex domain in Rn , that is, is convex, open, nonx the closure of in Rn and by @ D x n empty, and ¤ Rn . We denote by the topological boundary of . In the case of an unbounded domain, it will be convenient to add points at infinity to the boundary @. To do so, we consider Rn as an affine space in the projective space RP n and we denote by H1 D RP n n Rn the hyperplane at infinity. We then denote Q D Q n H1 . z n . Observe that @ D @ z the closure of in RP n and by @ by n For any two points x ¤ y in R , we denote by Œx; y the closed affine segment joining them. We also denote by R.x; y/ the affine ray starting at x and passing

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z y/ its closure in RP n . Finally we set through y and by R.x; Q 2 RP n : z y/ \ @ a .x; y/ D R.x; a ............. .............r. ......... ... ........... . . . . . . . . .. ......... .. ........ . . . . . . .. . . . . . . . . . . . . . . yr .. .... .. .... .. . . .... . . ... ..... .. ... ... r .... .. ... x ... ..... ... ........ .... .......... ................ ................... .........

We now define the Funk metric. Definition 2.1 (The Funk metric). The Funk metric on , denoted by F , is defined for x and y in , by F .x; x/ D 0 and by jx aj F .x; y/ D log jy aj if x ¤ y, where a D a .x; y/ 2 RP n . Here jp qj is the Euclidean distance between the points q and p in Rn . It is understood in this formula that if a 2 H1 , then F .x; y/ D 0. This is consistent with the convention 1=1 D 1. Let us begin with a few basic properties of the Funk metric. Proposition 2.2. The Funk metric in a convex domain ¤ Rn satisfies the following properties: (a) F .x; y/ 0 and F .x; x/ D 0 for all x; y 2 . (b) F .x; z/ F .x; y/ C F .y; z/ for all x; y; z 2 . (c) F is projective, that is, F .x; z/ D F .x; y/ C F .y; z/ whenever z is a point on the affine segment Œx; y. (d) The weak metric F is non symmetric, that is, F .x; y/ ¤ F .y; x/ in general. (e) The weak metric F is separating, that is, x ¤ y H) F .x; y/ > 0, if and only if the domain is bounded. (f) The weak metric F is unbounded. Property (a) and (b) say that F is a weak metric. Proof. Property (a) follows from the fact that y 2 Œx; a .x; y/, therefore jx a .x; y/j 1 jy a .x; y/j

Chapter 2. From Funk to Hilbert geometry

37

and we have equality if y D x. The triangle inequality (b) is not completely obvious. The classical proof by Hilbert (who wrote it for the Hilbert metric) is given in Appendix B and a new proof is given in Section 7. To prove (c), observe that if y 2 Œx; z and x ¤ y ¤ z, then a .x; y/ D a .x; z/ D a .y; z/. Denoting this common point by a, we have F .x; y/ C F .y; z/ D log

jy aj jx aj jx aj C log D log D F .x; z/: jy aj jz aj jz aj

Property (d) is easy to check, see Section 3 for more details. Property (e) follows immediately from the definition and the fact that a convex domain in Rn is unbounded if and only if it contains a ray, see [19], Remark 3.13. To prove (f), we recall that we always assume ¤ Rn , and therefore @ ¤ ;. Let x be a point in and a a point in @ and consider the open Euclidean segment .x; a/ contained in . For any sequence xn in this segment converging to a (with respect to the Euclidean metric), we have F .x; xn / D log jxjxaj ! 1 as n ! 1. n aj It is sometimes useful to see the Funk metric from other viewpoints. If x, y and z are three aligned points in Rn with z ¤ y, then there is a unique 2 R such that x D z C .y z/. We call this number the division ratio or affine ratio of x with respect to z and y and denote it suggestively by D zx=zy. Note also that > 1 if and only if y 2 Œx; z. We extend the notion of affine ratio to the case z 2 H1 by setting zx=zy D 1 if z 2 H1 . We then have F .x; y/ D log./; where is the affine ratio of x with respect to a D a .x; y/ and y. Proposition 2.3. Let x and y be two points in the convex domain and set a D a .x; y/. Let h W Rn ! R be an arbitrary linear form such that h.y/ ¤ h.x/. Then h.a/ h.x/ F .x; y/ D log : h.a/ h.y/ Proof. We first observe that a linear function h W Rn ! R is either constant on a given ray R.x; y/, or it is injective on that ray. If a D a .x; y/ 2 H1 , then h.a/ D lim h.x C t .y x// D h.x/ C t .h.y/ h.x// D ˙1; t!1

and the proposition is true since log.1=1/ D log.1/ D 0 D F .x; y/. If a 62 H1 , then the division ratio D ax=ay > 1 and we have .a x/ D .a y/ and F .x; y/ D log./. Since h W Rn ! R is linear, then .h.a/h.x// D .h.a/h.y// and the proposition follows at once. We now recall some important notions from convex geometry. For these and other classical results on convex geometry, the reader can consult the books by Eggleston [7], Fenchel [8], Valentine [26] and Rockafellar [22].

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Definition 2.4. Let Rn be a proper convex domain and a 2 @ be a (finite) boundary point. We assume that contains the origin 0. A supporting functional at a for is a linear function h W Rn ! R such that h.a/ D 1 and h.x/ < 1 for any point x in 1 . The hyperplane H D fz j h.z/ D 1g is said to be a support hyperplane at a for . If is unbounded, then the hyperplane at infinity H1 is also considered to be a support hyperplane. A basic fact is that any boundary point of a convex domain admits one or several supporting functional(s). Let us denote by S the set of all supporting functionals of . Then p .x/ D supfh.x/ j h 2 S g is the Minkowski functional of . This is the unique weak Minkowski norm such that D fx 2 Rn j p .x/ < 1g: We have the following consequences of the previous proposition: Corollary 2.5. Let Rn be a convex domain containing the origin and let h be a supporting functional for . We then have 1 h.x/ F .x; y/ log 1 h.y/ for any x; y 2 . Furthermore we have equality if and only if either a D a .x; y/ 2 H1 and h.x/ D h.y/, or a 62 H1 and h.a/ D 1. Proof. If h.x/ D h.y/ there is nothing to prove, we thus assume that h.x/ ¤ h.y/. Note that this condition means that the line through x and y is not parallel to the supporting hyperplane H D fh D 1g. We first assume that a D a .x; y/ 62 H1 , The hypothesis h.x/ ¤ h.y/ implies jxaj h.a/ ¤ h.y/. Because the function a 7! log jyaj is strictly monotone decreasing and h.a/ 1, we have by Proposition 2.3: 1 h.x/ h.a/ h.x/ log : F .x; y/ D log h.a/ h.y/ 1 h.y/ The equality holds if and only if h.a/ D 1. Suppose now that a 2 H1 ; this means that F .x; y/ D 0 and the ray R.x; y/ is contained in . In particular, we have h.x/ C h.y x/ D h.x C .y x// < 1

for all 0;

which implies h.y/ h.x/ D h.y x/ 0. Since we assumed h.x/ ¤ h.y/, we have h.y/ < h.x/ and therefore 1 h.x/ : F .x; y/ D 0 > log 1 h.y/ 1 The reader should not mistake the notion of supporting functional for that of support function h W Rn ! R defined as h .x/ D supfhx; yi j y 2 g.

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Corollary 2.6. If Rn is a convex domain containing the origin, then ° ± 1h.x/ F .x; y/ D max 0; suph2S log 1h.y/ : Proof. We first assume that F .x; y/ > 0. Then a D a .x; y/ 62 H1 . Let us denote by ˆ.x; y/ the right hand side in the above formula. Then Proposition 2.3 implies that F .x; y/ ˆ.x; y/ and Corollary 2.5 implies the converse inequality. If F .x; y/ D 0, then R.x; y/ < 0. For any supporting function h, we then have h.x C t .y x// D h.x/ C t.h.y/ h.x// < 1 for any t > 0. This implies that h.y/ h.x/ and it follows that ˆ.x; y/ D 0. Notice that for bounded domains, the previous formula reduces to 1 h.x/ F .x; y/ D sup log : 1 h.y/ h2S This can also be reformulated as follows (compare to Theorem 1 of [27]): Corollary 2.7. The Funk metric in a bounded convex domain Rn is given by dist.x; H / F .x; y/ D sup log ; dist.y; H / H where the supremum is taken over the set of all support hyperplanes H for and dist.x; H / is the Euclidean distance from x to H . The next consequence of Proposition 2.3 is the following relation between the division ratio of three aligned points in a convex domain and the Funk distances between them. Corollary 2.8. Let x, y and z be three aligned points in the convex domain Rn . Suppose that F .x; y/ > 0 and z D x C t .y x/ for some t 0. Then eF .x;y/ .eF .x;z/ 1/ ; eF .x;z/ .eF .x;y/ 1/ F .x; z/ D F .x; y/ log eF .x;y/ C t .1 eF .x;y/ / : tD

(2.1) (2.2)

Proof. Choose a supporting functional h for at the point a D a .x; y/ D a .x; z/. Then we have, from Corollary 2.5: eF .x;y/ D Therefore eF .x;z/ 1 D eF .x;z/

1 h.x/ 1 h.y/

1 h.z/ 1 h.x/

and

eF .x;z/ D

1 h.x/ : 1 h.z/

1 h.x/ h.z/ h.x/ 1 D ; 1 h.z/ 1 h.x/

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and likewise

eF .x;y/ 1 h.y/ h.x/ : D F .x;y/ 1 h.x/ e

Thus eF .x;y/ .eF .x;z/ 1/ h.z/ h.x/ h.z x/ D D t: D F .x;z/ F .x;y/ h.y x/ h.y/ h.x/ e .e 1/ This proves Equation (2.1). To prove the equation (2.2), we now resolve eF .x;z/ 1 eF .x;y/ 1 D t eF .x;y/ eF .x;z/ for eF .x;z/ . This gives us eF .x;z/ D

eF .x;y/ ; eF .x;y/ C t 1 eF .x;y/

which is equivalent to (2.2). It is useful to observe that computing the Funk distance between two points in is a one-dimensional operation. More precisely, if S D Œa1 ; a2 Rn is a compact segment in Rn containing the points x and y in its interior with y 2 Œx; a2 , we shall write jx a2 j FS .x; y/ D log : jy a2 j Although S is not an open set, FS .x; y/ clearly corresponds to the one-dimensional Funk metric in the relative interior of S. Proposition 2.9. The Funk distance between two points x and y in is given by F .x; y/ D inf fFS .x; y/ j S is a segment in containing x and yg : Proof. We identify S with a segment in R with b < x y < a and observe that the jxaj function a 7! log jyaj is strictly monotone decreasing. This result makes an analogy between the Funk metric and the Kobayashi metric in complex geometry, see [13]. It has the following immediate consequences: Corollary 2.10. (i) If 1 2 are convex subsets of Rn , then F1 F2 with equality if and only if 1 D 2 . (ii) Let 1 and 2 be two open convex subsets of Rn . Then, for every x and y in 1 \ 2 , we have F1 \2 .x; y/ D max.F1 .x; y/; F2 .x; y//: (iii) Let be a nonempty open convex subset of Rn , let 0 be the intersection of with an affine subspace of Rn , and suppose that 0 ¤ ;. Then, F0 is the metric induced by F on 0 as a subspace of .; F /.

Chapter 2. From Funk to Hilbert geometry

41

3 The reverse Funk metric Definition 3.1. The reverse Funk metric in a proper convex domain is defined as jy bj r ; F .x; y/ D F .y; x/ D log jx bj where b D a .y; x/. ................................ ........... ... ......... . . . . .. . . . . . . ........ . . . . . ... . . . . . . . . . . . . . . . . yr ..... .. .. .... ..... ..... . .. .... .. .. ... . . r .... . ..... x ... ....... .... . . . .... . r.. . b ..................................................

Figure 1. The reverse Funk metric.

Proposition 3.2. The reverse Funk metric in a convex domain ¤ Rn is a projective weak metric. It is unbounded and non-symmetric and it is separating if and only if the domain is bounded. The proof is a direct consequence of Proposition 2.2. An important difference between the Funk metric and the reverse Funk metric is the following: Proposition 3.3. Let be a bounded convex domain in Rn and x be a point in . Then the function y ! rF .x; y/ is bounded. Proof. Define x and ı by x D inf jx bj; [email protected]

and

ı D sup ja bj: a;[email protected]

Observe that ı is the Euclidean diameter of , thus ı < 1 since is bounded. We also have x > 0. The proposition follows from the inequality ı r F .x; y/ log : x In particular the reverse Funk metric rF is not bi-Lipschitz equivalent to the Funk metric F .

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4 Examples Example 4.1 (Polytopes). An (open) convex polytope in Rn is defined to be an intersection of finitely many half-spaces: D fx 2 Rn j j .x/ < sj ; 1 j kg; where j W Rn ! R is a nontrivial linear form for all j . The Funk distance between two points in such a polytope is given by ° ± s .x/ F .x; y/ D max 0; max1j k log sjj jj .y/ : The proof is similar to that of Corollary 2.6. As a special case, let us mention that the Funk metric in RnC is given by ° ± FRnC .x; y/ D max1in max 0; log yxii : Observe that the map x D .xi / 7! u D .ui /, where ui D log.xi /, is an isometry from the space .RnC ; FRnC / to Rn with the weak Minkowski distance ı.u; v/ D max1i n max ¹0; ui vi º: Example 4.2 (The Euclidean unit ball). The following is a formula for the Funk metric in the Euclidean unit ball B Rn : ! p jy xj2 jx ^ yj2 C jxj2 hx; yi ; (4.1) FB .x; y/ D log p jy xj2 jx ^ yj2 jyj2 C hx; yi p ! where jx ^ yj D jxj2 jyj2 hx; yi2 is the area of the parallelogram with sides 0x, ! 0y. y

x

a

a2 0

a1

Proof. If x D y, there is nothing to prove, so we assume that x ¤ y. Let us set a D aB .x; y/ D R.x; y/ \ @B. Using Proposition 2.3 with the linear form h.z/ D hy x; zi we get hy x; ai hy x; xi hy x; ai C jxj2 hx; yi D log : FB .x; y/ D log hy x; ai hy x; yi hy x; ai jyj2 C hx; yi

Chapter 2. From Funk to Hilbert geometry

43

So we just need to compute hy x; ai. This is an exercise in elementary Euclidean yx and geometry. Let us set u D jyxj a1 D hu; aiu;

a2 D a a1 :

Then a D a1 C a2 and a1 is a multiple of y x while a2 is the orthogonal projection of the origin O of Rn on the line through x and y. In particular the altitude of the triangle Oxy is equal to ja2 j, therefore Area.Oxy/ D

1 1 jx ^ yj D ja2 j jy xj: 2 2

Observe now that hu; ai > 0 and jaj2 D ja1 j2 C ja2 j2 D 1, we thus have hy x; ai2 D jy xj2 hu; ai2 D jy xj2 ja1 j2 D jy xj2 .1 ja2 j2 / D jy xj2 jx ^ yj2 : The desired formula follows immediately.

5 The geometry of balls in the Funk metric Since we are dealing with non-symmetric distances, we need to distinguish between forward and backward balls. For a point x in and > 0, we set B C .x; / D fy 2 B j F .x; y/ < g

(5.1)

and we call it the forward open ball (or right open ball) centered at x of radius . In a symmetric way, we set B .x; / D fy 2 B j F .y; x/ < g

(5.2)

and we call it the backward open ball (also called the left open ball) centered at x of radius . Note that the open backward balls of the Funk metric are the open forward balls of the reverse Funk metric, and vice versa. We define closed forward and closed backward balls by replacing the inequalities in (5.1) and (5.2) by non strict inequalities, and in the same way we define forward and backward spheres by replacing the inequalities by equalities. In Funk geometry, the backward and forward balls have in general quite different shapes and different properties. Proposition 5.1. Let be a proper convex open subset of Rn equipped with its Funk metric F , let x be a point in and let be a nonnegative real number. We have:

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• The forward open ball B C .x; / is the image of by the Euclidean homothety of center x and dilation factor .1 e /. • The backward open ball B .x; / is the intersection of with the image of by the Euclidean homothety of center x and dilation factor .e 1/, followed by the Euclidean central symmetry centered at x.

r

r

A forward and a backward ball in Funk geometry. The forward ball is always relatively compact in , while the closure of the backward ball may meet the boundary @ if its radius is large enough.

Proof. Let y ¤ x be a point in . If F .x; y/ D 0, then the ray R.x; y/ is contained in , and for any z on that ray, we have F .x; z/ D 0. Therefore the ray is also contained in B.x; /. If F .x; y/ D 0, then a D a .x; y/ 62 H1 and we have the following equivalent conditions for any point y on the segment Œx; a: jx aj jy xj:

y 2 B .x; / () log

Thus, for instance, if is the interior a Euclidean ball in Rn , then any forward ball for the Funk metric B C .x0 ; ı/ is also a Euclidean ball. However, its Euclidean center is not the center for the Funk metric (unless x0 is the center of ). Considering Example 4.2, if is the Euclidean unit ball and B C .x0 ; / is the Funk ball of radius and center x0 in , then y 2 B C .x0 ; / if and only if F .x0 ; y/ . Using

Chapter 2. From Funk to Hilbert geometry

45

Formula (4.1), we compute that this is equivalent to kyk2 2e hy; x0 i C e 2 kx0 k2 .1 e /2 : This set describes a Euclidean ball with center z0 D e x0 and Euclidean radius r D .1 e /. We deduce the following “local-implies-global” property of Funk metrics. The meaning of the statement is clear, and it follows directly from Proposition 5.1. Corollary 5.2. We can reconstruct the boundary @ of from the local geometry at any point of . Corollary 5.3. For any points x and x 0 in a convex domain equipped with its Funk metric and for any two positive real numbers ı and ı 0 , the forward balls B C .x; ı/ and B C .x 0 ; ı 0 / are either homothetic or a translation of each other. Proof. This follows from Proposition 5.1 and the fact that the set of Euclidean transformations which are either homotheties or translations form a group (sometimes called the the group of dilations, see e.g. [6]). Remark 5.4. The previous corollary also holds for backward balls B .x; ı/ of small enough radii. Remarks 5.5. In the case where the convex set is unbounded, the forward and backward open balls of its Funk metric are always noncompact. If is bounded. then for any x 2 and for large enough we have B .x; / D . This follows from Proposition 3.3. In particular, the closed backwards balls are not compact for large radii. The forward open balls are geodesically convex if and only if is strictly convex. Remark 5.6. The property for a weak metric on a subset of Rn to have all the right spheres homothetic is also shared by the Minkowski weak metrics on Rn . Indeed, it is easy to see that in a Minkowski weak metric, any two right open balls are homothetic. (Any two right spheres of the same radius are translates of each other, and it is easy to see from the definition that any two spheres centered at the same point are homothetic, the center of the homothety being the center of the balls.) Thus, Minkowski weak metrics share with the Funk weak metrics the property stated in Proposition 5.1. Busemann proved that in the setting of Desarguesian spaces, these are the only examples of spaces satisfying this property (see the definition of a Desarguesian space and the statement of this result in Section 6 of Chapter 1 in this volume ([19])). We state this as the following: Theorem 5.7 (Busemann [3]). A Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space.

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6 On the topology of the Funk metric Proposition 6.1. The topology induced by the Funk or reverse Funk metric in a bounded convex domain in Rn coincides with the Euclidean topology in that domain. Proof. The proof consists in comparing the balls in the Euclidean and the Funk (or reverse Funk) geometries. Let us fix a point x in . Then there exists 0 < x ƒx < 1 such that for any 2 @ we have x j xj ƒx : If we denote by B C .x; / the forward ball with center x and radius in the Funk metric, then Proposition 5.1 implies that y 2 @B C .x; / H) .1 e /x jy xj .1 e /ƒx : In other words, if EB.x; ı/ denotes the Euclidean ball with center x and radius ı, then B.x; .1 e /x / B C .x; / EB.x; .1 e /ƒx /:

E

(6.1)

This implies that the families of balls B C .x; / and EB.x; ı/ are sub-bases for the same topology. For the backward balls B .x; /, the second part of Proposition 5.1 implies the following y 2 @B .x; / H) .e 1/x jy xj .e 1/ƒx ; provided .e 1/ 1. This implies that for log.2/ we have B.x; .e 1/x / B .x; / EB.x; .e 1/ƒx /;

E

and therefore the family of backward balls B .x; / also generates the Euclidean topology.

For general convex domains, bounded or not, we have the following weaker result on the topology: Proposition 6.2. For any convex domain in Rn , F is a continuous functions on . Proof. We first consider the case where is bounded. Suppose first that x and y are distinct points in and let xn , yn be sequences in converging to x and y respectively. Taking subsequences if necessary, we may assume that xn ¤ yn for all n. Then an D a .xn ; yn / is well defined and this sequence converges to a D a .x; y/.

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Chapter 2. From Funk to Hilbert geometry

Since a ¤ y we have

jxn an j lim F .xn ; yn / D lim log n!1 n!1 jyn an j jx aj D log jy aj D F .x; y/:

Assume now that x D y and let xn ; yn 2 be sequences converging to x such that xn ¤ yn for all n. We have jxn an j F .xn ; yn / D log jyn an j j.yn an / C .xn yn /j D log jyn an j jxn yn j log 1 C : jyn an j Since yn 2 converges to a point x in , we have ı D [email protected] jyn bj1 < 1. We then have F .xn ; yn / log .1 C ıjxn yn j/ ! 0; since jxn yn j ! 0. If is unbounded, we set R D \ EB.x; R/ where EB.x; R/ is the Euclidean ball of radius R centered at the origin. It is easy to check that FR converges uniformly to F on every compact subset of as R ! 1. The continuity of F follows therefore from the proof for bounded convex domains. For the next result we need some more definitions: Definition 6.3. Let ı be a weak metric defined on a set X . A sequence fxk g in X is forward bounded if sup ı.xk ; xm / < 1 where the supremum is taken over all pairs k, m satisfying m k. Note that this definition corresponds to the usual notion in the case of a usual (symmetric) metric space. We then say that the weak metric space .X; ı/ is forward proper, or forward boundedly compact if every forward bounded sequence has a converging subsequence. The sequence fxk g is forward Cauchy if lim sup ı.xk ; xm / D 0:

k!1 mk

The weak metric space .X; ı/ is forward complete if every forward Cauchy sequence converges. We define backward properness and backward completeness in a similar way.

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Proposition 6.4. The Funk metric in a convex domain Rn is forward proper (and in particular forward complete) if and only if is bounded. The Funk metric is never backward complete. Proof. For a convex domain, the ball inclusions (6.1) immediately imply that forward complete balls are relatively compact; this implies forward properness. If is unbounded, then it contains a ray and such a ray contains a divergent sequence fxk g such that F .xk ; xm / D 0 for any m k therefore F is not complete. To prove that the Funk metric is never backward complete, we consider an affine segment Œa; b 2 Rn with a ¤ b and such that Œa; b \ @ D fa; bg. Set xk D b C k1 .a b/. If m k, then F .xk ; xm / D log

jxm aj ; jxk aj

which converges to 0 as k ! 1. Since the sequence fxk g has no limit in , we conclude that F is not backward complete. Remark 6.5. The previous proposition also says that in a bounded convex domain, the Funk metric is forward complete and the reverse Funk metric is not.

7 The triangle inequality and geodesics In this section, we prove the triangle inequality for the Funk metric and give a necessary and sufficient condition for the equality case. We also describe all the geodesics of this metric.

7.1 On the triangle inequality Theorem 7.1. If x, y and z are three points in a proper convex domain , then the triangle inequality (7.1) F .x; y/ C F .y; z/ F .x; z/ holds. Furthermore we have equality F .x; y/ C F .y; z/ D F .x; z/ if and only if the three points Q (7.2) a .x; y/; a .y; z/; a .x; z/ 2 @ are aligned in RP n . Before proving this theorem, let us first recall a few additional definitions from convex geometry: Let Rn be a convex domain. Then it is known that its closure x is also convex. A convex subset D x is a face of x if for any x; y 2 D and any

Chapter 2. From Funk to Hilbert geometry

49

0 < < 1 we have .1 /x C y 2 D H) Œx; y D: x are also considered to be faces. A face D x is called proper if The empty set and x D ¤ and D 6D ;. A face D is said to be exposed if there is a supporting hyperplane x Recall that a support hyperplane is a hyperplane H H for such that D D H \ . that meets @ and H \ D ;. It is easy to prove that every proper face is contained in an exposed face. In fact every maximal proper face is exposed. x is an exposed point of if fxg is an exposed face, that is, if there A point x 2 @ x D fxg. If is bounded, then x is the exists a hyperplane H Rn such that H \ closure of the convex hull of its exposed points (Straszewicz’s Theorem). x is an extreme point if x n fxg is still a convex set. Such a point A point x 2 x is the convex hull of its belongs to the boundary @ and if is bounded, then extreme points (Krein–Milman’s Theorem). Every exposed point is an extreme point, but the converse does not hold in general. The following result immediately follows from the definitions: Lemma 7.2. The following are equivalent conditions for a convex domain Rn : (i) Every boundary point is a extreme point. (ii) Every boundary point is an exposed point. (iii) The boundary @ does not contain any non-trivial segment. If one of these conditions holds, then is said to be strictly convex. The following result will play an important role in the proof of Theorem 7.1: Lemma 7.3. Let be bounded convex domain and x, y, z three points in . Then the following are equivalent: Q such that (a) There exists a proper face D @ a .x; y/; a .y; z/; a .x; z/ 2 D: (b) The three points a .x; y/; a .y; z/ and a .x; z/ are aligned in RP n . Proof. Let us set a D a .x; y/, b D a .y; z/ and c D a .x; z/. If x, y and z are aligned, then a D b D c. Otherwise, a, b and c belong to the 2-plane … containing x, y, z. Therefore if a, b, c belong to a proper face D, then those three points are contained in the interval … \ D. This proves the implication (a) H) (b) The converse implication (b) H) (a) is obvious. Proof of Theorem 7.1. We first consider F .x; z/ D 0. In this case the inequality (7.1) is trivial and we have c 2 H1 . We then have equality in (7.1) if and only if F .x; y/ D F .y; z/ D 0, which is equivalent to a 2 H1 and b 2 H1 (the points a and b are as in the proof of the previous lemma). It then follows from Lemma 7.3 that a, b, c lie on some line (at infinity).

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We now consider the case F .x; z/ > 0, that is, c 62 H1 . Choose a supporting functional h at the point c (that is, h.c/ D 1). We then have from Corollary 2.5: 1 h.x/ F .x; z/ D log 1 h.z/ 1 h.y/ 1 h.x/ C log D log 1 h.y/ 1 h.z/ F .x; y/ C F .y; z/: This proves the triangle inequality. Using again Corollary 2.5, we see that we have equality if and only if 1 h.x/ F .x; y/ D log 1 h.y/ and

1 h.y/ F .y; z/ D log ; 1 h.z/

and this holds if and only if one of the following cases holds: Case 1. We have a 62 H1 and b 62 H1 . In that case, h.a/ D h.b/ D 1 D h.c/. The three points a, b, c belong to the face D D @ \ fh D 1g and we conclude by Lemma 7.3 that a, b, c lie on some line . Case 2. We have a 2 H1 and b 62 H1 . In that case h.b/ D 1 D h.c/ and h.x/ D h.y/. This implies that the line through x and y is parallel to the hyperplane fh D 1g and therefore the point Q z y/ \ H1 @ a 2 R.x; belongs to the support hyperplane fh D 1g. Since h.b/ D 1, the three points a, b, c Q \ fh D 1g and we conclude by Lemma 7.3. belong to the face D D @ Case 3. We have a 62 H1 and b 2 H1 . The argument is the same as in Case 2. To complete the proof, we need to discuss the case a 2 H1 and b 2 H1 . In this case, we would have h.x/ D h.y/ and h.y/ D h.z/ and this is not possible. Indeed, we have c D x C .z x/ for some and the equality h.z/ D h.x/ would lead to the contradiction 1 D h.c/ D h.x C .z x// D h.x/ C .h.z/ h.x// D h.x/ < 1: We thus proved in all cases that the equality holds in (7.1) if and only if the points a, b and c are aligned in RP n . Corollary 7.4. Let x and z be two points in a proper convex domain Rn . Suppose that a .x; z/ 2 @ is an exposed point. Then for any point y 62 Œx; z we have F .x; z/ < F .x; y/ C F .y; z/.

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7.2 Geodesics and convexity in Funk geometry We now describe geodesics in Funk geometry. Let us start with a few definitions. Definitions 7.1. A path in a weak metric space .X; d / is a continuous map W I ! X , where I is an interval of the real line. The length of path W Œa; b ! X is defined as Length./ D sup

N 1 X

d..ti /; .tiC1 //;

iD0

where the supremum is taken over all subdivisions a D t0 < t1 < < tN D b. Note that in the case where the weak metric d is non-symmetric, the order of the arguments is important. The path W Œa; b ! X is a geodesic if d..a/; .b// D Length. /. The weak metric space .X; d / is said to be a weak geodesic metric space if there exists a geodesic path connecting any pair of points. It is said to be uniquely geodesic if this geodesic path is unique up to reparametrization. A subset A X is said to be geodesically convex if given any two points in A, any geodesic path joining them is contained in A. Lemma 7.5. The path W Œa; b ! X is geodesic if and only if for any t1 , t2 , t3 in Œa; b satisfying t1 t2 t3 we have d..t1 /; .t3 // D d..t1 /; .t2 // C d..t2 /; .t3 //. The proof is an easy consequence of the definitions. Q Let us now consider a proper convex domain Rn and a proper face D @. For any point p 2 , we denote by x p C v/ \ D ¤ ;g: Cp .D/ D fv 2 Rn j v D 0 or R.p;

(7.3)

x p C v/ is the extended ray through p and p C v in RP . (Recall that Here R.p; the projective space RP n is considered here as a completion of the Euclidean space Rn obtained by adding a hyperplane at infinity; the completion of the ray is then its topological completion.) Observe that Cp .D/ is a cone in Rn at the origin, its translate p C Cp .D/ is the cone over D with vertex at p. We then have the following n

Theorem 7.6. Let W Œ0; 1 ! be a path in a proper convex domain of Rn . Then Q such is a geodesic for the Funk metric in if and only if there exists a face D @ that for any t1 < t2 in Œ0; 1 we have .t2 / .t1 / 2 C.t1 / .D/: In particular if a .x; y/ 2 @ is an exposed point, then there exists a unique (up to reparametrization) geodesic joining x to y, and this geodesic is a parametrization of the affine segment Œx; y. Proof. This is a direct consequence of Theorem 7.1 together with Lemma 7.3 and Lemma 7.5.

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For smooth curves we have the following Corollary 7.7. Let W Œ0; 1 ! be a C 1 path in a proper convex domain of Rn . Q such that Then is a Funk geodesic if and only if there exists a face D @ P .t / 2 C.t / .D/ for any t 2 Œa; b.

D

A typical smooth geodesic in Funk geometry: all tangents to the curve meet the same face D @.

For a subset of Rn , equipped with a (weak) metric F , we have two notions of convexity: affine convexity, saying that for every pair of points in , the affine (or Euclidean) geodesic joining them is contained in , and geodesic convexity, saying that for every pair of points in , the F -geodesic joining them is contained in From the preceding results, we have the following consequence on geodesic convexity of subsets for the Funk metric. Corollary 7.8. Let be a bounded convex domain of Rn . Then the following are equivalent: (1) is strictly convex. (2) is uniquely geodesic for the Funk metric. (3) A subset A is geodesically convex for the Funk metric if and only if A is affinely convex. (4) The forward open balls in are geodesically convex with respect to the Funk metric F . Proof. (1) H) (2) immediately follows from Theorem 7.6. (2) H) (3) is obvious and (3) H) (4) follows from Proposition 5.1.

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53

q

x p y

a

b

We now prove (4) H) (1) by contraposition. Let us assume that is not strictly convex, so that there exists a non-trivial segment Œa; b @. On can then find a supporting functional h for such that h.a/ D h.b/ D 1. Let us chose a segment Œx; y such that a .x; y/ 2 Œa; b and another segment Œp; q such that p 2 Œx; y and h.q/ D h.p/. We also assume x ¤ p ¤ y and F .p; q/ > ı WD F .p; x/ C F .p; y/. Observe that we then have h.x/ < h.p/ D h.q/ < h.y/. If a .x; q/ 2 Œa; b and a .q; y/ 2 Œa; b, then the proof is finished since in this case h.x/ h.q/ h.x/ D log D F .x; q/ C F .q; y/: F .x; y/ D log h.y/ h.q/ h.y/ Since x; y 2 B C .p; ı/ while q 62 B C .p; ı/, we conclude that the forward ball B C .p; ı/ is not geodesically convex. If a .x; q/ 62 Œa; b or a .q; y/ 62 Œa; b, we let c D a .x; y/ and we consider the Euclidean homothety f W Rn ! Rn centered at c with dilation factor < 1. Let p 0 D f .p/, q 0 D f .q/; x 0 D f .x/; y 0 D f .y/ and q 0 D f .q/. It is now clear that if > 0 is small enough, then a .x 0 ; q 0 / 2 Œa; b and a .q 0 ; y 0 / 2 Œa; b. It is also clear that one can find a number ı 0 such that x 0 ; y 0 2 B C .p 0 ; ı 0 / while q 0 62 B C .p 0 ; ı 0 /. The previous argument shows then that F .x 0 ; y 0 / D F .x 0 ; q 0 / C F .q 0 ; y 0 / and therefore B C .p 0 ; ı 0 / is not geodesically convex. Remark 7.9. Note the formal analogy between Corollary 7.8 and the corresponding result concerning the geodesic segments of a Minkowski metric on Rn : if the unit ball of a Minkowski metric is strictly convex, then the only geodesic segments of this metric are the affine segments.

8 Nearest points in Funk geometry Let Rn be a convex set equipped with its Funk metric F . Definition 8.1. Let x be a point in and let A be a subset of . A point y in A is said to be a nearest point, or a foot (in Buseman’s terminology), for x on A if F .x; y/ D F .x; A/ WD inf F .x; z/: z2A

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It is clear from the continuity of the function y 7! F .x; y/ that for any closed non-empty subset A and any x 2 , there exists a nearest point y 2 A. This point need not be unique in general. Proposition 8.2. For a proper convex domain Rn , the following properties are equivalent: a) is strictly convex. b) For any closed convex subset A and for any x 2 , there is a unique nearest point y 2 A. Proof. let x be a point in and assume r D F .x; A/ > 0. Suppose that y and z are two nearest points of A for x. For any point w on the segment Œy; z we have F .x; w/ r because the closed ball BxC .x; r/ is convex. Since A is also assumed to be convex, we have w 2 A and therefore F .x; w/ r. We conclude that F .x; w/ D r, that is, w 2 @BxC .x; r/. From Proposition 5.1, we know that if is strictly convex, then BxC .x; r/ is also strictly convex and we conclude that y D z. It follows that we have a unique nearest point on A for x. This proves (a) H) (b). To prove (b) H) (a), we assume by contraposition that is strictly convex. Again from Proposition 5.1, we know that the forward ball BxC .x; r/ is not strictly convex. In particular @B C .x; r/ contains a non trivial segment A D Œy; z. Any point in the convex set A is a nearest point to x and this completes the proof. Proposition 8.3. Let A be an affinely convex closed subset of a proper convex domain Rn and let x 2 n A. A point y 2 A is a nearest point in A for x if and only if either F .x; y/ D 0 or there exists a hyperplane … Rn which contains y, which separates A and x and which is parallel to a support hyperplane H for at a D a .x; y/. Proof. We assume F .x; y/ > 0 (otherwise, there is nothing to prove). First, suppose there exists a hyperplane … Rn containing y and separating A from x and which is parallel to a support hyperplane H for at a. Let h be the corresponding supporting functional. Then we have, from our hypothesis, h.x/ < h.y/ D inf h.z/: z2A

From Proposition 2.3 and Corollary 2.5 we then have 1 h.x/ 1 h.x/ D inf log F .x; A/; F .x; y/ D log z2A 1 h.y/ 1 h.z/ therefore y is a nearest point on A for x. To prove the converse, we assume that y is a nearest point on A for x. Set r D F .x; y/ D F .x; A/, then, by definition, the forward open ball B C .x; r/ and the set A are disjoint. Since both sets are affinely convex, there exists a hyperplane … that separates them. Note that … is then a support hyperplane at y for the ball B C .x; r/.

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We conclude from Proposition 5.1 that … is parallel to a support hyperplane H for at a. There are several possible notions of perpendicularity in metric spaces. The following definition is due to Busemann (see [2], page 103). Definition 8.4 (Perpendicularity). Let A be a subset of and p a point in A. A geodesic W I ! is said to be perpendicular to A at p if the following two properties hold: (1) p D .t0 / for some t0 2 I , (2) for every t 2 I , p is a nearest point for .t / on A. From the previous results we have the following Corollary 8.5. Let x be a point in a convex domain and a 2 @ be a boundary point. If … Rn is a hyperplane containing x, then the ray Œx; a/ is perpendicular to … \ if and only if … is parallel to a support hyperplane Ha of at a. If b 2 @ is another boundary point, then the line .a; b/ is perpendicular to …\ if and only if … \ .a; b/ ¤ ; and … is parallel to both a support hyperplane Ha at a and a support hyperplane Hb at b.

9 The infinitesimal Funk distance In this section, we consider a convex domain Rn and a point p in . We define a weak distance ˆp D ˆ;p on Rn as the limit ˆp .x; y/ D lim

t&0

F;p .p C tx; p C ty/ : t

Theorem 9.1. The weak metric ˆp at a point p in Rn is a Minkowski weak metric in Rn . Its unit ball is the translated domain p D p. Proof. Choose a supporting functional h for and set 1 h.p C tx/ t h.y x/ D log 1 C : 'h .t/ D log 1 h.p C ty/ 1 h.p/ t h.y/ The first two derivatives of this functions are .1 h.p// h.y x/ ; 'h0 .t/ D .1 h.p/ th.x// .1 h.p/ t h.y// and 'h00 .t / D

..1 h.p//.h.x/ C h.y// 2t h.x/h.y// .1 h.p// h.y x/ .1 h.p/ th.x//2 .1 h.p/ t h.y//2

:

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We have in particular 'h0 .0/ D

h.y x/ ; 1 h.p/

and the second derivative is uniformly bounded in some neighborhood of p. More precisely, given a relatively compact neighborhood U Ux of p, one can find a constant C which depends on U but not on h such that j'h00 .t/j C; for any x; y 2 U and jt j 1 and any support function h. We have, from Taylor’s formula, h.y x/ 'h .t/ D t C t 2 .x; y; h/; 1 h.p/ where j.x; y; h/j C . Using Corollary 2.6 we have F;p .p C tx; p C ty/ D sup 'h .t / D sup h

h

t h.y x/ C O.t 2 /; 1 h.p/

where the supremum is taken over the set S of all support functions for . Therefore ˆp .x; y/ D sup

h2S

h.y x/ : 1 h.p/

We then see that ˆp .x; y/ is weak Minkowski distance (see Chapter 1 in this Handbook). ˆp .0; y/ 1 () sup

h2S

h.y/ 1 1 h.p/

() h.y/ 1 h.p/ for all support functions h of () h.p C y/ 1 for all h x p; () y 2 this means that the unit ball of ˆp is the translate of by p. Remark 9.2. The Funk metric of a convex domain is in fact Finslerian, and the previous theorem means that the Finslerian unit ball at any point p coincides with the domain itself with the point p as its center. This is why the Funk metric was termed tautological in [17]. The Finslerian approach to Funk geometry is developed in [24].

10 Isometries It is clear from its definition that the Funk metric is invariant under affine transformation. Conversely, we have the following:

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57

Proposition 10.1. Let 1 and 2 be two bounded convex domains in Rn . Assume that there exists a Funk isometry f W U1 ! U2 , where Ui is an open convex subset of i , i D 1; 2. If 2 is strictly convex, then f is the restriction of a global affine map of Rn that maps 1 to 2 . Proof. Let x and y be two distinct points in U1 . Then, for any z 2 Œx; y, we have F2 .f .x/; f .y// F2 .f .x/; f .z// F2 .f .z/; f .y// D F1 .x; y/ F1 .x; z/ F1 .z; y/ D 0: Since 2 is assumed to be strictly convex, this implies that f .z/ belongs to the line through f .x/ and f .y/. We can thus define the real numbers t and s by z D x C t .y x/

and

f .z/ D f .x/ C s.f .y/ f .x//:

It now follows from Corollary 2.8 and the fact that f is an isometry that s D t . We thus have proved that for any x; y 2 U1 and any t 2 Œ0; 1 we have f .x C t .y x// D f .x/ C t .f .y/ f .x//:

(10.1)

This relation easily implies that f is the restriction of a global affine mapping. We immediately deduce the following: Corollary 10.2. The group of Funk isometries of a strictly convex bounded domains Rn coincides with the subgroup of the affine group of Rn leaving invariant. Remark 10.3. The conclusion of the corollary may fail for unbounded domains. For instance, if is the upper half plane fx2 > 0g in R2 , then F .x; y/ D maxf0; log.x2 =y2 /g and any map f W ! of the type f .x1 ; x2 / D .ax1 C b; .x2 /, where a ¤ 0 and W R ! R is arbitrary, is an isometry.

11 A projective viewpoint on Funk geometry In this section we consider the following generalization of Funk geometry: We say that a subset U RP n is convex if it does not contain any full projective line and if the intersection of any projective line L RP n with U is a connected set. If U and are connected domains in RP n with U , and if x, y are two distinct points in . We denote by a .x; y/ 2 @ and ! .x; y/ 2 @U the boundary points on the line L through x and y appearing in the order a, y, x, !.

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Definition 11.1. The relative Funk metric for U is defined by F;U .x; x/ D 0 and by jy !j jx aj ; F;U .x; y/ D log jx !j jy aj if x ¤ y. The relative Funk metric is a projective weak metric, it is invariant under projective transformations in the sense that if f W RP n ! RP n is a projective transformation, then F;U .x; y/ D Ff ./;f .U / .f .x/; f .y//: Observe also that if U Rn is a proper convex domain, then F;U .x; y/ D F .x; y/ C rFU .x; y/: Lemma 11.2. In the case U D Rn , we have F;U .x; y/ D F .x; y/: Proof. If U D Rn , then its boundary is the hyperplane at infinity H1 and thus jy!j D 1 for any x; y 2 . jx!j Recall that there is no preferred hyperplane in projective space. Therefore, the classical Funk geometry is a special case of the relative Funk geometry where the englobing domain U RP n is the complement of a hyperplane. Such a set U is sometimes called an affine patch.

12 Hilbert geometry Definition 12.1. The Hilbert metric in a proper convex domain Rn is defined as 1 .F .x; y/ C rF .y; x// ; 2 where is considered as a subset of an affine patch U RP n . H .x; y/ D

This metric is a projective weak metric. Note that for x ¤ y we have jy bj jx aj 1 ; H .x; y/ D log 2 jx bj jy aj where a D a .x; y/ and b D a .y; x/. The expression inside the logarithm is the cross ratio of the points b, x, y, a, therefore the Hilbert metric is invariant by projective transformations.

Chapter 2. From Funk to Hilbert geometry

59

a...r................... .... ............ ... .......... . . . . . . . . ... . . . . . . . . . . . . . . . . ... ...... . . . . . . . . . . . . r y .. . .. .... .. ... .... ..... . .. .... .. ... .. . ... r . ..... x .. ...... ... .......r .... . . . .......... . .. b ....................................... Figure 2. The Hilbert metric.

Note also that the Hilbert metric coincides with half of the relative Funk metric of the domain with respect to itself: 1 F; .x; y/; 2 In the case where is the Euclidean unit ball, the Hilbert distance coincides with the Klein model (also called the Beltrami–Cayley–Klein model) of hyperbolic space. We refer to Sections 2.3–2.6 in [25] for a nice introduction to Klein’s model. In the case of a convex polytope defined by the linear inequalities j .x/ < sj , 1 j k, we have si j .y/ sj j .x/ 1 log : H .x; y/ D max si j .x/ sj j .y/ 1i;j k 2 H .x; y/ D

Applying our investigation on Funk geometry immediately gives a number of results on Hilbert geometry. In particular, applying Proposition 2.2 we get: Proposition 12.2. The Hilbert metric in a convex domain ¤ Rn satisfies the following properties: (a) H .x; y/ 0 and H .x; x/ D 0 for all x; y 2 . (b) H .x; z/ H .x; y/ C H .y; z/ for all x; y; z 2 . (c) H is projective, that is, H .x; z/ D H .x; y/ C H .y; z/ whenever z is a point on the affine segment Œx; y. (d) The weak metric H is symmetric, that is, H .x; y/ D H .y; x/ for any x and y. (e) The weak metric H is separating, that is, x ¤ y H) H .x; y/ > 0, if and only if the domain does not contain any affine line. (f) The weak metric H is unbounded. The proof of this proposition easily follows from the definitions and from Proposition 2.2.

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Proposition 12.3. If the convex domain ¤ Rn does not contain any affine line, then H is a metric in the classical sense. Furthermore, it is complete. A convex domain which does not contain any affine line is called a sharp convex domain. From Theorem 7.1, we deduce the following necessary and sufficient condition for the equality case in the triangle inequality: Theorem 12.4. Let x, y and z be three points in a proper convex domain . We have F .x; y/ C F .y; z/ D F .x; z/ if and only if both triples of boundary points a .x; y/; a .y; z/; a .x; z/ and a .y; x/; a .z; y/; a .z; x/ are aligned in RP n . From Theorem 7.6, we obtain: Theorem 12.5. Let W Œ0; 1 ! be a path in a sharp convex domain of Rn . Then is a geodesic for the Hilbert metric in if and only if there exist two faces D ; D C Q such that for any t1 < t2 in Œ0; 1 we have .t2 / .t1 / 2 C.t / .D C / and @ 1 .t1 / .t2 / 2 C.t2 / .D /. Recall that Cp .D/ is the cone at p on the face D, see Equation (7.3). If is smooth, then it is geodesic if and only if .t P / 2 C.t / .D C / and P .t / 2 C.t / .D / for any t 2 Œ0; 1. We then have the following characterization of smooth geodesics in Hilbert geometry which we formulate only for bounded domains for convenience: Corollary 12.6. Let W Œ0; 1 ! be a path of class C 1 in a bounded convex domain of Rn . Then is a geodesic for the Hilbert metric in if and only if there exist two Q such that for any t , the tangent line to the curve at t meets the faces D ; D C @ boundary @ on D C [ D . Corollary 12.7 (compare [10]). Assume that is strictly convex, or more generally that all but possibly one of its proper faces are reduced to points. Then the Hilbert geometry in is uniquely geodesic.

13 Related questions In this section we briefly discuss some recent developments related to the idea of the Funk distance. Yamada recently introduced what he called the Weil–Petersson Funk metric on Teichmüller space (see [27], and see [15] in this volume). The definition is analogous to one of the definitions of the Funk metric, using in an essential way the noncompleteness of the Weil–Petersson metric on Teichmüller space. One can wonder whether there are Funk-like metrics associated to other interesting known symmetric

Chapter 2. From Funk to Hilbert geometry

61

metrics. This geometry bears some analogies with Thurston’s metric on Teichmüller space and the Funk metric. One can ask for a study of the Thurston metric which parallels the study of the Funk metric (that is, study its balls, its convexity properties, orthogonality and projections, etc.). It should also be of interest to study the geometric properties of the reverse Funk metric, that is, the metric rF on an open convex set defined by rF .x; y/ D F .y; x/. Let us recall that the reverse Funk metric is not equivalent to the Funk metric in any reasonable sense, see Remark 6.5. This is also related to the fact that the forward and backward open balls at some point can be very different, as we already noticed. We note in this respect that the reverse metrics of the Thurston weak metric and of the Weil–Petersson Funk weak metric that we mentioned are also very poorly understood. Finally, there is another symmetrization of the Funk metric, besides the Hilbert metric, namely, its max-symmetrization, defined as S.x; y/ D maxf.F .x; y/; F .y; x/g; and it should be interesting to study its properties. Note that the max-symmetrization of the Thurston metric is an important metric on Teichmüller space, known as the length spectrum metric.

A Menelaus’ Theorem An elementary proof of the triangle inequality for the Funk metric is given by Zaustinsky in [28], and it is recalled in the next appendix. This proof is based on the classical Menelaus’ Theorem.2 For the convenience of the reader, we give a statement and a proof of this result in the present appendix. To state Menelaus’ Theorem, we recall the notion of division ratio of three aligned points. Consider three points A, B, P in Rn with A ¤ B. Then P belongs to the line through A and B if and only if P D tB C .1 t /A for some uniquely defined t 2 R. The number t is called the division ratio or the affine ratio of P relative to . The division ratio is invariant under any affine B and A. We denote it by t D AP AB transformation. Note that if both A ¤ B and A ¤ P , then AP Dt AB

()

PB t 1 D : PA t

For instance P is the midpoint of ŒB; A if and only if AP D 12 or, equivalently, AB PB D 1. Note that the sign is an important component of the division ratio, and in PA 2 This theorem, in the Euclidean and in the spherical case, is quoted by Ptolemy (2nd century A.D) and it is due to Menelaus (second century A.D.) Its proof is contained in Menelaus’ Spherics. No Greek version of Menelaus’ Spherics survived, but there are Arabic versions; cf. the forthcoming English edition [21] from the Arabic original of al-HarawNı (10th century).

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fact we have jP Aj AP D˙ AB jB Aj with a minus sign if and only if A lies between B and P . Proposition A.1 (Menelaus’ Theorem). Let ABC be a non-degenerate Euclidean triangle and let A0 , B 0 , C 0 be three arbitrary points on the lines containing the sides BC , AC , AB. Assume that A0 ¤ C , B 0 ¤ A and C 0 ¤ B. Then, the points A0 , B 0 , C 0 are aligned if and only if we have A0 B B 0 C C 0 A D C1: A0 C B 0 A C 0 B

A

B C

C B

A

Proof. Although a purely geometric proof is possible, it is somewhat delicate to correctly handle the signs of the division ratios throughout the arguments. We follow below a more algebraic approach. It will be convenient to assume that A, B and C are points in Rn with n 3 and to assume that the origin 0 2 Rn does not belong to the plane containing the three points A, B, C . By hypothesis, the point C 0 lies on the line through A and B, therefore C 0 D A C .1 /B;

with

1 C 0A D : 0 C B

0

AB 1 Likewise, we have A0 D B C .1 /C and B 0 D C C .1 /A with A 0C D 1 B0C 0 0 0 and B 0 A D . Now the point C lies on the line through A and B if and only if

Chapter 2. From Funk to Hilbert geometry

63

there exists 2 R such that C 0 D A0 C .1 /B 0 . We then have C 0 D .B C .1 /C / C .1 /.C C .1 /A/ D .1 /.1 /A C B C ..1 / C .1 //C D A C .1 /B: Our hypothesis implies that A, B, C correspond to three linearly independent vectors in Rn , therefore the latter identity implies

D .1 /.1 /;

.1 / D

and

..1 / C .1 // D 0:

Thus, .1 /.1 / D D 1 1

and we conclude that C 0 is aligned with A0 and B 0 if and only if . 1/. 1/. 1/ A0 B B 0 C C 0 A D D 1: A0 C B 0 A C 0 B

Remark. Using similar arguments, we can also prove Ceva’s Theorem. Both theorems are dual to each other. Let us recall the statement: Proposition A.2 (Ceva’s Theorem). Let ABC be a non-degenerate Euclidean triangle and let A0 , B 0 , C 0 be three arbitrary points on the lines containing the sides BC , AC , AB such that A0 ¤ C , B 0 ¤ A and C 0 ¤ B. Then, the lines AA0 , BB 0 and C C 0 are concurrent or parallel if and only if we have A0 B B 0 C C 0 A D 1: A0 C B 0 A C 0 B

A B

C

B A

P

C

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Proof. Let us give the main step of the proof of Ceva’s Theorem. Suppose that the lines AA0 , BB 0 and C C 0 meet at a point P . Then we can find r; s and t in R such that P D tA C .1 t /A0 D sB C .1 s/B 0 D rC C .1 r/C 0 : We also have as before A0 D B C .1 /C , B 0 D C C .1 /A and C 0 D

A C .1 /B. This implies P D tA C .1 t /B C .1 t /.1 /C D .1 s/.1 /A C sB C .1 s/C D .1 r/ A C .1 r/.1 /B C rC: By uniqueness of the barycentric coordinates with respect to the triangle ABC , we have t D .1 s/.1 / D .1 r/ ; s D .1 t / D .1 r/.1 /; r D .1 t/.1 / D .1 s/: Therefore, we have . 1/. 1/. 1/ r t s A0 B B 0 C C 0 A D D D 1: A0 C B 0 A C 0 B

s r t We leave it to the reader to discuss the case where the lines AA0 , BB 0 and C C 0 are parallel and to prove the converse direction.

B The classical proof of the triangle inequality for the Funk metric The triangle inequality is proved in Section 7. Note that it also easily follows from Corollary 2.6 (see also [27]), and it is also a consequence of the Finslerian description of the Funk metric (see Chapter 3 ([24]) in this volume). In this appendix, we present the classical proof of the triangle inequality following Zaustinsky [28]. This proof is similar to the original proof of the triangle inequality for the Hilbert distance, as given by D. Hilbert in [11], although the proof in the case of the Hilbert distance is a bit simpler and does not use the Menelaus theorem. We now prove the triangle inequality for the Funk metric following [28] (see p. 85). Let x, y, z be three points in . In view of Property (b) in Proposition 2.2, we may assume that they are not collinear. Let a, b, c, d , e, f be the intersections with @ of the lines xz, yx and zy, using the notation of the figure concerning the order of intersections.

Chapter 2. From Funk to Hilbert geometry

65

p

a

d

y

e

a z

b

x

f

y c

b

From the invariance of the cross ratio from the perspective at p, we have jx a0 j jb 0 y 0 j jx aj jb yj D 0 jy a0 j jb 0 xj jy aj jb xj and

jy 0 a0 j jb 0 zj jy cj jd zj D : jz cj jd yj jz a0 j jb 0 y 0 j

Multiplying both sides of these two equations, we get jx a0 j jb 0 zj jb xj jd yj jx aj jy cj D : jy aj jz cj jz a0 j jb 0 xj jb yj jd zj The three points b, b 0 and d lie on the sides of the triangle xyz and are aligned, therefore we have by Menelaus’ Theorem (Theorem A.1): jb 0 zj jb xj jd yj D1 jb 0 xj jb yj jd zj and ja0 xj jd xj jf yj D 0 : jd yj jf zj ja zj This gives jx a0 j jx aj jx aj jy cj D ; 0 ja cj jz cj jz a j jz aj

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and the inequality is strict unless a D a0 . This inequality is equivalent to the triangle inequality for the Funk metric3 . Acknowledgement. The first author is partially supported by the French ANR project FINSLER.

References [1]

H. Busemann, Local metric geometry. Trans. Amer. Math. Soc. 56 (1944), 200–274.

[2]

H. Busemann, The geometry of geodesics. Academic Press, New York 1955; reprinted by Dover in 2005.

[3]

H. Busemann, Spaces with homothetic spheres. J. Geometry 4 (1974), 175–186.

[4]

H. Busemann, Recent synthetic differential geometry. Ergeb. Math. Grenzgeb. 54, Springer-Verlag, Berlin 1970.

[5]

H. Busemann and P. J. Kelly, Projective geometry and projective metrics. Academic Press, New York 1953.

[6]

H. S. M. Coxeter, Introduction to geometry. John Wiley & Sons, New York 1961.

[7]

H. G. Eggleston, Convexity. Cambridge Tracts in Math. Math. Phys. 47, Cambridge University Press, New York 1958.

[8]

W. Fenchel, Convex cones, sets and functions. Mimeographed lecture notes, Princeton University, 1956.

[9]

P. Funk, Über Geometrien, bei denen die Geraden die Kürzesten sind. Math. Ann. 101 (1929), 226–237.

[10] P. de la Harpe, On Hilbert’s metric for simplices. In Geometric group theory, Vol. 1 London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge 1993. [11] D. Hilbert, Grundlagen der Geometrie. B. G. Teubner, Stuttgart 1899, several later editions revised by the author, and several translations. [12] D. Hilbert, Mathematische Probleme. Göttinger Nachrichten 1900 (1900), 253–297; reprinted in Archiv der Mathematik und Physik (3) 1 (1901), 44–63 and 213–237; English version “Mathematical problems”, translated by M. Winston Newson, Bull. Amer. Math. Soc. 8 (1902), 437– 445 and 478–479; the English translation was also reprinted in “Mathematical developments arising from Hilbert problems”, Proc. Sympos. Pure Math. XXVII, Part 1, F. Browder (ed.), Amer. Math. Soc., Providence, RI, 1974, and in Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436. [13] S. Kobayashi, Intrinsic metrics on complex manifolds. Bull. Amer. Math. Soc. 73 (1967), 347–349 [14] R. D. Nussbaum, Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations. Differential Integral Equations 7 (1994), no. 5–6, 1649–1707. 3 Observe that the argument shows that the inequality is strict for all x, y, z unless @ contains a Euclidean segment; compare with Theorem 7.1.

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[15] K. Ohshika, H. Miyachi and S. Yamada, Weil–Petersson Funk metric on Teichmüller space. In Handbook of Hilbert geometry, European Mathematical Society, Zürich 2014, 339–352. [16] A. Papadopoulos, Hilbert’s fourth problem. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 391–431. [17] A. Papadopoulos and M. Troyanov, Weak Finsler structures and the Funk weak metric. Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 419–437. [18] A. Papadopoulos and M. Troyanov, Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry. Expo. Math. 27 (2009), no. 2, 109–124. [19] A. Papadopoulos and M. Troyanov, Weak Minkowski spaces. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich, 2014, 11–32. [20] A. Papadopoulos and S.Yamada, Funk and Hilbert geometries in spaces of constant curvature. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 353–379. [21] R. Rashed andA. Papadopoulos, Critical edition with English translation and mathematical commentary of Menelaus’ Spherics, based on the Arabic text of al-HarawNı. To appear. [22] R. T. Rockafellar, Convex analysis. Princeton Mathematical Series 28, Princeton University Press, Princeton, NJ, 1970. [23] Z. Shen, Lectures on Finsler geometry. World Scientific, Singapore 2001. [24] M. Troyanov, Funk and Hilbert geometries from the Finslerian viewpoint. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 69–110. [25] W. P. Thurston Three-dimensional geometry and topology. Vol. 1, edited by Silvio Levy, Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ, 1997. [26] F. A. Valentine, Convex sets. McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York 1964. [27] S.Yamada. Convex bodies in Euclidean and Weil–Petersson geometries. Proc. Amer. Math. Soc. 142 (2014), no. 2, 603–616. [28] E. M. Zaustinsky, Spaces with non-symmetric distance. Mem. Amer. Math. Soc. 34 (1959).

Chapter 3

Funk and Hilbert geometries from the Finslerian viewpoint Marc Troyanov

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . 3 The tautological Finsler structure . . . . . . . . . . . . . 4 The Hilbert metric . . . . . . . . . . . . . . . . . . . . . 5 The fundamental tensor . . . . . . . . . . . . . . . . . . 6 Geodesics and the exponential map . . . . . . . . . . . . 7 Projectively flat Finsler metrics . . . . . . . . . . . . . . 8 The Hilbert form . . . . . . . . . . . . . . . . . . . . . . 9 Curvature in Finsler geometry . . . . . . . . . . . . . . . 10 The curvature of projectively flat Finsler metrics . . . . . 11 The flag curvature of the Funk and the Hilbert geometries 12 The Funk–Berwald characterization of Hilbert geometries A On Finsler metrics with constant flag curvature . . . . . . B On the Schwarzian derivative . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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69 72 75 81 83 84 88 92 94 97 100 104 105 106 107

1 Introduction The hyperbolic space Hn is a complete, simply connected Riemannian manifold with constant sectional curvature K D 1. It is unique up to isometry and several concrete models are available and well known. In particular, the Beltrami–Klein model is a realization in the unit ball Bn Rn in which the hyperbolic lines are represented by the affine segments joining pairs of points on the boundary sphere @Bn and the distance between two points p and q in Bn is half of the logarithm of the cross ratio of a, p with b, q, where a and b are the intersections of the line through p and q with @Bn . We refer to [1] for a historical discussion of this model. In 1895, David Hilbert observed that the Beltrami–Klein construction defines a distance in any convex set U Rn , and this metric still has the properties that affine segments are the shortest

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curves connecting points. More precisely, if the point x belongs to the segment Œp; q, then d.p; q/ D d.p; x/ C d.x; q/: (1.1) Metrics satisfying this property are said to be projective and Hilbert’s IVth problem asks to classify and describe all projective metrics in a given domain U Rn , see Chapter 15 of this volume [42]. The Hilbert generalization of Klein’s model to an arbitrary convex domain U Rn is no longer a Riemannian metric, but it is a Finsler metric. This means that the distance between two points p and q in U is the infimum of the length of all smooth curves ˇ W Œ0; 1 ! U joining these two points, where the length is given by an integral of the type Z 1 P //dt: `.ˇ/ D F .ˇ.t /; ˇ.t (1.2) 0

Here, F W T U ! R is a sufficiently regular function called the Lagrangian. Geometrical considerations lead us to assume that F defines a norm on the tangent space at any point of U; in fact it is also useful to consider non-symmetric and possibly degenerate norms (which we call weak Minkowski norms, see [46]). General integrals of the type (1.2) are the very subject of the classical calculus of variations and Finsler geometry is really a daughter of that field. The early contributions (in the period 1900–1920) in Finsler geometry are due to mathematicians working in the calculus of variations1 , in particular Bliss, Underhill, Landsberg, Hamel, Carathéodory and his student Finsler. Funk is the author of a quite famous book on the calculus of variations that contains a rich chapter on Finsler geometry [26]. The name “Finsler geometry” has been proposed (somewhat improperly) by Élie Cartan in 1933. The 1950 article [12] by H. Busemann contains some interesting historical remarks on the early development of the subject, and the 1959 book by H. Rund [51] gives a broad overview of the development of Finsler geometry during its first 50 years; this book is rich in references and historical comments. In 1908, A. Underhill and G. Landsberg introduced a notion of curvature for twodimensional Finsler manifolds that generalizes the classical Gauss curvature of Riemannian surfaces, and in 1926 L. Berwald generalized this construction to higher dimensional Finsler spaces [7], [33], [62]. This invariant is today called the flag curvature and it generalizes the Riemannian sectional curvature. In 1929, Paul Funk proved that the flag curvature of a Hilbert geometry in a (smooth and strongly convex) domain U R2 is constant K D 1 and Berwald extended this result in all dimensions and refined Funk’s investigation in several aspects, leading to the following characterization of Hilbert geometry [8], [24]: Theorem (Funk–Berwald). Let F be a smooth and strongly convex Finsler metric defined in a bounded convex domain U Rn . Suppose that F is complete with 1Although Bernhard Riemann already considered this possible generalization of his geometry in his Habilitation Dissertation, he did not pursue the subject and Finsler geometry did not emerge as a subject before the early twentieth century.

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constant flag curvature K D 1 and that the associated distance d satisfies (1.1), then d is the Hilbert metric in U. The completeness hypothesis means that every geodesic segment can be indefinitely extended in both directions; in other words, every geodesic segment is contained in a line that is isometric to the full real line R. In fact, Funk and Berwald assumed the Finsler metric to be reversible, that is, F .p; / D F .p; /. But they also needed the (unstated) hypothesis that F is forward complete, meaning that every oriented geodesic segment can be extended as a ray. Observe that reversibility together with forward completeness implies the completeness of F , and therefore the way we characterize the Hilbert metrics in the above theorem is slightly stronger than the Funk–Berwald original statement. Note that completeness is necessary: it is possible to locally construct reversible (incomplete) Finsler metrics for which the condition (1.1) holds and which are not restrictions of Hilbert metrics. The Funk–Berwald Theorem is quite remarkable, and it has been an important landmark in Finsler geometry. Our goal in this chapter is to develop all the necessary concepts and tools to explain this statement precisely. The actual proof is given in Section 12. The rest of the chapter is organized as follows. In Section 2 we explain what a Finsler manifold is and give some basic definitions in the subject with a few elementary examples. In Section 3, we introduce a very natural example of a Finsler structure on a convex domain, discovered by Funk, and which we called the tautological Finsler structure. We also compute the distances and the geodesics for the tautological structure. In Section 4, we introduce Hilbert’s Finsler structure as the symmetrization of the tautological structure and compute its distances and geodesics. In Section 5, we introduce the fundamental tensor gij of a Finsler metric (assuming some smoothness and strong convexity hypothesis). In Section 6, we compute the geodesic equations and we introduce the notion of spray and the exponential map. In Section 7, we discuss various characterizations of projectively flat Finsler metrics, that is, Finsler metrics for which the affine lines are geodesics. In Section 8, we introduce the Hilbert differential form and the notion of Hamel potential, which is a tool to compute distances in a general projectively flat Finsler space. In Section 9, we introduce the curvature of a Finsler manifold based on the classical Riemannian curvature of some associated (osculating) Riemannian metric. The curvature of projectively flat Finsler manifolds is computed in Sections 10 and 11. The characterization of Hilbert geometries is given in Section12 and the chapter ends with two appendices: one on further developments of the subject and one on the Schwarzian derivative. We tried to make this chapter as self-contained as possible. The material we present here is essentially built on Berwald’s paper [8], but we have also used a number of more recent sources. In particular we found the books [53], [54] by Z. Shen and [15] by S.-S. Chern and Z. Chen quite useful.

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2 Finsler manifolds We start with the main definition of this chapter: Definition 2.1. 1. A weak Finsler structure on a smooth manifold M is a lower semi-continuous function F W TM ! Œ0; 1 such that for every point x 2 M , the restriction Fx D FjTx M is a weak Minkowski norm, that is, it satisfies the following properties: i) F .x; 1 C 2 / F .x; 1 / C F .x; 2 /, ii) F .x; / D F .x; / for all 0, for any x 2 M and 1 ; 2 2 Tx M . 2. If F W TM ! Œ0; 1/ is finite and continuous and if F .x; / > 0 for any ¤ 0 in the tangent space Tx M , then one says that F is a Finsler structure on M . We shall mostly be interested in Finsler structures, but extending the theory to the case of weak Finsler structures can be useful in some arguments. The notion of weak Finsler structure appears for instance in [19], [29], [44]. Another situation where weak Finsler structures appear naturally is the field of sub-Finsler geometry. The function F itself is called the Lagrangian of the Finsler structure; it is also called the (weak) metric, since it is used to measure the length of a tangent vector. The weak Finsler metric is called reversible if F .x; / is actually a norm, that is, if it satisfies F .x; / D F .x; / for all points x and any tangent vector 2 Tx M . The Finsler metric is said to be of class C k if the restriction of F to the slit tangent bundle TM 0 D f.x; / 2 TM j ¤ 0g TM is a function of class C k . If it is C 1 on TM 0 , then one says F is smooth. Some of the most elementary examples of Finsler structures are the following: Examples 2.2. i) A weak Minkowski norm F0 W Rn ! R defines a weak Finsler structure F on M D Rn by F .x; / D F0 ./: See [46] for a discussion of weak Minkowski norms. This Finsler structure is constant in x and is thus translation-invariant. Such structures are considered to be the flat spaces of Finsler geometry. ii) A Riemannian metric g on the manifold M defines a Finsler structure on M by p F .x; / D gx .; /.

iii) If g is a Riemannian metric on M and 2 1 .M / is a 1-form whose g-norm is everywhere smaller than 1, then one can define a smooth Finsler structure on M by p F .x; / D gx .; / C x ./:

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Such structures are called Randers metrics. iv) If is an arbitrary 1-form on the Riemannian manifold .M; g/, then one can define two new Finsler structures on M by p p F1 .x; / D gx .; / C jx ./j; F2 .x; / D gx .; / C maxf0; x ./g: v) If F is a Finsler structure on M , then the reverse Finsler structure F W TM ! Œ0; 1/ is defined by F .x; / D F .x; /: Additional examples will be given below. A number of concepts from Riemannian geometry naturally extend to Finsler geometry, in particular one defines the length of a smooth curve ˇ W Œ0; 1 ! M as Z 1 P F .ˇ.s/; ˇ.s//dt: `.ˇ/ D 0

We then define the distance dF .x; y/ between two points p and q to be the infimum of the length of all smooth curves ˇ W Œ0; 1 ! M joining these two points (that is, ˇ.0/ D p and ˇ.1/ D q). This distance satisfies the axioms of a weak metric, see [46]. Together with the distance comes the notion of completeness. Definition 2.3. A sequence fxi g M is a forward Cauchy sequence if for any " > 0, there exists an integer N such that d.xi ; xiCk / < " for any i N and k 0. The Finsler manifold .M; F / is said to be forward complete if every forward Cauchy sequence converges. We similarly define backward Cauchy sequence by the condition d.xi Ck ; xi / < ", and the corresponding notion of backward complete. The Finsler manifold .M; F / is a complete Finsler manifold if it is both backward and forward complete. A weak Finsler structure on the manifold M can also be seen as a “field of convex sets” sitting in the tangent bundle TM of that manifold. Indeed, given a Finsler structure F on the manifold M , we define its domain DF TM to be the set of all vectors with finite F -norm. The unit domain DF is the bundle of all tangent unit balls D f.x; / 2 TM j F .x; / < 1g: The unit domain contains the zero section in TM and its restriction x D \ Tx M to each tangent space is a bounded and convex set. It is called the tangent unit ball of F at x, while its boundary x D f 2 Tx M j F .x; / D 1g Tx M is called the indicatrix of F at x. We know from Theorem 3.12 of [46] that the Lagrangian F W TM ! R can be recovered from TM via the formula F .x; / D infft > 0 j 1t 2 g:

(2.1)

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Sometimes, a weak Finsler structure is defined by specifying its unit domain, and the Lagrangian is then obtained from (2.1). Let us give two elementary examples; more can be found in [43]: Example 2.4. Given a bounded open set 0 Rn that contains the origin, one naturally defines a Finsler structure on Rn by parallel transporting 0 . That is, T Rn D Rn Rn is defined as D f.x; / 2 Rn Rn j 2 0 g: The space Rn equipped with this Finsler structure is characterized by the property that its Lagrangian F is invariant with respect to the translations of Rn , i.e. F is independent of the point x: F .x; / D F0 ./: Such a Finsler structure on Rn is of course the same thing as a Minkowski norm, see [46]. Example 2.5. Let F be an arbitrary weak Finsler structure on a manifold M with unit domain TM . If Z W M ! TM is a continuous vector field such that F .x; Z.x// < 1, for any point x (equivalently Z.M / ), a new weak Finsler structure can be defined as Z D f.x; / 2 TM j 2 .x Z.x//g:

(2.2)

Here, x Z.x/ is the translate of x Tx M by the vector Z.x/. The corresponding Lagrangian is given by FZ .x; / D infft > 0 j 1t 2 .x Z.x//g; and for ¤ 0, it is computable from the identity

(2.3)

C Z.x/ D 1: F x; FZ .x; /

(2.4)

This weak Finsler structure FZ is called the Zermelo transform of F with respect to the vector field Z. We end this section with a word on Berwald spaces. Recall first that in Riemannian geometry, the tangent space of the manifold at every point is isometric to a fixed model, which is a Euclidean space. Furthermore, using the Levi-Civita connection, one defines parallel transport along any piecewise C 1 curve, and this parallel transport induces an isometry between the tangent spaces at any point along the curve. In a general Finsler manifold, neither of these facts holds. This motivates the following definition: Definition 2.6. A weak Finsler manifold .M; F / is said to be Berwald if there exists a torsion free linear connection r (called an associated connection) on M whose

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associated parallel transport preserves the Lagrangian F . That is, if W Œ0; 1 ! M is a smooth path connecting the point x D .0/ to y D .1/ and P W Tx M ! Ty M is the associated r-parallel transport, then F .y; P .// D F .x; / for any 2 Tx M . This definition is a slight generalization of the usual definition, compare with [15], Proposition 4.3.3. It is known that the connection associated to a smooth Berwald metric can be chosen to be the Levi-Civita connection of some Riemannian metric [37], [60].

3 The tautological Finsler structure Definition 3.1. Let us consider a proper convex set U Rn . This will be our ground manifold. The tautological weak Finsler structure Ff on U is the Finsler structure for which the unit ball at a point x 2 U is the domain U itself, but with the point x as center. The unit domain of the tautological weak Finsler structure is thus defined as D f.x; / 2 T U j 2 .U x/g T U D U Rn ; and the Lagrangian is given by

® ¯ Ff .x; / D infft > 0 j 2 t .U x/g D inf t > 0 j x C t 2 U :

Equivalently, Ff is given by Ff .x; / D 0 if the ray x C RC is contained in U, and

Ff .x; / > 0 and

xC

2 @U Ff .x; /

otherwise. The convex set U can be recovered from the Lagrangian as follows: U D fz 2 Rn j Ff .x; z x/ < 1g:

(3.1)

By construction, this formula is independent of x. The tautological weak structure has been introduced by Funk in 1929, see [24]. We will often call it the Funk weak metric on U (whence the index f in the notation Ff ). Remark 3.2. If the convex set U contains the origin, then the tautological weak structure Ff is the Zermelo transform of the Minkowski norm with unit ball U for the position vector field Zx D x.

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Example 3.3. Suppose U is the Euclidean unit ball fx 2 Rn j kxk < 1g, where k k is the Euclidean norm. Then 2 x C Ff 2 @U () x C Ff D 1: Rewriting this condition as Ff2 .1 kxk2 / 2F hx; i kk2 D 0; the non-negative root of this quadratic equation is p hx; i C hx; i2 C .1 kxk2 /kk2 ; Ff .x; / D .1 kxk2 /

(3.2)

which is the Lagrangian of the tautological Finsler structure in the Euclidean unit ball. Observe that this is a Randers metric. Example 3.4. Consider a half-space H D fx 2 Rn j h; xi < g Rn where is a non-zero vector and 2 R; here h ; i is the standard scalar product in Rn . We then have D E 2 @H () ; x C D ; xC F F which implies h; i ;0 : (3.3) Ff .x; / D max h; xi We now compute the distance between two points in the tautological Finsler structure: Theorem 3.5. The tautological distance in a proper convex domain U Rn is given by ja pj ; (3.4) %f .p; q/ D log ja qj where a is the intersection of the ray starting at p in the direction of q with the boundary @U: a D @U \ .p C RC .q p// : If the ray is contained in U, then a is considered to be a point at infinity and %f .p; q/ D 0. Definition 3.6. The distance (3.4) is called the Funk metric in U. In his paper [24], Funk introduced the tautological Finsler structure while the distance (3.4) appears as an interesting geometric object in the 1959 memoir [67] by Eugene Zaustinsky, a student of Busemann. We refer to [44], [67] and Chapter 2 of this volume [47] for expositions of the Funk geometry.

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Proof of Theorem 3.5. We follow the proof in [44]. Consider first the special case where U D H is the half-space fx 2 Rn j h; xi < g Rn for some vector ¤ 0. Then the tautological Lagrangian is given by (3.3) and the length of a curve ˇ joining p to q is Z 1 ² ³ P h; ˇ.s/i max 0; dt `.ˇ/ D s h; ˇ.s/i 0 ³ ² Z 1 . h; ˇ.s/i/0 D dt max 0; j h; ˇ.s/ij 0 ² ³ h; pi ; max 0; log h; qi P with equality if and only if h; ˇ.s/i has almost everywhere constant sign. It follows that the tautological distance in the half-space H is given by ²

%f .p; q/ D max 0; log

h; pi h; qi

³

:

(3.5)

We can rewrite this formula in a different way. Suppose that the ray LC starting at p in the direction of q meets the hyperplane @H at a point a. Then h; pi D and h; pi D h; a pi D ja pj h; i; where D

ap . japj

We also have h; qi D ja qj h; i; therefore ja pj : %f .p; q/ D log ja qj

(3.6)

If the ray LC is contained in the half-space U, then %f .p; q/ D 0 and the above formula still holds in the limit sense if we consider the point a to be at infinity. For the general case, we will need two lemmas on general tautological Finsler structures. Lemma 3.7. Let U1 and U2 be two convex domains in Rn . If F1 and F2 are the Lagrangians of the corresponding tautological structures, then U1 U2 () F1 .x; / F2 .x; / for all .x; / 2 T U. Proof. We have indeed F1 .x; / D infft > 0 j 2 t .U1 x/g infft > 0 j 2 t .U2 x/g D F2 .x; /: We also have the following reproducing formula for the tautological Lagrangian.

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Lemma 3.8. Let U be a convex domain in Rn and p; q 2 U. If q D p C t for some 2 Rn , and t 0, then Ff .p; / Ff .q; / D : (3.7) 1 tFf .p; / Proof. If D 0, then there is nothing to prove. Assume ¤ 0 and denote by LpC and C LC q the rays in the direction starting at p and q. We obviously have Lp U if and C only if Lq U. But this means that Ff .p; / D 0 () Ff .q; / D 0; so Equation (3.7) holds in this case. If on the other hand the rays are not contained in U, then we have LpC \ @U D LC q \ @U: We denote by a D a.x; / this intersection point; then by definition of F we have aDqC

DpC : Ff .q; / Ff .p; /

Since q D p C t , we have

(3.8)

1 D t ; Ff .q; / Ff .p; / from which (3.7) follows, since ¤ 0. To complete the proof of Theorem 3.5, we need to compute the tautological distance between two points p and q in a general convex domain U 2 Rn . We first compute the length of the affine segment Œp; q which we parametrize as yx ; ˇ.t / D p C t ; D jy xj where 0 t jy xj. Let us denote as above by a the point LpC \ @U where LpC is the ray with origin p in the direction q. Then Ff .p; / D

1 ; ja pj

and using (3.7) we have P // D Ff .p C t ; / D Ff .ˇ.t /; ˇ.t

1

Ff .p; / 1 japj : D D t ja pj t 1 tFf .p; / 1 japj

The length of ˇ is then Z jqpj dt 1 1 log : D log `.ˇ/ D ja pj t ja pj jq pj ja pj 0

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But q 2 Œp; a, therefore ja pj jq pj D ja qj and we finally have

`.ˇ/ D log This proves that

ja pj : ja qj

ja pj %f .p; q/ log : ja qj In fact we have equality. To see this, choose a supporting hyperplane for U at a, and let H be the corresponding half-space containing U (recall that a hyperplane in Rn is said to support the convex set U if it meets the closure of that set and U is contained in one of the half-space bounded by that hyperplane). Using Lemma 3.7 and Equation (3.6), we obtain ja pj %f .p; q/ log : ja qj The proof of Theorem 3.5 follows now from the two previous inequalities. Remark 3.9. 1. From Equation (3.8), one sees that a p D . Ff .q;/

Ff .p;/

and a q D

Therefore, the Funk distance can also be written as Ff .q; q p/ : %f .p; q/ D log Ff .p; q p/

(3.9)

2. The proof of Theorem 3.5 given here is taken from [44]. Another interesting proof is given in [66]. Proposition 3.10. The tautological distance %f in a proper convex domain U satisfies the following properties. a) The distance %f is projective, that is, for any point z 2 Œp; q we have %f .p; q/ D %f .p; z/ C %f .z; p/: b) %f is invariant under affine transformations. c) %f is forward complete. d) %f is not backward complete. The proof is easy, see also Chapter 2 of this volume [47] for more on Funk geometry. Proposition 3.11. The unit speed linear geodesic starting at p 2 U in the direction 2 Tp U is the path .1 es / ˇp; .s/ D p C (3.10) Ff .p; /

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Proof. Let us define ˇ by (3.10); we have then ap D

; F .p; /

Therefore,

and

%f .p; ˇ.s// D log

a ˇ.s/ D

es : F .p; /

ja pj 1 D log s ja ˇ.s/j e

D s:

We can also prove the proposition using the fact that (3.10) parametrizes an affine segment and is therefore a minimizer for the length. We then only need to check that the curve has unit speed. Indeed we have P ˇ.s/ D

es ; Ff .p; /

and using Equation (3.7), we thus obtain

.1 es / es F pC ; Ff .p; / Ff .p; / Ff .p; / es D s / .1e Ff .p; / 1 F .p; / Ff .p;/ f

P Ff .ˇ.s/; ˇ.s// D

D 1: The tautological Finsler structure Ff in a convex domain U is not reversible. We can thus define the reverse tautological Finsler structure Ff to be the Finsler structure whose Lagrangian is defined as Ff .p; / D Ff .p; /: We then have the following Proposition 3.12. The distance %f associated to the reverse tautological Finsler structure in a proper convex domain U Rn is given by %f .p; q/

jb qj D %f .q; p/ D log ; jb pj

(3.11)

where b is the intersection of the ray starting at q in the direction p with @U. The proof is obvious. Observe in particular that %f is also a projective metric.

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4 The Hilbert metric Definition 4.1. The Hilbert Finsler structure Fh in a convex domain U is the arithmetic symmetrisation of the tautological Finsler structure: 1 .Ff .p; / C Ff .p; //: 2

Fh .x; / D

By construction, the Hilbert Finsler structure is reversible. Its tangent unit ball at a point p 2 U is obtained from the tautological unit ball by the following procedure from convex geometry: first, take the polar dual of .U p/, then symmetrize this convex set and finally take again the polar dual of the result. This procedure is called the harmonic symmetrization of U based at p, see [45] for the details. Example 4.2. Symmetrizing the metric (3.2) we obtain the Hilbert metric in the unit ball Bn : p .1 kxk2 /kk2 C hx; i2 : (4.1) Fh .x; / D .1 kxk2 / Observe that this is a Riemannian metric. We shall prove later on that it has constant sectional curvature K D 1. This metric is the Klein model for hyperbolic geometry. Using the fact that the affine segment joining two points p and q in U has minimal length for both the Funk metric Ff and its reverse Ff , it is easy to prove the following Proposition 4.3. The Hilbert distance %h in a convex domain U Rn is obtained by symmetrizing the Funk distance (3.4) in that domain:

%h .p; q/ D

1 1 ja pj jb qj .%f .p; q/ C %f .q; p// D log : 2 2 ja qj jb pj

b p

@U q

a

Using (3.9), one can also write the Hilbert distance as

%h .p; q/ D

Ff .q; q p/ Ff .p; p q/ 1 : log 2 Ff .p; q p/ Ff .q; p q/

We also have the following properties:

(4.2)

(4.3)

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Proposition 4.4. The Hilbert distance %h in a proper convex domain U satisfies the following a) The distance %h is projective: for any point z 2 Œp; q we have %h .p; q/ D %h .p; z/ C %h .z; p/: b) %h is invariant under projective transformations. c) %h is (both forward and backward) complete. The proof is elementary. For an introduction to Hilbert geometry, we refer to Sections 28 and 50 in [14] and Section 18 in [13]. We now describe the geodesics in Hilbert geometry. Proposition 4.5. The unit speed linear geodesic starting at p 2 U in the direction 2 Tp U is the path ˇp; .s/ D p C '.s/ ; (4.4) where ' is given by '.s/ D

.es es / : Ff .p; /es C Ff .p; /es

(4.5)

Proof. To simplify notation we write F D Ff .p; / and F D Ff .p; /. From (3.8) we have a p D F and b q D F , therefore

1 1 ja pj jb ˇ.s/j 1 C F '.s/ %h .p; ˇ.s// D log D log : 2 ja ˇ.s/j jb pj 2 1 F '.s/ From (4.5), we have .F es C F es / C .F es F es / 1 C F '.s/ D 1 F '.s/ .F es C F es / .F es F es / D

.F C F / e s D e 2s ; .F C F / es

and we conclude that %h .p; ˇ.s// D s: Corollary 4.6. The metric balls in a Hilbert geometry are convex sets. Proof. We see from the previous proposition that the ball of radius r around the point p 2 U is the set of points z 2 U such that er Ff .p; z p/ C er Ff .p; p z/ .er er /; which is a convex set. Note that another proof of this proposition is given in Chapter 2 of this volume [47].

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5 The fundamental tensor Our goal in this section is to write down and study an ordinary differential equation for the geodesics in a Finsler manifold .M; F /. To this aim we need to assume the following condition: Definition 5.1. A Finsler metric F on a smooth manifold M is said to be strongly convex if it is smooth on the slit tangent bundle TM 0 and if the vertical Hessian of F 2 at a point .p; / 2 T 0 M 1 @2 ˇˇ gp; . 1 ; 2 / D F 2 .p; C u1 1 C u2 2 / (5.1) ˇ 2 @u1 @u2 u1 Du2 D0 is positive definite. The bilinear form (5.1) is called the fundamental tensor of the Finsler metric. This condition is called the Legendre–Clebsch condition in the calculus of variations. Geometrically it means that the indicatrix at any point is a hypersurface of strictly positive Gaussian curvature. Classical Finsler geometry has sometimes the reputation of being an “impenetrable forest of tensors”,2 and we shall need to venture a few steps into this wilderness. It will be convenient to work with local coordinates on TM ; more precisely, if U M is the domain of some coordinate system x 1 ; x 2 ; : : : ; x n , then any vector 2 T U can be written as D y i @[email protected] i . The 2n functions on T U given by x 1 ; : : : ; x n ; y 1 ; : : : ; y n are called natural coordinates on T U . The restriction of the Lagrangian on T U is thus given by a function of 2n variables F .x; y/. The Legendre-Clebsch condition states that F .x; y/ is smooth (although C 3 would suffice) on fy ¤ 0g and that the matrix given by 1 @2 F 2 .x; y/ (5.2) gij .x; y/ D 2 @y i @y j is positive definite for any y ¤ 0. The fundamental tensor shares some formal properties with a Riemannian metric, but it is important to remember that it is not defined on U (nor on the manifold M ), but on the slit tangent bundle T 0 U . Manipulating tensors in Finsler geometry needs to be done with care. In general, a tensor on a Finsler manifold .M; F / is a field of multilinear maps on TM which smoothly depends on a point .x; y/ 2 TM 0 (and not only on a point x 2 M as in Riemannian geometry). In the case of the fundamental tensor, note that3 gy .u; v/ D gx;y .u; v/ D gij .x; y/ui v j ; where u D ui .x/ @[email protected] i and v D vj .x/ @[email protected] are elements in Tx M . 2 This comment on the subject goes back to the paper [12] by Busemann. The first sentence of this nice paper is “The term Finsler space evokes in most mathematicians the picture of an impenetrable forest whose entire vegetation consists of tensors.” A quick glance at [51] will probably convince the reader. 3 We use the summation convention: terms with repeated indices are summed from 1 to n.

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Given a point .x; y/ 2 TM 0 , we have a canonical element in Tx M , namely the vector y itself, and we can evaluate a given tensor on that canonical vector. In particular we have the following basic fact. Lemma 5.2. The Lagrangian and the fundamental tensor are related by q q F .x; y/ D gy .y; y/ D gij .x; y/y i y j : Proof. Recall that if f W Rn n f0g ! R is a smooth positively homogenous function of degree r on Rn , that is, f .y/ D r f .y/ for 0, then its partial derivatives are @f positively homogenous functions of degree .r 1/ and r f .y/ D y i @y i (see [46]).

Applying this fact twice to the function y 7! 12 F 2 .x; y/ proves the lemma.

Observe that the same argument also shows that the fundamental tensor gij .x; y/ is 0-homogenous with respect to y. This type of argument plays a central role in Finsler geometry and anyone venturing into the subject will soon become an expert in recognizing how homogeneity is being used (sometimes in a hidden way) in calculations, or she will quit the subject.

6 Geodesics and the exponential map We now consider a curve ˇ W Œa; b ! M of class C 1 in the Finsler manifold .M; F /. Recall that its length is defined as Z b P `.ˇ/ D F .ˇ.s/; ˇ.s//ds: (6.1) a

We also define the energy of the curve ˇ by Z b P E.ˇ/ D F 2 .ˇ.s/; ˇ.s//ds:

(6.2)

a

The following basic inequality between these functionals holds: Lemma 6.1. For any curve ˇ W Œa; b ! M of class C 1 , we have `.ˇ/2 .b a/ E.ˇ/; P with equality if and only if ˇ has constant speed, i.e. t 7! F .ˇ.s/; ˇ.s// is constant. P Proof. Let us denote by f .s/ D F .ˇ.s/; ˇ.s// the speed of the curve. We have by the Cauchy–Schwarz inequality Z b 1=2 Z b 1=2 p Z b p 1 f .s/ds 12 ds v.t /2 ds D .b a/ E.ˇ/: `.ˇ/ D a

a

a

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Furthermore, the equality holds if and only if f .s/ and 1 are collinear as elements of L2 .a; b/, that is, if and only if the speed f .s/ is constant. Corollary 6.2. The curve ˇ W Œa; b ! M is a minimal curve for the energy if and only if it minimizes the length and has constant speed. Definition 6.3. The curve ˇ W Œa; b ! M is a geodesic if it is a critical point of the energy functional. Sometimes the terminology varies, and the term geodesic also designates a critical or a minimal curve for the length functional. Note that our notion imposes the restriction that ˇ has constant speed. From the point of view of calculations, this is an advantage since the length is invariant under forward reparametrization. We now seek to derive the equation satisfied by the geodesics. By the classical theory of the calculus of variations, a curve s 7! ˇ.s/ in some coordinate domain U M is a critical point of the energy functional (6.2) if and only if the following Euler–Lagrange equations @F 2 d @F 2 D ds @y @x

. D 1; : : : ; n/

(6.3)

P hold, where .x.s/; y.s// D .ˇ.s/; ˇ.s//. Using the fundamental tensor, one writes F 2 .x; y/ D gij .x; y/y i y j , therefore @gij i j @F 2 D y y @x @x

(6.4)

and @gij i j @F 2 D y y C gi y i C gj y j : @y @y Observe that the sums gi y i and gj y j coincide. Using the homogeneity of F 2 in y and the fact that F is of class C 3 on y ¤ 0, we obtain @3 F 2 @3 F 2 @gij i i y D y D y i D 0: @y @y @y i @y j @y i @y @y j Therefore, @F 2 D 2gi y i : @y Differentiating this expression with respect to s, we get @gi i dx j dy i @gi i dy j d @F 2 D 2 y y C 2g C 2 : i ds @y @x j ds ds @y j ds

(6.5)

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We have, as before, xP i D y i , we obtain

@gi @y j

y i D 0 (because gi is 0-homogenous), therefore using

d @F 2 @gi D 2 j y i y j C 2gi yP i : ds @y @x

(6.6)

From Equations (6.3), (6.5) and (6.6), we obtain for D 1; : : : ; n, X 1 X @gij @gi i j gi yP i D 2 y y : 2 @x @x j i

i;j

Multiplying this identity by g k .x; y/, where g k gi D ıik , and summing over , one gets 1 X k @gij @gi i j k g 2 j y y : (6.7) yP D 2 @x @x i;j;

Proposition 6.4. The C 2 curve ˇ.s/ 2 .M; F / is a geodesic for the Finsler metric F if only if in local coordinates, we have xR k C ijk xP i xP j D 0;

(6.8)

where ˇ.s/ D .x 1 .s/; : : : ; x n .s// is a local coordinate expression for the curve and

1 @gj @gij @gi D g k C ; 2 @x j @x i @x are the formal Christoffel symbols of F . ijk .x; y/

Proof. A direct calculation shows that Equation (6.8) is equivalent to Equation (6.7) with y k D xP k . Another way to write the geodesic equation is to introduce the functions 1 k .x; y/ y i y j 2 ij 1 @gi @gij D g k 2 j y i y j 4 @x @x @F 1 k @2 F j y ; D g @y @x j @x 2

G k .x; y/ D

so the geodesic equation can be written as yP k C 2G k .x; y/ D 0;

xP k D y k

.k D 1; : : : n/:

(6.9)

The vector field on T U defined by G D yk

@ @ 2G k .x; y/ k k @x @y

(6.10)

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87

is in fact independent on the choice of the coordinates x j on U and is therefore globally defined on the tangent bundle TM . This vector field is called the spray of the Finsler manifold. A significant part of a Finsler geometry is contained in the behavior of its spray (see [53]). Observe that a curve ˇ.s/ 2 M is geodesic if and only if its lift P .ˇ.s/; ˇ.s// 2 TM is an integral curve for the spray G. Observe the rather surprising analogy between the equation (6.8) and the Riemannian geodesic equation. This is due to the fact that the 0-homogeneity of gi;j .x; y/ in y implies that all the non-Riemannian terms in the calculation of the Euler–Lagrange equation end up vanishing. The important difference between the Finslerian and the Riemannian cases lies in the fact that the formal Christoffel symbols ijk are not functions of the coordinates x i only. Equivalently, the spray coefficients are generally not quadratic polynomials in the coordinates y i . In fact, it is known that the spray coefficients G k .x; y/ of a Finsler metric F are quadratic polynomials in the coordinates y i if and only if F is Berwald. On a smooth manifold M with a strongly convex Finsler metric, one can define an exponential map as it is done in Riemannian geometry. Because the coefficients G k .x; y/ of the geodesic equation are Lipschitz continuous, for any point p 2 M and any vector 2 Tp M , there is locally a unique solution to the geodesics equation s 7! .s/ 2 M;

< s < ;

with initial conditions .0/ D p, P .0/ D . Observe that .s/ D .s/;

0;

so if is small enough, then .1/ is well defined and we denote it by expp ./ D .1/: We then have the following Theorem 6.5. The set Op of vectors 2 Tp M for which expp ./ is defined is a neighborhood of 0 2 Tp M . The map expp W Op ! M is of class C 1 in the interior of Op and its differential at 0 is the identity. In particular, the exponential is a diffeomorphism from a neighborhood of 0 2 Tp M to a neighborhood of p in M . If .M; F / is connected and forward complete, then expp is defined on all of Tp M and is a surjective map expp W Tp M ! M: The coefficients G k .x; y/ in the geodesic equation are in general not C 1 at y D 0, therefore the proof of this theorem is more delicate than its Riemannian counterpart. See [6] for the details. Finally note that expp has no reason to be of class C 2 . In fact a result of AkbarZadeh states that expp is a C 2 map near the origin if and only if .M; F / is Berwald.

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7 Projectively flat Finsler metrics Definition 7.1. 1. A Finsler structure F on a convex set U Rn is projectively flat if every affine segment Œp; q U can be parametrized as a geodesic. 2. A Finsler manifold .M; F / is locally projectively flat if each point admits a neighborhood that is isometric to a convex region in Rn with a projectively flat Finsler structure. Examples 7.2. Basic examples are the following. a) Minkowski metrics are obviously projectively flat. b) The Funk and the Hilbert metrics are projectively flat Finsler metrics. In particular the Klein metric (4.1) in Bn is a projectively flat Riemannian metric. c) The canonical metric on the sphere S n RnC1 is locally projectively flat. Indeed, the central projection of the half-sphere S n \fxn < 0g on the hyperplane fxn D 1g with center the origin maps the great circles in S n on affine lines on that hyperplane (in cartography, this map is called the gnomonic representation of the sphere). In formulas, the map Rn ! S n is given by x 7! p.x;1/ 2 , and 1Ckxk

the metric can be written as

p .1 C kxk2 /kk2 hx; i2 : Fx .x; / D .1 C kxk2 /

(7.1)

This is the projective model for the spherical metric. A classic result, due to E. Beltrami, says that a Riemannian manifold is locally projectively flat if and only if it has constant sectional curvature, see Theorem 10.6 below. In Finsler geometry, there are more examples and the Finsler version of Hilbert’s IVth problem is to determine and study all conformally flat (complete) Finsler metrics in a convex domain. A first result on projectively flat Finsler metrics is given in the next proposition: Proposition 7.3. A smooth and strongly convex Finsler metric F on a convex domain U Rn is projectively flat if only if its spray coefficients satisfy G k .x; y/ D P .x; y/ y k for some scalar function P W T U ! R. Proof. The proof is standard, see e.g. [15]. Suppose that the affine segments are geodesics. This means that ˇ.s/ D p C '.s/ satisfy the geodesic equation (6.9) for any p 2 U , 2 U and some (unknown) function '.s/. Along ˇ we have k y k D xP k D '.s/ P ;

k yP k D '.s/ R ;

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89

therefore Equation (6.9) can be written as k C 2'P 2 .s/G k .ˇ.s/; / D 0; '.s/ R

which implies that G k .x; y/ D P .x; y/ y k with P .x; / D

'.0/ R : 2'.0/ P 2

(7.2)

The function P .x; / in the previous proposition is called the projective factor. It can be computed from (7.2) if the geodesics are explicitly known. For instance we have the Proposition 7.4. The projective factor of the tautological Finsler structure Ff in the convex domain U is given by 1 Ff .x; y/: 2 For the Hilbert Finsler structure Fh , we have Pf .x; y/ D

Ph .x; y/ D

1 .Ff .x; y/ Ff .x; y//: 2

Proof. The geodesics for Ff are given by ˇ.s/ D p C '.s/y with '.s/ D

1 .1 es /: Ff .p; /

Therefore, Pf .x; y/ D

'.0/ R 1 D Ff .p; /: 2 2'.0/ P 2

The geodesics for the Hilbert metric Fh .x; y/ D curves ˇ.s/ D p C '.s/y with '.s/ D

1 .F 2 f

.x; y/ C Ff .x; y// are the

.es es / : Ff .p; /es C Ff .p; /es

The derivative of this function is '.s/ P D and '.s/ R D 4

2.Ff .p; / C Ff .p; // ; .Ff .p; /es C Ff .p; /es /2

.Ff .p; / C Ff .p; //.Ff .p; /es Ff .p; /es / : .Ff .p; /es C Ff .p; /es /3

Therefore, Ph .x; y/ D

'.0/ R 1 D .Ff .x; y/ Ff .x; y//: 2 2'.0/ P 2

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It is clear that the distance associated to a projectively flat metric is projective in the sense of Definition 2.1 in [46]. It is also clear that the sum of two projective (weak) distances is again projective. This suggests that one can write down a linear condition on F which is equivalent to projective flatness. This is the content of the next proposition which goes back to the work [31] of G. Hamel. Proposition 7.5. Let F W T U ! R be a smooth Finsler metric on the convex domain U. The following conditions are equivalent. (a) F is projective flat. @F @2 F (b) y k k m m D 0, for 1 m n. @x @x @y 2 @2 F @ F D , for 1 j; m n. (c) @x j @y m @y j @x m @2 F @2 F (d) y k m k D y k k m : @x @y @x @y Proof. The Euler–Lagrange equation for the length functional (6.1) is 0D

d @F @F @2 F @2 F @F k D x P yP k ; @x m dt @y m @x m @x k @y m @y k @y m

and this can be written as @F @2 F @2 F k y P D yk : @x m @x k @y m @y k @y m Now F is projectively flat if and only if x.t / D p C t is a solution (recall that the length is invariant under reparametrization), which is equivalent to yP k D 0. The equivalence (a) , (b) follows. To prove (b) ) (c), we differentiate (b) with respect to y j . This gives Aj m D y k and thus 0D

@2 F @2 F @3 F C D 0; m j @x @y @y j @x m @y j @x k @y m

@2 F @2 F 1 .Aj m Amj / D j m j m ; 2 @x @y @y @x

which is equivalent to (c). (c) ) (d) is obvious. Finally, to prove (d) ) (b), we use the homogeneity of F in y. We have F .x; y/ D @F y k @y k , therefore condition (d) implies 2 2 @F k @ F k @ F D y ; D y @x m @x m @y k @x k @y m

which is equivalent to (b).

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Remark 7.6. In [50] A. Rapcsàk gave a generalization of the previous conditions for the case of a pair of projectively equivalent Finsler metrics, that is, a pair of metrics having the same geodesics up to reparametrization. Example 7.7. We know that the tautological (Funk) Finsler metric Ff in a convex domain U is projectively flat. Using the previous proposition we have more examples: i) the reverse Funk metric Ff .x; / D Ff .x; /, ii) the Hilbert metric Fh .x; / D 12 .Ff .x; / C Ff .x; //, and

iii) the metric Ff .x; / C F0 ./, where F0 is an arbitrary (constant) Minkowski norm are projectively flat. More generally if F1 , F2 are projectively flat, then so is the sum F1 C F2 . Assuming either F1 or F2 to be (forward) complete, the sum is also a (forward) complete solution to Hilbert’s IVth problem. The following result gives a general formula computing the projective factor of a projectively flat Finsler metric: Lemma 7.8. Let F .x; y/ be a smooth projectively flat Finsler metric on some domain in Rn . Then the following equations hold: 2FP D y k and

@F @x k

(7.3)

@F @P @F D P m C F m; m @x @y @y

(7.4)

where P .x; y/ is the projective factor. Proof. Along a geodesic .x.s/; y.s//, we have yP k D 2G k .x; y/ D 2P .x; y/y k ;

xP k D y k :

Since the Lagrangian F .x.s/; y.s// is constant in s, we obtain the first equation 0D

@F @F @F @F @F dF D k y k C k yP k D k y k 2P k y k D k y k 2PF: ds @x @y @x @y @x

Differentiating this equation and using (b) in Proposition 7.5, we obtain

@F @P 2 P m CF m @y @y

@ @F D m yk k @y @x

D

2 @F @F k @ F C y D 2 m: m k m @x @x @x @y

A first consequence is the following description of Minkowski metrics: Corollary 7.9. A strongly convex projectively flat Finsler metric on some domain in Rn is the restriction of a Minkowski metric if and only if the associated projective factor P .x; y/ vanishes identically.

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@F Proof. Suppose F .x; y/ is locally Minkowski, then @x m D 0 and therefore the spray k k coefficient satisfies P .x; y/y D G .x; y/ D 0. Conversely, if P .x; y/ 0, then @F the second equation in the lemma implies that @x m D 0.

Another consequence is the following result about the projective factor of a reversible Finsler metric. Corollary 7.10. Let F .x; y/ be a strongly convex projectively flat Finsler metric. Suppose F is reversible, then its projective factor satisfies P .x; y/ D P .x; y/: Proof. This is obvious from the first equation in Lemma 7.8.

8 The Hilbert form We now introduce the Hilbert 1-form of a Finsler manifold and show its relation with Hamel’s condition: Definition 8.1. The Hilbert 1-form on a smooth Finsler manifold .M; F / is the 1-form on TM 0 defined in natural coordinates as @F ! D j dx j D gij .x; y/y i dx j : @y From the homogeneity of F , we have F .x; / D !.x; /. Note also that !. / D 0 for any vector 2 T TM 0 that is tangent to some level set F .x; y/ D const: These two conditions characterize the Hilbert form which is therefore independent of the choice of coordinates, see also [18]. Observe also that the length of a smooth non-singular curve ˇ W Œa; b ! M is Z b Z Q D `.ˇ/ D F .ˇ/ !; a

ˇQ

where ˇQ W Œa; b ! TM 0 is the natural lift of ˇ. We then have the Proposition 8.2 ([4], [16]). The smooth Finsler metric F W T U ! R on the convex domain U is projectively flat if and only if the Hilbert form is dx -closed, that is, if we have @2 F dx ! D i j dx i ^ dx j D 0: @x @y Equivalently, F is projectively flat if and only if the Hilbert form is dx -exact. This means that there exists a function h W T U0 ! R such that ! D dx h D

@h dx j : @x j

Chapter 3. Funk and Hilbert geometries from the Finslerian viewpoint

93

Proof. The first assertion is a mere reformulation of the Hamel Condition (c) in Proposition7.5. The second assertion is proved using the same argument as in the proof of the Poincaré Lemma. Indeed, using Condition (c) from Proposition 7.5, we compute:

d @F d t !.tx;y/ D t i .tx; y/dx i dt dt @y @F @2 F D i .tx; y/dx i C t k i x k dx i @y @x @y @F @2 F D i .tx; y/dx i C t i k x k dx i @y @x @y @F D dx x k k .tx; y/ D dx !.tx;y/ .x/ : @y Suppose now that 0 2 U and set Z 1 Z h.x; y/ D !.tx;y/ .x/dt D 0

0

then by the previous calculation we have Z 1 Z dx h D dx .!.tx;y/ /dt D 0

1

0

1

xk

@F .tx; y/ dt; @y k

d .t !.tx;y/ /dt D !.x;y/ : dt

We shall call such a function h a Hamel potential of the projective Finsler metric F . Observe that a Hamel potential is well defined up to adding a function of y. The above proof shows that one can chose h.x; y/ to be 0-homogenous in y. This potential allows us to compute distances. Corollary 8.3. The distance dF associated to the projective Finsler metric F is given by dF .p; q/ D h.q; q p/ h.p; q p/; where h.x; y/ is a Hamel potential for F . Proof. Let ˇ.t / D p C t .q p/, (t 2 Œ0; 1). Then Z 1 Z P dF .p; q/ D F .ˇ; ˇ/dt D ! D h.q; q p/ h.p; q p/: 0

ˇQ

Example 8.4. If Ff is the tautological Finsler structure in U, then from the previous corollary and Equation (3.9) we deduce that a Hamel potential is given by h.x; y/ D log.Ff .x; y//: If one prefers a 0-homogenous potential, then a suitable choice is

Ff .p; y/ h.x; y/ D log ; Ff .x; y/

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where p 2 U is some fixed point. Remark 8.5. A consequence of this example is that the tautological Finsler structure in a domain U satisfies the following equation: @Ff @Ff D Ff j : j @x @y

(8.1)

Indeed, since h.x; y/ D log.Ff .x; y// is a Hamel potential, the Hilbert form is @Ff 1 @Ff dx j D ! D dx h D dx j ; @y j Ff @x j from which (8.1) follows immediately. This equation plays an important role in the study of projectively flat metrics with constant curvature, see e.g. [9]. An intrinsic discussion of geodesics based on the Hilbert form is given in [3], [16], [17], [18]. Remark 8.6. J. C. Álvarez Paiva and G. Berck gave a nice generalization of Proposition 8.2 above to a Hamel-type characterization of higher dimensional Lagrangians whose minimal submanifolds are k-flats, see Theorem 4.1 in [4].

9 Curvature in Finsler geometry A notion of curvature for Finsler surfaces already appeared in the beginning of the last century. This notion was extended in all dimensions by L. Berwald in 1926 [7]. This curvature is an analogue of the sectional curvature in Riemannian geometry and it is best explained using the notion of osculating Riemannian metric introduced by Varga [64]. See also Auslander [5] and the book of Rund [51], p. 84. Definition 9.1. 1. Let .M; F / be a smooth manifold with a strongly convex Finsler metric and let .x0 ; y0 / 2 TM 0 . A vector field V defined in some neighborhood O M of the point x0 is said to be a geodesic extension of y0 if Vx0 D y0 and if the integral curves of V are geodesics of the Finsler metric F (in particular V does not vanish throughout the neighborhood O). 2. The osculating Riemannian metric gV of F in the direction of V is the Riemannian metric on O defined by the Fundamental tensor at the point .x; y/ D .x; Vx / 2 T O 0 . In local coordinates we have gV D gij .x/dx i dx j D gij .x; V .x//dx i dx j D

1 @2 F .x; Vx / i j dx dx : 2 @y i @y j

Let us fix a point .x0 ; y0 / 2 TM 0 and a geodesic extension of y0 . We shall denote by RiemV the .1; 3/ Riemann curvature tensor of the osculating metric gV . Recall

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Chapter 3. Funk and Hilbert geometries from the Finslerian viewpoint

that RiemV .X; Y /Z D .rX rY rY rX rŒX;Y /Z; where r is the Levi-Civita connection of gV . Definition 9.2. The Riemann curvature of gV is the field of endomorphisms ((1,1)tensor) RV W TM ! TM defined as RV .W / D RiemV .W; V /V: A basic fact of Finsler geometry is the following result: Proposition 9.3. Let .M; F / be a smooth manifold with a strongly convex Finsler metric and let .x; y/ be a point in TM 0 . Then the Riemann curvature RV at .x; y/ 2 TM 0 is well defined independently of the choice of a geodesic extension V of y. Proof. Choose some local coordinate system and let us write Ri k .x; y/ for the components of the tensor RV : RV D Ri k .x; y/ dx k ˝

@ : @x i

Then we have the formula Ri k D 2

2 i @G i @2 G i j @G i @G j j @ G y C 2G ; @y j @y k @x k @x j @y k @y j @y k

(9.1)

where the G i D G i .x; y/ are the spray coefficients of F . We refer to Lemma 6.1.1 and Proposition 6.2.2 in [54] for a proof. Formula (9.1) is also obtained in [15], p. 43, where it is seen as a consequence of the structure equations for the Chern connection, see also [53], Proposition 8.4.3. Since the spray coefficients G i .x; y/ only depend on the fundamental tensor and its partial derivatives, it follows that Ri k depends only on .x; y/ D .x; Vx / 2 TM 0 and not on the choice of a geodesic field extending y. This proposition implies that we can write Ry D RV D Ri k .x; y/ dx k ˝

@ @x i

for the Riemann curvature at a point .x; y/ 2 TM 0 , where V is an arbitrary geodesic extension of y. We then have the following important Corollary 9.4. Let Tx M be a 2-plane containing the non-zero vector y 2 Tx M . Choose a local geodesic extension V of y, then the sectional curvature K gV . / of for the osculating Riemannian metric gV is independent of the choice of V . Proof. By definition of the sectional curvature in Riemannian geometry, we have K gV . / D

gV .RV .W /; W / ; gV .y; y/ gV .W; W / gV .y; W /2

(9.2)

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where W 2 and y D Vx are linearly independent (and thus D spanfV; W g), and RV is the Riemann curvature of gV . This quantity is independent of the geodesic extension V , by the previous proposition. Definition 9.5. The pair .y; / with 0 ¤ y 2 is called a flag in M , and KgV . / is called the flag curvature of .y; /, and denoted by K.y; /. The vector y 2 is sometimes suggestively called the flagpole of the flag. In local coordinates, the flag curvature can be written as K.y; / D

Rmk .x; y/ w k w m ; F 2 .x; y/gij .x; y/w i w j .grs .x; y/w r y s /2

(9.3)

where W D w k @[email protected] 2 and y are linearly independent and Rmk D gim Ri k . Example 9.6. The flag curvature of a Minkowski metric is zero. Indeed, the fundamental tensor is constant and coincides with any osculating metric which is thus flat. Note that conversely, there are many examples of Finsler metrics with vanishing flag curvature which are not locally isometric to a Minkowski metric. The first example has been given in Section 7 of [8]. We also have a notion of Ricci curvature: Definition 9.7. The Ricci curvature of the Finsler metric F at .x; y/ 2 TM 0 is defined as n X K.y; i / Ric.x; y/ D Trace.Ry / D F 2 .x; y/ iD2

where e1 ; e2 ; : : : ; en 2 Tx M is an orthonormal basis relative to the inner product gy y such that e1 D F .x;y/ . The geometric meaning of the Finslerian flag curvature presents both similarities and striking differences with its Riemannian counterpart. The Riemann curvature Ry plays an important role in the Finsler literature. It appears naturally in the second variation formula for the length of geodesics and in the theory of Jacobi fields (see [54], Lemma 6.1.1, or [51], Chapters IV.4 and IV.5). This leads to natural formulations of comparison theorems in Finsler geometry which are similar to their Riemannian counterparts. In particular we have • In 1952, L.Auslander proved a Finsler version of the Cartan–Hadamard Theorem. If the Finsler manifold .M; F / is forward complete and has non-positive flag curvature, then the exponential map is a covering map [5], [39]. • Auslander also proved a Bonnet–Myers Theorem: A forward complete Finsler manifold .M; F / with Ricci curvature Ric.x; y/ .n 1/F 2 .x; y/ for all .x; y/ 2 TM 0 is compact with diameter , see [5], [39].

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97

• In 2004, H.-B. Rademacher proved a sphere theorem: a compact, simply-connected Finsler manifold of dimension n 3 such that F .p; / F .p; / for any .p; / 2 TM and with flag curvature satisfying

2 1 0.

References [1]

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[10] J.-P. Benzécri, Sur les variétés localement affines et projectives. Bull. Soc. Math. France 88 (1960), 229–332. [11] R. Bryant, Projectively flat Finsler 2-spheres of constant curvature. Selecta Math. (N.S.) 3 (1997), no. 2, 161–203. [12] H. Busemann, The geometry of Finsler spaces. Bull. Amer. Math. Soc. 56, (1950), 5–16. [13] H. Busemann, The geometry of geodesics. Academic Press, New York 1955; reprinted by Dover in 2005. [14] H. Busemann and P. J. Kelly, Projective geometry and projective metrics. Academic Press, New York 1953. [15] S.-S. Chern and Z. Shen, Riemann–Finsler geometry. Nankai Tracts in Mathematics 6, World Scientific Publishing, Hackensack, NJ, 2005. [16] M. Crampin, Some remarks on the Finslerian version of Hilbert’s fourth problem. Houston J. Math. 37 (2011), no. 2, 369–391. [17] M. Crampin, T. Mestdag and D.J. Saunders, Hilbert forms for a Finsler metrizable projective class of sprays. Differential Geom. Appl. 31 (2013), no. 1, 63–79. [18] M. Crampon, The geodesic flow of Finsler and Hilbert geometries. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 161–206. [19] G. De Cecco and G. Palmieri, LIP manifolds: from metric to Finslerian structure. Math. Z. 218 (1995), 223–237. [20] P. Dazord, Variétés finslériennes de dimension paire ı-pincées. C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A496–A498. [21] D. Egloff, Some new developments in Finsler geometry. Ph.D. Thesis, University of Fribourg, 1995. [22] D. Egloff, Uniform Finsler Hadamard manifolds. Ann. Inst. H. Poincaré Phys. Théor. 66 (1997), no. 3, 323–357. [23] P. Finsler, Über Kurven und Flächen in allgemeinen Räumen. O. Füssli, Zürich 1918. [24] P. Funk, Über Geometrien, bei denen die Geraden die Kürzesten sind. Math. Ann. 101 (1929), 226–237. [25] P. Funk, Über zweidimensionale Finslersche Räume, insbesondere über solche mit geradlinigen Extremalen und positiver konstanter Krümmung. Math. Z. 40 (1936), no. 1, 86–93. [26] P. Funk, Variationsrechnung und ihre Anwendung in Physik und Technik. Grundlehren Math. Wiss. 94 Springer-Verlag, Berlin 1970. [27] E. Guo, X. Mo and X. Wang, On projectively flat Finsler metrics. In Differential geometry: Under the influence of S.-S. Chern, Adv. Lect. Math. (ALM) 22, Higher Education Press, Beijing, International Press, Somerville, MA, 2012, 77–88. [28] R. Guo, Characterizations of hyperbolic geometry among Hilbert geometries. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 147–158. [29] S. V. Ivanov, Volumes and areas of Lipschitz metrics. St. Petersburg Math. J. 20 (2009), no. 3, 381–405.

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Chapter 5

The horofunction boundary and isometry group of the Hilbert geometry Cormac Walsh

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The horofunction boundary . . . . . . . . . . . . . . . . . . . 2.1 Busemann points . . . . . . . . . . . . . . . . . . . . . . 2.2 Horoballs . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The horofunction boundary of the Hilbert geometry . . . . . . . 3.1 The horofunction boundary of the reverse-Funk geometry 3.2 The horofunction boundary of the Funk geometry . . . . . 3.3 The horofunction boundary of the Hilbert geometry . . . . 4 Isometries of the Hilbert metric . . . . . . . . . . . . . . . . . 4.1 Simplicial geometries . . . . . . . . . . . . . . . . . . . 4.2 Polyhedral geometries . . . . . . . . . . . . . . . . . . . 4.3 The general case . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

127 129 130 132 135 135 136 138 140 140 142 142 145

1 Introduction Building on the work of Busemann, Gromov defined in [12] a certain compactification of a metric space. Recall that the Busemann function of a geodesic ray is the limiting function B ./ WD lim d.; .t // d.b; .t //; t!1

where b is an arbitrary base-point. Gromov’s idea was to generalise this notion by considering all possible limits of d.; z/d.b; z/ as z heads off to infinity in the metric space, not necessarily along a geodesic ray. Such a limit is called a horofunction, and may be thought of as a “point at infinity” of the metric space. By adjoining the horofunctions to the metric space, under mild conditions, one compactifies it. The set of horofunctions thus forms a “boundary at infinity” of the metric space.

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The attractiveness of this boundary arises in part from its generality: whereas the definition of other boundaries, such as the hyperbolic boundary of a Gromov hyperbolic space, or the ideal boundary of a CAT(0) space, require the metric space to have certain properties, the horofunction boundary exists for any metric space. The horofunction boundary is often closely related to these other boundaries however. For example, in the CAT(0) case it is actually the same [2], and in the Gromov hyperbolic case, the hyperbolic boundary can be recovered from the horofunction boundary by quotienting out by a certain equivalence relation [8], [30], [23]. The horofunction boundary has found diverse applications. Rieffel [21] has used it in his work on quantum metric spaces. It also appears to be the right general setting for Patterson–Sullivan measures [5]. In this chapter, we will consider the horofunction boundary of the Hilbert geometry. Let D be a bounded open convex subset of a finite-dimensional real linear space. Given a pair of distinct points x and y in D, let w and z be points in the usual boundary of D such that w, x, y, and z lie in this order along a straight line; see Figure 1. The Hilbert distance between x and y is defined to be the logarithm of the cross ratio of these four points: Hil.x; y/ WD log

jzxj jwyj ; jzyj jwxj

where j j denotes any norm on the linear space. (Be aware that some authors use a different convention here and divide by a factor of 2.)

w

x

y z

Figure 1. Definition of the Hilbert distance.

The horofunction boundary of this metric was first studied by Karlsson, Metz, and Noskov [15], who determined it in the case where D is an open simplex. The general case was considered in [25]. In that paper, use was made of the fact that the Hilbert metric is the symmetrisation of the Funk metric: Hil.x; y/ D Funk.x; y/ C RFunk.x; y/; where Funk.x; y/ WD log

jzxj ; jzyj

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 129

and RFunk.x; y/ WD Funk.y; x/ D log

jwyj : jwxj

The horofunction boundaries of the Funk and reverse-Funk geometries were studied separately, and the results were combined to find the horofunction boundary of the Hilbert geometry. It turns out that the horofunction boundary of the reverse-Funk geometry is just the usual boundary of the convex domain D. That of the Funk geometry, however, is more closely related to the usual boundary of the polar body D B of the domain. We describe these boundaries in detail in Section 3, and say how they combine to form the Hilbert-geometry horofunction boundary. Observe that, since every collineation preserves cross-ratios, any collineation that leaves the domain D invariant is an isometry of the Hilbert metric on D. There are examples known, however, [9], [18], [17], [4] where there exist other isometries as well, namely, cross-sections of non-Lorentzian symmetric cones. For these domains, the projective action of Vinberg’s -map, or equivalently, the inverse map in the associated Jordan algebra, is an isometry but not a collineation, and the isometry group is generated by this map and the collineations. It was proved in [29] that there are no other examples. The horofunction boundary is useful for studying isometry groups. This is because such groups have a natural action by homeomorphisms on the boundary, and often this action is easier to understand than the action on the space itself. The horofunction boundary was used in this way in [28] to determine the group of isometries of Teichmüller space with Thurston’s metric. It was also used in [17] to determine the isometry group of the Hilbert geometry when the domain D is polyhedral. In Section 4, we discuss in more detail the isometry group of the Hilbert geometry and how the horofunction boundary may be used to study it. Before that, however, we recall in the next section the definition of the horofunction boundary and some of its properties. To visualize horofunctions, it is often useful to consider their horoballs, that is, their sublevel sets. Intuitively, a horoball looks like a large ball of the metric whose center is at infinity. We will make this precise by showing that a sequence of points converges to a horofunction if and only if sequences of closed balls of the right size with those points as centers converge to horoballs of the horofunction, in a certain topology.

2 The horofunction boundary Since we will be dealing with non-reversible metrics, we will discuss the horofunction boundary in this setting. The development will be similar to that in [28]. Let .X; d / be a non-reversible metric space, for example the Funk or reverseFunk geometry. We consider X to be equipped with the topology coming from the

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symmetrised metric dsym .x; y/ WD d.x; y/ C d.y; x/, which for the two examples just mentioned is the Hilbert metric. Associate to each point z 2 X the function z W X ! R, z .x/

WD d.x; z/ d.b; z/;

where b 2 X is an arbitrary fixed base-point. One can show that the map W X ! C.X /; z 7! z is injective and continuous. Here, C.X / is the space of continuous real-valued functions on X, with the topology of uniform convergence on bounded sets of dsym . We define the horofunction boundary of .X; d / to be X.1/ WD cl .X / n .X /; where cl is the topological closure operator. The elements of this set are the horofunctions of .X; d /. Although this definition appears to depend on the choice of base-point, one may verify that horofunction boundaries coming from different base-points are homeomorphic, and that corresponding horofunctions differ by only an additive constant. Recall that a geodesic in a non-reversible metric space .X; d / is a map from a closed interval of R to X such that d..s/; .t // D t s, for all s and t in the domain satisfying s < t . We make the following assumptions. (I) the metric dsym is proper, that is, its closed balls are compact; (II) there exists a geodesic in .X; d / between any two given points; (III) for any point x and sequence xn in X , we have d.xn ; x/ ! 0 if and only if d.x; xn / ! 0. Both the Funk and reverse-Funk metrics satisfy these assumptions. Indeed, for (II), one may take the straight line segment between the two points, parameterised appropriately. Assumption (I) implies that uniform convergence on bounded sets is equivalent to uniform convergence on compact sets. Moreover, for elements of cl .X /, the latter is equivalent to pointwise convergence, by the Ascoli–Arzelà theorem, since elements of this set are equi-Lipschitzian with respect to dsym . Again from the Ascoli–Arzelà theorem, the set cl .X/ is compact. We call it the horofunction compactification. Under assumptions (I), (II), and (III), one may show that is an embedding of X into C.X /, in other words, that it is a homeomorphism from X to its image in C.X /. From now on we identify X with its image.

2.1 Busemann points Define an almost-geodesic to be a path W T ! X such that T RC contains 0 and does not contain sup T , and such that, for any > 0, ˇ ˇ ˇd..0/; .s// C d..s/; .t // t ˇ < ; for s; t 2 T large enough, with s t. This definition has been modified from the original in [21], where T was required to be an unbounded subset of RC containing 0.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 131

We must relax the condition that T be unbounded because, in the reverse-Funk metric, one may approach the boundary along a path of bounded length. See Figure 2 for an illustration. This modification is very minor.

b

xn

Figure 2. Finite distance to the boundary. Funk.xn ; b/ and RFunk.b; xn / are bounded above, even though the sequence xn approaches the boundary.

Rieffel proved that every almost-geodesic converges. In his setting, it will be to a horofunction, but in ours it may be to a point of X . Every point of X is trivially the limit of an almost-geodesic under our definition. We say that a horofunction is a Busemann point if it is the limit of an almost-geodesic. Non-Busemann points of the horofunction boundary have been found for normed spaces [24] and various finitely-generated groups with their word metrics [26], [27], [31]. We define the detour cost for any two horofunctions and in X.1/ to be H.; / D sup inf d.b; x/ C .x/ ; W 3 x2W \X

where the supremum is taken over all neighbourhoods W of in X [ X.1/. This concept appears in [1]. An equivalent definition is H.; / D inf lim inf d.b; .t // C ..t // ; t!sup T

where the infimum is taken over all paths W T ! X converging to . The proof of the following proposition is easy. Proposition 2.1. For all horofunctions , , and , (i) H.; / 0; (ii) H.; / H.; / C H.; /. Proposition 2.2. Assume that .X; d / satisfies assumptions (I), (II), and (III), and let be a horofunction. Then, H.; / D 0 if and only if is Busemann. The proof of this proposition may be found in [28]. There, it was assumed that d.b; xn / converges to infinity for a sequence xn in X whenever dsym .b; xn / does.

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Although this is not true for the reverse-Funk metric, it was only needed to ensure that the almost-geodesic constructed is defined on all of RC . Since the definition of almost-geodesic has been modified in the current setting, this is no longer necessary. Propositions 2.1 and 2.2 together say that, on the set of Busemann points, the detour cost is a weak metric, sometimes also called an extended pseudo quasi-metric. By symmetrising, one may obtain a genuine metric on the set of Busemann points XB .1/. For Busemann points and in XB .1/, define ı.; / WD H.; / C H.; /: This construction first appeared in [1], Remark 5.2. See [28] for a proof of the following proposition. Proposition 2.3. The function ı W XB .1/ XB .1/ ! Œ0; 1 is a (possibly 1-valued) metric. We call ı the detour metric. Note that we can partition XB .1/ into disjoint subsets such that ı.; / is finite for each pair of Busemann points and lying in the same subset. We call these subsets the parts of the horofunction boundary of .X; d /, and ı is a genuine metric on each one. Consider an isometry g from one metric space .X; d / to another .X 0 ; d 0 /. We can extend g continuously to the horofunction boundary X.1/ of X as follows: g./.x 0 / WD .g 1 .x 0 // .g 1 .b 0 //; 0

0

0

(2.1)

0

for all 2 X.1/ and x 2 X . Here b is the base-point of X .

2.2 Horoballs To visualize horofunctions, it is often useful to consider their horoballs, that is, their sublevel sets. We denote the sublevel set of a function f at height ˛ by slv.f; ˛/ WD fx 2 X j f .x/ ˛g. Intuitively, a horoball looks like a large ball of the metric whose center is at infinity. To make this precise, we define a topology on the set of closed subsets of X , called the Painlevé–Kuratowski topology. In this topology, a sequence of closed sets .Cn /n2N is said to converge to a closed set C if the upper and lower closed limits (also known as the limit superior and limit inferior) of the sequence both equal C . These limits are defined to be, respectively, \ [ cl Ci Ls Cn WD n0

and Li Cn WD

\

i>n

cl

[ i0

Cni j .ni /i2N is an increasing sequence in N :

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 133

An alternative characterisation of convergence is that .Cn /n2N converges to C if and only if each of the following hold: • for each x 2 C , there exists xn 2 Cn for all n large enough, such that .xn /n converges to x; • if .Cnk /k2N is a subsequence of the sequence of sets and xk 2 Cnk for each k 2 N, then convergence of .xk /k2N to x implies that x 2 C . See the book [3] for more about this and other topologies on sets of closed sets. For each z 2 X and ˛ 2 R, we denote by B.z; ˛/ WD fx 2 X j d.x; z/ ˛g the right closed ball about z of radius ˛. In [15], it was shown that in a geodesic metric space each horoball slv.; ˛/ of a horofunction that is the limit of a sequence zn takes the form Ls B.zn ; d.b; zn / C ˛/. Here we improve this result by establishing a necessary and sufficient condition for convergence to a horofunction in terms of convergence of balls in the Painlevé–Kuratowski topology. Lemma 2.4. Assume .X; d / satisfies (I), (II), and (III). Let x 2 X, and let 2 X.1/ be a horofunction. Then, for all 0 small enough, there exists y 2 X such that .x/ .y/ D d.x; y/ D . Proof. Choose a sequence .zn / in X converging to . Since no subsequence of .zn / converges to x, we have that .zn / eventually remains outside any sufficiently small open ball fz 2 X j d.x; z/ < g about x. So, for n large enough, we may choose a point yn on a geodesic from x to zn such that d.x; zn / d.yn ; zn / D d.x; yn / D . But some subsequence of .yn / converges to a point y 2 X, and, taking limits, we get the result. Lemma 2.5. Assume .X; d / satisfies (I), (II), and (III). Let 2 X.1/ be a horofunction. Then, slv.; / is continuous in the Painlevé–Kuratowski topology. Proof. Let ˛ 2 R, and let .˛n / be a sequence in R converging to ˛. Let .nk / be an increasing sequence in N, and let .xk / be a sequence of points in X converging to some point x 2 X , such that xk 2 slv.; ˛nk / for all k 2 N. So, .x/ D lim .xk / lim ˛nk D ˛: k!1

k!1

Therefore, x 2 slv.; ˛/. Now let y 2 slv.; ˛/. For any c 2 R, we use the notation c C WD max.c; 0/. By Lemma 2.4, for n large enough, there exists yn 2 X such that .y/ .yn / D d.y; yn / D .˛ ˛n /C : Observe that .˛ ˛n /C is a non-negative sequence converging to zero. It follows that .yn / converges to y, and that yn 2 slv.; ˛n / for all n large enough. The Painlevé–Kuratowski topology can be used to define a topology on the space of lower-semicontinuous functions on X as follows. Recall that the epigraph of a

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function f on X is the set f.x; ˛/ 2 X R j f .x/ ˛g. A sequence of lowersemicontinuous functions is declared to be convergent in the epigraph topology if the associated epigraphs converge in the Painlevé–Kuratowski topology on X R. For proper metric spaces the epigraph topology is identical to another topology called the Attouch–Wets topology. Lemma 2.6. Make assumption (I), which is that .X; dsym / is a proper metric space. Let be a real-valued lower-semicontinuous function on X, and let n be a sequence of such functions that is equi-Lipschitzian with respect to dsym . Then, n converges to in the epigraph topology if and only if it converges to uniformly on bounded sets. Proof. This is a consequence of Lemma 7.1.2 of [3] and Proposition 7.1.3 of [3], and the fact that, for proper metric spaces, Attouch–Wets convergence is equivalent to epigraph convergence. We will also need the following result relating convergence of functions to convergence of their sublevel sets. Recall that a proper metric space is always separable. Proposition 2.7 ([3], Theorem 5.3.9). Let f be a lower-semicontinuous function in a separable metric space, and let fn be a sequence of such functions. Then, fn converges to f in the epigraph topology if and only if there exists, for all ˛ 2 R, a sequence ˛n in R converging to ˛ such that slv.fn ; ˛n / converges to slv.f; ˛/ in the Painlevé–Kuratowski topology. Now we can prove our result concerning the convergence of balls. Proposition 2.8. Let zn be a sequence in X. Then, zn converges to a point in the horofunction boundary if and only if, for each ˛ 2 R, the sequence of balls B.zn ; d.b; zn / C ˛/ converges to slv.; ˛/ in the Painlevé–Kuratowski topology. Proof. Suppose that the balls converge as stated. Observe that slv.

z ; ˛/

D B.z; d.b; z/ C ˛/

for all z 2 X and ˛ 2 R:

So, by Proposition 2.7, zn converges to in the epigraph topology, which is equivalent, by Lemma 2.6, to uniform convergence on bounded sets. Now suppose that zn converges to in the horofunction boundary, and let ˛ 2 R. Choose ˇ < ˛. From Lemma 2.6 and Proposition 2.7, we get that there exists a sequence ˇn in R converging to ˇ such that B.zn ; d.b; zn / C ˇn / converges to slv.; ˇ/. For n large enough, B.zn ; d.b; zn / C ˛/ contains B.zn ; d.b; zn / C ˇn /. Therefore, Li B.zn ; d.b; zn / C ˛/ contains slv.; ˇ/. Since this is true for all ˇ < ˛, and slv.; / is continuous by Lemma 2.5, we get Li B.zn ; d.b; zn / C ˛/ slv.; ˛/: The upper bound on the upper closed limit is proved in a similar manner.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 135

Figure 3 illustrates the convergence of balls to horoballs in the case of the Hilbert metric on the 2-simplex.

Figure 3. Sequences of balls converging to horoballs in the Hilbert geometry.

3 The horofunction boundary of the Hilbert geometry In this section, we study the horofunction boundary of the Hilbert metric on a bounded open convex set D. As mentioned before, we will consider the horofunction boundaries of the Funk and reverse-Funk geometries separately, and then combine the information to get the boundary of the Hilbert geometry. Unless otherwise stated, results come from [25].

3.1 The horofunction boundary of the reverse-Funk geometry Observe that the function RFunk.; /, defined on D D extends continuously to a function on D cl D. So, for all z 2 cl D, we may define on D the function rz ./ WD RFunk.; z/ RFunk.b; z/. The following proposition shows that the horofunction compactification of the reverse-Funk geometry is basically the same as the compactification obtained by taking the closure of the domain in the usual topology. We use @D WD cl DnD to denote the boundary of D in the usual topology. Proposition 3.1. Let D be a bounded open convex set. The set of horofunctions in the reverse-Funk geometry on D is BRF WD frx j x 2 @Dg. A sequence in D converges to rx 2 BRF if and only if it converges in the usual sense to x. Since straight line segments are geodesic in the reverse-Funk geometry, it is clear that every horofunction in this geometry is Busemann. The detour cost in this geometry was calculated in [17]. Recall that a convex subset E of a convex set D is said to be an extreme set if the endpoints of any line segment

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in D are contained in E whenever any relative interior point of the line segment is. The relative interiors of the extreme sets of a convex set D partition D. When we consider RFunk etc. on convex sets other than D, we use a subscript to specify the set. Proposition 3.2 ([17], Proposition 4.3). Let x and y be in the usual boundary @D of D. Then, the detour cost in the reverse-Funk geometry is HRF .rx ; ry / D RFunkD .b; x/ C RFunk F .x; y/ RFunkD .b; y/; if y is in the smallest extreme set F of cl D containing x, and is infinity otherwise. It follows immediately that the detour metric in this geometry is given by ıRF .rx ; ry / D HilF .x; y/; when x and y are in the relative interior of the same extreme set F of cl D, and is infinity otherwise. Thus, the parts of the horofunction boundary are just the relative interiors of the proper extreme sets of cl D.

3.2 The horofunction boundary of the Funk geometry The horofunctions of the Funk geometry can be best understood by using convex duality because they take a simpler form in the dual space. Consider first the cone over D, in other words, ˚ C WD .p; 1/ 2 RnC1 j p 2 D and > 0 : Here we are assuming that D lives within Rn . We identify D in the obvious way with a cross-section of this cone. One may extend the function FunkD .; / from D to C in the following way. For x 2 RnC1 and y 2 C , define the gauge ˚ M.x=yI C / WD inf > 0 j x C y : Here C is the partial ordering on RnC1 induced by C , that is, x C y if y x 2 cl C . One can show [16], page 29, that log M.x=yI C / D FunkD .x; y/ for all x and y in D. For each z 2 D, the map z has the following extension to the whole of RnC1 : z .x/

WD log

M.x=zI C / ; M.b=zI C /

for all x 2 RnC1 :

Define the (closed) dual cone of C : C WD fu 2 RnC1 j hu; xi 0 for all x 2 C g: Here h; i denotes the standard Euclidean inner product. Observe that, by the Hahn– Banach separation theorem, x C y if and only if hu; xi hu; yi for all u 2 C . It

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 137

follows that M.x=yI C / D

hu; xi u2C nf0g hu; yi sup

for all x 2 RnC1 and y 2 C .

(3.1)

So, we see that M.=zI C / is a convex function for fixed z. Therefore, the same is true for the following function defined on RnC1 : jC;z ./ WD

M.=zI C / D exp B M.b=zI C /

z ./:

Recall that the Legendre–Fenchel transform of a convex function f W RnC1 ! R [ f1g is the function f W RnC1 ! R [ f1g defined by f .y/ WD sup hy; xi f .x/ for all y 2 RnC1 : x2RnC1

The Legendre–Fenchel transform is a bijection from the set of proper lower-semicontinuous convex functions to itself, and is in fact a homeomorphism in the Attouch–Wets topology. Using (3.1), one may calculate without much difficulty the Legendre–Fenchel transform of jC;z . We use IE to denote the characteristic function of a set E, that is, the function taking value 0 on E and C1 everywhere else. For any open cone U in RnC1 and any point x 2 U , define ZU;x WD U \ fu 2 RnC1 j M.b=xI C /hu; xi 1g: Proposition 3.3. Let C RnC1 be an open cone. For all z 2 C , we have that jC;z is the characteristic function of the set ZC;z So the set ZC;z on which jC;z is zero is the intersection of C with a half-space bounded by a hyperplane on which h; zi is constant. This constant is the largest possible such that ZC;z lies within fu 2 RnC1 j hu; bi 1g. This is illustrated in Figure 4.

M.b=z/h; zi D 1

h; bi D 1

C . Figure 4. The support of jC;x

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As z approaches the boundary of D, the hyperplane becomes more and more tilted. In the limit, the zero set will lie entirely within the (usual) boundary of C . We consider the limit of the zero sets in the Painlevé–Kuratowski topology. Proposition 3.4. A sequence zn in D converges to a point in the horofunction boundary of the Funk geometry if and only if ZC;zn converges in the Painlevé–Kuratowski topology. We have the following simple description of the Busemann points. Proposition 3.5. A function W D ! R is a Busemann point of the Funk geometry if and only if it is the restriction to D of a function of the form log IZE;x , where E is an open convex cone such that E is a proper extreme set of C , and x 2 E. It is clear from this proposition that in some cases there may be horofunctions that are not Busemann. This will happen for instance when there is a subset F of C that is not extreme but is the limit of extreme sets. In this case, the function log IZF;b will be, by Proposition 3.4, a horofunction, but will not be, by Proposition 3.5, a Busemann point. The next result shows that this phenomenon is the only way that non-Busemann points may arise. Proposition 3.6. All horofunctions of the Funk geometry on a domain D are Busemann if and only if the set of extreme sets of the polar of D is closed in the Painlevé– Kuratowski topology. Recall that the polar D B of D may be identified with a cross-section of C . The detour metric in the Funk geometry was calculated in [17]. Proposition 3.7. Let D log IZE;x and D log IZF;y be two Busemann points of the Funk geometry. The distance between them in the detour metric is ıF .; / D HilE .x; y/;

if the extreme sets E and F of C are equal, and is infinity otherwise. Thus, there is a part for each extreme set of the polar of D.

3.3 The horofunction boundary of the Hilbert geometry It is evident from the expression of the Hilbert metric as the symmetrisation of the Funk metric that every horofunction in the Hilbert geometry is the sum of one in the Funk geometry and one in the reverse-Funk geometry. The following proposition shows that there is a unique such decomposition of each Hilbert horofunction.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 139

Proposition 3.8. A sequence converges to a point in the Hilbert-geometry horofunction boundary if and only if it converges to a horofunction in both the Funk and reverseFunk geometries. Proof. This follows from the proof of Theorem 1.3 in [25]. Note that, combined with Proposition 3.1, this implies that every sequence that converges to a Hilbert horofunction also converges to a point in the usual boundary. This generalises a result of [14], where is was shown that every Hilbert-geometry geodesic converges to such a point. We also have the following description of the Busemann points. Theorem 3.9. Let h D rx C f be a Hilbert horofunction written as the sum of a reverse-Funk horofunction rx and a Funk horofunction f . Then, h is a Busemann point if and only if f is. Combining this with Proposition 3.6, we get the following. Corollary 3.10. All horofunctions of the Hilbert geometry on a domain D are Busemann if and only if the set of extreme sets of the polar of D is closed in the Painlevé– Kuratowski topology. It remains to determine which combinations of Funk and reverse-Funk horofunctions give Hilbert horofunctions. The answer is given in the next theorem. Recall that every point in @C is a supporting functional of C , and so defines an exposed face of C . Theorem 3.11. Let rx be a reverse-Funk horofunction, and let f WD log IZE;z be a Busemann point of the Funk geometry. Then, the function rx C f is a Busemann point of the Hilbert geometry if and only if the extreme set E of C is contained in the exposed face of C defined by x. Proof. This is just a restatement of Theorem 1.1 of [25] using the description given above of the Funk Busemann-points in the dual space. Indeed, by [25], Lemma 3.13, the tangent cone .C; x/ of C at x is dual to the exposed face of C defined by x. Also, by [25], Lemma 3.14, an open cone T can be obtained from another U by taking tangent cones if and only if T is an extreme set of U . The final ingredient is that the extreme sets of an exposed face of a closed cone are exactly the extreme sets of the cone that are contained within the exposed face. Using Propositions 3.2 and 3.7, one may find the detour metric of the Hilbert geometry.

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Theorem 3.12 ([17]). Let D rp C log IZE;x and D rq C log IZF;y be two Busemann points of the Hilbert geometry. Then, the distance between them in the detour metric of the Hilbert geometry is ıH .; / D HilG .p; q/ C HilE .x; y/;

(3.2)

when p and q are in the relative interior of the same extreme set G of cl D, and the extreme sets E and F of C are equal. Otherwise, it is infinity. So, we see that each part of the horofunction boundary of a Hilbert geometry is the direct product of two lower-dimensional Hilbert geometries with the `1 -sum distance.

4 Isometries of the Hilbert metric De la Harpe [9] was the first to consider the isometry group of the Hilbert geometry. Let P n D Rn [ P n1 be real n-dimensional projective space, and suppose that D is contained within the open cell Rn inside P n . Let Coll.D/ be the set of collineations, that is, elements of PGL.n C 1; R/ preserving D. As de la Harpe observed, each element of Coll.D/ is an isometry since collineations preserve the cross-ratios.

4.1 Simplicial geometries However, not every isometry is a collineation, as we will now see. Think of the nsimplex as being a cross-section of the positive cone RnC1 C . Hilbert’s projective metric on the interior of this cone is given by xi yi Hil.x; y/ D log max C log max : i i yi xi Define V to be the n-dimensional linear space RnC1 = , where x y if x D y C ˛.1; 1; : : : ; 1/ for some ˛ 2 R, and equip V with the variation norm: kxkvar WD max xi min xi : i

i

We see that the Hilbert metric on the simplex is isometric to the normed space V via the map ! RnC1 ; log W int RnC1 C

.xi /i 7! .log xi /i

that takes logarithms coordinate-wise. This isometry was first observed by Nussbaum [20] and de la Harpe [9]. It was shown by Foertsch and Karlsson [14] that simplices are the only Hilbert geometries isometric to normed spaces. The result is also stated in [6], p. 35, and [7], p. 313.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 141

Theorem 4.1. A finite-dimensional Hilbert geometry on a bounded open convex set is isometric to a normed space if and only if the domain is a simplex. Using this correspondence, and the Masur–Ulam theorem, the isometry group of the simplicial Hilbert geometry was determined for n D 2 in [9] and for arbitrary n in [17]. Let nC1 be the group of coordinate permutations on V , let W V ! V be the isometry given by .x/ D x for x 2 V , and identify the group of translations in V with Rn . Theorem 4.2 ([11], [17]). If D is an open n-simplex with n 2, then Coll.D/ Š Rn Ì nC1 and Isom.D/ Š Rn Ì nC1 ; where nC1 D nC1 hi. The same result was obtained independently in [11], although the authors there were not aware that the geometry they were considering is isometric to a normed space. Consequently, their proof is somewhat longer. So, we see that in the case of the simplex the isometry group differs from the collineation group. Indeed, the latter is a subgroup of index two of the former. Let us to the coordinate-wise look in more detail at the map , which corresponds on int RnC1 C reciprocal map O W int RnC1 ! int RnC1 C C ;

.xi /i 7!

1 xi

i

:

(4.1)

The projective action of this map is an isometry but not a collineation. This is shown in Figure 5 for the case of n D 2.

Figure 5. An isometry of the triangle that is not a collineation. The solid straight line on the left is mapped to the curve on the right.

As observed in [9], the vertices are blown up to edges, the edges are collapsed to vertices, and a straight line in the interior is mapped to a straight line if and only if one of its ends approaches a vertex of the triangle.

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4.2 Polyhedral geometries At the opposite extreme from the case of simplices, we have the following. Theorem 4.3 ([9]). If the closure of D is strictly convex, then Isom.D/ D Coll.D/. It was also shown in the same paper that the only two-dimensional polyhedra with non-collineation isometries are the triangles. This was generalised to arbitrary dimension in the following theorem. Theorem 4.4 ([17]). If .D; Hil/ is a polyhedral Hilbert geometry, then Coll.D/ differs from Isom.D/ if and only if D is an open n-simplex, with n 2. The proof involves studying the action of an isometry on the parts of the horofunction boundary, in particular, on the parts where one or other of the `1 factors is trivial, that is, consists of a single point. There is one such part for every vertex on the polyhedron, and one for every facet. It is shown that every isometry either maps vertex parts to vertex parts and facet parts to facet parts, or it interchanges them. Isometries of the former type are shown to extend continuously to the Euclidean boundary of the domain and to be collineations, whereas isometries of the latter type are shown to only exist on simplices. The above theorem verifies, for the case of polyhedral Hilbert geometries, some conjectures of de la Harpe, namely that the Hilbert isometry group Isom.D/ is a Lie group, and that its identity component coincides with that of Coll.D/. Observe that Coll.D/ is naturally a Lie group since it is a closed subgroup of PGL.n C 1; R/. A consequence of these conjectures is that Isom.D/ acts transitively on D if and only if Coll.D/ does, the latter happening exactly when D is the cross-section of an homogeneous cone. The only polyhedra that occur as cross-sections of such cones are the simplices. De la Harpe enumerates all possible cross-sections of homogeneous cones up to dimension four [9].

4.3 The general case Dropping the polyhedral assumption, can one find other domains with isometries that are not collineations? Consider again the reciprocal map O defined in (4.1) on the interior of the nC1-dimensional positive cone. One way of seeing that it is an isometry of Hilbert’s projective metric is to observe that it is order-reversing, homogeneous of degree 1, and involutive, meaning that it is its own inverse. The following proposition then implies that O is gauge-reversing, that is, nC1 M..x/= O .y/I O RnC1 C / D M.y=xI RC /

for all x; y 2 int RnC1 C :

Proposition 4.5 ([19], [20]). Let g W C ! C 0 be a bijection between two open cones in a finite-dimensional vector space. Then, g is gauge-reversing if and only if it is order-reversing and homogeneous of degree 1 and its inverse is order-reversing.

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 143

It is clear that gauge-reversing bijections are isometries of Hilbert’s projective metric. However, gauge-reversing maps do not just exist on the positive cone, they exist on all symmetric cones, of which the positive cone is an example. Recall [10] that a proper open cone C in a finite-dimensional real vector space is called symmetric if it is homogeneous, meaning that its group of linear automorphisms acts transitively on it, and self dual, meaning that C D C ? , where C ? now denotes the open dual. One defines the characteristic function on C to be Z e hy;xi dy for all x 2 C :

.x/ D C?

This map is homogeneous of degree minus the dimension of the cone, which implies that Vinberg’s -map, C ! C ? ; x 7! x WD r log .x/; is homogeneous of degree 1. It is known [13] that on symmetric cones the map is order-reversing. In fact, it was shown in [13] that this property of the -map characterises the symmetric cones among the homogeneous cones. One may easily verify that the map O in (4.1) is the -map for the positive cone int RnC1 C . The symmetric cones have been completely classified. Each such cone is the product of one or more of the following irreducible cones: the positive definite Hermitian matrices Herm.n; E/, with n 3, where the set of entries E can be the reals R, the complex numbers C, or the quaternions H; the positive definite Hermitian 3 3 matrices Herm.3; O/ with octonion entries; the Lorentz cone (or ice-cream cone) ˚ ƒn D .x1 ; : : : ; xn / 2 Rn j x1 > 0 and x12 x22 xn2 > 0 ; for some n 2. For the reasons discussed above, the -map on a symmetric cone is an isometry of Hilbert’s projective metric. Its projective action is therefore an isometry of the Hilbert geometry on a cross-section D of the cone. This isometry is not a collineation except when the symmetric cone is a Lorentz cone. It was conjectured in [17] that Isom.D/ and Coll.D/ differ if and only if the cone generated by D is symmetric and not Lorentzian, in which case the isometry group was conjectured to be generated by the collineations and the isometry coming from the -map. This was known to be true for the cone of positive-definite Hermitian matrices [18], and has recently been established for general symmetric cones [4]. The conjecture has now been proved in general [29]. An important step in the proof is the following. Theorem 4.6 ([29]). An open convex cone in a finite-dimensional real vector space admits a gauge-reversing map if and only if it is symmetric. Establishing the homogeneity of such a cone admitting a gauge-reversing map is not difficult. One uses Rademacher’s theorem to get that the map is differentiable

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almost everywhere. At each point of differentiability, the negative of the differential is a linear automorphism of the cone. This gives a large supply of automorphisms with which to prove homogeneity. Proving the self-duality of the cone is more complicated. The first step is to use arguments similar to those in the proof of homogeneity to show there is a map g W C ! C that is gauge-reversing and involutive, and fixes the base-point b. One then wishes to define a positive-definite bilinear form with respect to which C is self-dual. This is done by exploiting the fact that certain Funk horofunctions, namely those associated to extreme rays of C , are the logarithms of linear functionals. Each of these horofunctions is a singleton part and so gets mapped by g to a singleton part of the reverse-Funk horofunction boundary, that is, a reverse-Funk horofunction of the form rx , where x is an extremal generator of C , in other words, a non-zero element of an extremal ray of cl C . We now define A.y; x/ WD M.x=bI C / exp B rx B g.y/ D M.x=g.y/I C /; for all y 2 C and extremal generators x of C . The equality holds because of the definition of rx . From what we have just seen, exp B rx B g is the restriction to C of a linear functional. This lets us extend the definition of A to all y in V in such a way that it is linear in this variable. Further work is then required to extend the definition to all x in V and verify that A has the right properties. One may also be interested in gauge-preserving maps,that is, maps g W C ! C satisfying M.g.x/=g.y/I C / D M.x=yI C / for all x and y in C . Such maps are also clearly isometries of Hilbert’s projective metric. The following proposition parallels Proposition 4.5. Proposition 4.7 ([19], [20]). Let g W C ! C 0 be a bijection between two open cones in a finite-dimensional vector space. Then, g is gauge-preserving if and only if it is order-preserving and homogeneous and its inverse is order-preserving. These maps are well understood. Theorem 4.8 ([19], [22]). Let C and C 0 be open cones in a finite-dimensional vector space, and let g W C ! C 0 be gauge-preserving bijection. Then, g is the restriction to C of a linear isomorphism. Here is a simple proof using horofunctions. Proof. Choose the base-points b 2 C and b 0 2 C 0 such that g.b/ D b 0 . Since g is gauge-preserving, it preserves the Funk metric, and therefore maps Funk horofunctions to Funk horofunctions according to (2.1). In particular, singleton horofunctions are mapped to singleton horofunctions. Recall that there is a singleton horofunction of the Funk geometry for each extremal ray of the closed dual cone; in fact the horofunction is just the logarithm of a suitably normalised element of the ray. So for

Chapter 5. The horofunction boundary and isometry group of the Hilbert geometry 145

every extremal generator x of C , there is an extremal generator x 0 of .C 0 / satisfying hx; yi D hx 0 ; g.y/i, for all y 2 C . However, by choosing the right number of extremal generators of C we obtain a basis for the dual space, that is, a linear coordinate system. It follows that g is the restriction to C of a linear map. So the projective action of every gauge-preserving map is a collineation. The conjecture referred to above now follows from combining Theorems 4.6 and 4.8 with the following fact proved in Theorem 1.3 of [29]: every isometry of a Hilbert geometry arises as the projective action of either a gauge-preserving map or a gauge-reversing map. Acknowledgement. This work is partially supported by the joint RFBR-CNRS grant number 05-01-02807.

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[14] A. Karlsson and T. Foertsch, Hilbert metrics and Minkowski norms. J. Geometry 83 (2005), no. 1, 22–31. [15] A. Karlsson, V. Metz, and G. Noskov, Horoballs in simplices and Minkowski spaces. Int. J. Math. Math. Sci. (2006), Art. ID 23656. [16] B. Lemmens and R. Nussbaum, Nonlinear Perron–Frobenius theory. Cambridge Tracts in Math. 189, Cambridge University Press, Cambridge 2012. [17] B. Lemmens and C. Walsh, Isometries of polyhedral Hilbert geometries. J. Topol. Anal. 3 (2011), no. 2, 213–241. [18] L. Molnár, Thompson isometries of the space of invertible positive operators. Proc. Amer. Math. Soc. 137 (2009), no. 11, 3849–3859. [19] W. Noll and J. J. Schäffer, Orders, gauge, and distance in faceless linear cones; with examples relevant to continuum mechanics and relativity. Arch. Rational Mech. Anal. 66 (1977), no. 4, 345–377. [20] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 75 (391), 1988. [21] M. A. Rieffel, Group C -algebras as compact quantum metric spaces. Doc. Math. 7 (2002), 605–651. [22] O. S. Rothaus, Order isomorphisms of cones. Proc. Amer. Math. Soc. 17 (1966), 1284–1288. [23] P. A. Storm, The barycenter method on singular spaces. Comment. Math. Helv. 82 (2007), no. 1, 133–173. [24] C. Walsh, The horofunction boundary of finite-dimensional normed spaces. Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 3, 497–507. [25] C. Walsh, The horofunction boundary of the Hilbert geometry. Adv. Geom. 8 (2008), no. 4, 503–529. [26] C. Walsh, Busemann points of Artin groups of dihedral type. Internat. J. Algebra Comput. 19 (2009), no. 7, 891–910. [27] C. Walsh, The action of a nilpotent group on its horofunction boundary has finite orbits. Groups Geom. Dyn. 5 (2011), no. 1, 189–206. [28] C. Walsh, The horoboundary and isometry group of Thurston’s Lipschitz metric. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, European Mathematical Society, Zürich 2014, 327–353. [29] C. Walsh, Gauge-reversing maps on cones and Thompson isometries. Preprint, arXiv:1312.7871. [30] C. Webster and A. Winchester, Boundaries of hyperbolic metric spaces. Pacific J. Math. 221 (2005), no. 1, 147–158. [31] C. Webster and A. Winchester, Busemann points of infinite graphs. Trans. Amer. Math. Soc. 358 (2006), no. 9, 4209–4224.

Chapter 6

Characterizations of hyperbolic geometry among Hilbert geometries Ren Guo

Contents 1 Introduction . . . . . . 2 Reflections . . . . . . 3 Perpendicularity . . . 4 Ptolemaic inequality . 5 Curvature . . . . . . . 6 Median . . . . . . . . 7 Isometry group . . . . 8 Ideal triangles . . . . . 9 Spectrum: a conjecture References . . . . . . . . .

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1 Introduction The Hilbert metric is a canonical metric associated to an arbitrary bounded convex domain in Rn or in a projective space. It was introduced by David Hilbert in 1894 as an example of a metric for which the Euclidean straight lines are geodesics. Hilbert geometry generalizes Klein’s model of hyperbolic geometry. Let K be a bounded open convex set in Rn .n 2/. The Hilbert metric dK on K is defined as follows. For any x 2 K, let dK .x; x/ D 0. For distinct points x, y in K, assume the straight line passing through x, y intersects the boundary @K at two points a, b such that the order of these four points on the line is a, x, y, b as in Figure 1. Denote the cross-ratio of the points by Œx; y; b; a D

kb xk ka yk ; kb yk ka xk

where k k is the Euclidean norm of Rn . Then the Hilbert metric is dK .x; y/ D

1 logŒa; x; y; b: 2

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a

x

y

b

Figure 1. Hilbert metric.

The metric space .K; dK / is called a Hilbert geometry. A Euclidean straight line in K is a geodesic under the metric dK . When K is the unit open ball °

.x1 ; : : : ; xn / 2 Rn j

Pn

±

iD1

xi2 < 1 ;

.K; dK / is Klein’s model of hyperbolic geometry. Since the cross-ratio is invariant under a perspectivity P of center O 2 Rn [ f1g, .K; dK / and .P .K/; dP .K/ / are isometric as Hilbert geometries. In particular, when K is enclosed by an ellipsoid (ellipse when n D 2), i.e., °

K D .x1 ; : : : ; xn / 2 Rn j

Pn

iD1

xi2 ai2

±

2; properly means there exists a projective hyperplane which does not intersect the closure of , or, equivalently, there is an affine chart in which appears as a relatively compact set; • d is the distance on defined, for two distinct points x, y, by 1 logŒa; b; x; y; 2 where a and b are the intersection points of the line .xy/ with the boundary @ (chosen as in Figure 1) and Œa; b; x; y denotes the cross ratio of the four points: jayj=jbyj . if we identify the line .xy/ with R [ f1g, it is defined by Œa; b; x; y D jaxj=jbxj d .x; y/ D

2.2 (Regular) Finsler metrics and their geodesic flow Definition 2.1. A Finsler metric on a manifold M is a field of (non-necessarily symmetric) norms on M , that is, a function F W T M ! Œ0; C1/ such that: • F .x; u/ D F .x; u/, .x; u/ 2 T M , > 0; • F .x; u C v/ 6 F .x; u/ C F .x; v/, x 2 M , u; v 2 Tx M .

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b y x

a

Figure 1. The Hilbert distance.

A Finsler metric defines a (non-symmetric) distance dF on M : the distance between two points x and y of M is the minimal Finsler length LF .c/ of a C 1 curve c W Œ0; 1 ! M from x to y, that is, Z 1 F .c.t P // dt: dF .x; y/ D inf LF .c/ D inf c

c

0

We will say that a Finsler metric F is regular if F is C 2 and the boundaries of its unit balls B.x; 1/ D fu 2 Tx M; F .x; u/ < 1g; x 2 M have positive definite Hessian (for some (hence any) fixed Euclidean metric on Tx M ). This is the minimal assumption we have to make for local geodesics to exist: as in Riemannian geometry, those are C 1 curves defined through a second order differential equation. Geodesics have constant speed, that is, F .c.t P // is constant, and if x D c.t / and y D c.t 0 / are points on the curve which are close enough, then the curve c W Œt; t 0 ! M is the shortest path from x to y: Z t0 F .c.s// P ds: dF .x; y/ D t

We say that the Finsler metric is complete if geodesics exist for all times. This is always the case if the manifold is compact. If the Finsler metric is complete, we can define the geodesic flow of the metric as the flow of the second-order differential equation which defines geodesics. In this way, the geodesic flow is defined on the tangent bundle T M . But, since geodesics have constant speed, this flow preserves the subbundles T ˛ M D fu 2 T M; F .u/ D ˛g;

˛ > 0:

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The most natural space to study the geodesic flow is thus the unit tangent bundle T 1 M . However, this space depends on the metric, so we will prefer considering this flow as defined on the homogeneous tangent bundle HM D T M X f0g=RC , identifying it with the unit tangent bundle: a point of the homogeneous tangent bundle consists of a base point on the manifold and a tangent direction at this point that we denote by w D .x; Œ/; its image ' t .w/ by the geodesic flow is obtained by following during the time t the unit speed geodesic leaving x in the direction Œ. Geodesic flows are important examples in dynamics, especially when the manifold is not “too big”, so we can expect strong recurrence properties. For example, the geodesic flow of a compact negatively-curved Riemannian manifold has strong hyperbolic properties, such as the Anosov property. The main goal of this chapter is to study these kind of properties for the geodesic flow of compact quotients of some Finsler geometries, with an emphasis on Hilbert geometries.

2.3 Hilbert geometries and their isometries A Hilbert geometry .; d / is an example of a Finsler manifold. The Finsler norm of u 2 Tx is given by the formula 1 1 juj C ; F .x; u/ D 2 jxuC j jxu j where u˙ are the intersection points of the line generated by u. It is Riemannian if and only if is an ellipsoid [50], in which case .; d / is the Beltrami model of the hyperbolic space. Various volume forms can be associated to a Finsler metric. However, all natural volumes that could be associated to Hilbert geometries are equivalent (see L. Marquis’ contribution in this volume [46]. So let us fix the volume Vol once and for all as

u−

x

u

Figure 2. The Finsler metric.

u+

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being the Busemann–Hausdorff volume: this is the Hausdorff measure of .; d /; equivalently, it is the volume such that the unit Finsler ball has the same volume as the Euclidean unit ball. Our main interest will lie in those Hilbert geometries which have enough symmetries to admit compact quotients. The rest of this part recalls some facts about the existence of such quotients. 2.3.1 Regular Hilbert geometries. From the formula, we see that F is regular if the boundary @ of is C 2 with positive definite Hessian1 . Geodesics coincide with projective lines. Thus, it is easy to see the geodesic flow on H : we just have to follow lines…. However, apart from the case of the ellipsoid, a regular Hilbert geometry has essentially no quotients: Theorem 2.2 (É. Socié-Méthou [51]). The isometry group of a regular Hilbert geometry .; d / is compact, unless is an ellipsoid. Hence, there is no interesting space where to study their geodesic flows from a global point of view. That is why we have to consider less regular Hilbert geometries. 2.3.2 Geodesics. For a general Hilbert geometry, geodesics can be defined metrically: a geodesic segment is a metric isometry c W Œ0; T ! from R to .; d / and a geodesic is an isometry from R to . It is not difficult to see that projective lines are still geodesics but there might be others, even locally [22]. This happens as soon as the boundary of the convex set contains two nonempty open segments which are in the same 2-dimensional subspace but not in the same supporting hyperplane. 2.3.3 Isometries. The automorphism group Aut./ consisting of those projective transformations preserving is an important subgroup of isometries. Indeed, we expect Aut./ to be the full isometry group in most cases: Conjecture. Aut./ D Isom.; d / unless .; d / is symmetric, in which case Aut./ is a subgroup of index 2 of Isom.; d /. Homogeneous Hilbert geometries are those whose automorphism group acts transitively. They have been described by M. Koecher and E. B. Vinberg in the fifties and sixties [37], [52]. Among them, symmetric ones are those which are self-dual. They fall into three classes: the simplices, the hyperbolic space and the symmetric spaces of the groups SL.n; K/, with K D R; C; H and n > 3 or the exceptional group E6.26/ (see [24] for example). The last conjecture is confirmed in some cases: 1 The Hessian is computed in some affine chart equipped with a Euclidean metric; its positive definiteness does not depend on the choice of the chart and the metric. See Section 8.5 for more on this.

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• for strictly convex sets: it is a consequence of the uniqueness of geodesics (P. de la Harpe [22]); • for polytopes: Aut./ D Isom.; d / unless is a simplex, in which case Aut./ has index 2 in Isom.; d / (P. de la Harpe [22] for the 2-dimensional case, B. Lemmens and C. Walsh [40] for the general case) • for symmetric Hilbert geometries: A. Bosché [14] proved that Aut./ has index 2 in Isom.; d /. A careful and general study of Aut./ is made in Marquis’ contribution (Chapter 8, [46]). In particular, he describes and classifies automorphisms in terms of the dynamics of their action on . 2.3.4 Divisible Hilbert geometries Definition 2.3. A quotient M D = of a Hilbert geometry .; d / by a discrete subgroup of Aut./ is called a convex projective manifold (or orbifold in case has torsion). A Hilbert geometry .; d / or the convex set is said to be divisible if it admits a compact quotient by a discrete subgroup of Aut./. Remark that by Selberg’s lemma, if < Aut./ has finite type, in particular when acts cocompactly on , then has a finite index subgroup without torsion. In this case, the quotient M D = is a smooth manifold and the group is isomorphic to the fundamental group 1 .M / of M via a faithful linear representation 1 .M / ! SL.n C 1; R/. Socié-Méthou’s theorem asserts that the hyperbolic space is the only regular divisible Hilbert geometry. Among homogeneous Hilbert geometries, only the symmetric ones and their products are divisible [53]. Remark that among symmetric Hilbert geometries, the hyperbolic space is the only one to be strictly convex or with C 1 boundary. Apart from the homogeneous cases, the existence of divisible Hilbert geometries has been a long-standing question. From a general point of view, the following result of Y. Benoist is essential: Theorem 2.4 (Y. Benoist [11]). Let .; d / be a divisible Hilbert geometry, divided by a discrete subgroup < Aut./. The following propositions are equivalent: • the convex set is strictly convex; • the boundary @ is C 1 ; • the metric space .; d / is Gromov-hyperbolic; • the group is Gromov-hyperbolic.

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Recall that a geodesic metric space .X; d / is said to be Gromov-hyperbolic if there is some ı > 0 such that any geodesic triangle xyz X of vertices x; y; z 2 X is ı-thin, that is, for any point p on the side Œxz, minfd.p; Œxy/; d.x; Œyz/g 6 ı: When the metric space is Gromov-hyperbolic for the constant ı, we say it is ı-hyperbolic. A finitely-generated group is Gromov-hyperbolic if its Cayley graph with respect to some finite set of generators is Gromov-hyperbolic for the word metric. This property does not depend on the generating set, but the constant ı of hyperbolicity does depend on it. 2.3.5 Strictly convex divisible Hilbert geometries. Examples of strictly convex divisible Hilbert geometries are now available in all dimensions. They are obtained by deformations of compact hyperbolic manifolds following an idea of [32]. In low dimensions, examples can be obtained using Coxeter groups. First examples were provided by these means [33]. Y. Benoist [10] also constructed in this way strictly convex divisible Hilbert geometries which are not quasi-isometric to the hyperbolic space. M. Kapovich [34] found such examples in all dimensions. We refer to Marquis’ contribution [46] for a deeper study of this question. Assume M D = is a compact quotient manifold of a strictly convex Hilbert geometry. Then all elements g 2 are hyperbolic isometries of .; d /, which means the following: • g fixes exactly two points xgC and xg on the boundary @ and acts as a translation (for the Hilbert metric) on the line .xg xgC /, which is called the axis of g; x X fxg g, we have limn!C1 g n y D xgC ; • the point xgC is attractive: for any y 2 C n x y D xg . • the point xgC is repulsive: for any y 2 Xfx g g, we have lim n!C1 g

As elements of SL.n C 1; R/, the elements g 2 are biproximal: their biggest and smallest eigenvalues (in modulus) 0 and n are simple, that is, the corresponding eigenspaces are 1-dimensional; these eigenspaces are the fixed points xgC and xg ; the translation distance on the axis is 12 log n0 . Let us end this paragraph with a result about the group which divides the Hilbert geometry that will be crucial to deduce dynamical rigidity results. It reads as follows: Theorem 2.5 (Y. Benoist [7]). Suppose that .; d / is a strictly convex Hilbert geometry, divided by a discrete subgroup < Aut./. The group is Zariski-dense in SL.n C 1; R/, unless is an ellipsoid. Recall that the Zariski-closure of a subgroup of SL.n C 1; R/ is the smallest algebraic subgroup G of SL.n C 1; R/ which contains . We then say that is Zariski-dense in G.

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The hypothesis of strict convexity in the last theorem is actually unnecessary, but the proof in this case is far more involved [9]. This last theorem will be useful through the following characterization of Zariskidense subgroups of semisimple Lie groups, which is also due to Y. Benoist. Let us explain it in the context of the Lie group SL.n C 1; R/. To each element g in SL.n C 1; R/, we associate the vector log.g/ D .log 0 .g/; : : : ; log n .g// 2 RnC1 ; where 0 .g/ > 1 .g/ > > n .g/ denote the moduli of the eigenvalues of g. For a subgroup of SL.n C 1; R/, let log D flog g; g 2 g. Theorem 2.6 (Y. Benoist, [8]). Let be a subgroup of SL.n C 1; R/. If is Zariskidense P in SL.n C 1; R/, then the subgroup generated by log is dense in the subspace f i xi D 0g.

2.4 The geodesic flow 2.4.1 Definition. For general Hilbert geometries, we consider the geodesic flow ' t W H ! H following projective lines. Let X W H ! TH be the vector field which generates this flow. If we choose an affine chart and a Euclidean metric j j on it, in which is a bounded open convex set, then X can be written as X D mX e , where X e is the generator of the Euclidean geodesic flow, because X and X e have the same orbits. A direct computation gives m.w/ D 2

1 1 C C jxw j jxw j

1

D2

jxw C j jxw j ; jw C w j

w D .x; Œ/ 2 H ;

where w C (resp. w ) denotes the intersection point of @ with the ray leaving x in the direction (resp. ). This link between X and X e , that is, the flatness of Hilbert geometries, is crucial in extending some differential objects in Section 3. The vector field X and the geodesic flow ' t have the same regularity as the boundary of . If M D = is a quotient manifold, the geodesic flow is defined on HM by projection. To study its (local) properties, we will often work directly on H where geodesics are lines and computations can be made. For a compact quotient, we can expect from Benoist’s theorem strong differences for the geodesic flow on HM between the strictly convex (with C 1 boundary) and non-strictly convex (with non-C 1 boundary) cases. We can give easy illustrations of this difference: • at a non-C 1 point of the boundary, there are asymptotic geodesics whose distance does not go to 0; • if I is an open segment in the boundary @, the distance between two geodesic rays ending at some points of I stays bounded.

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We do not know much more than these observations in the non-strictly convex case. On the contrary, more attention has been paid to the strictly convex case, where the flow happens to exhibit strong hyperbolicity properties. This chapter is dedicated to this case, that is, the study of the geodesic flow of a compact quotient M D = with strictly convex (and by Theorem 2.4 its boundary is of class C 1 ). 2.4.2 Closed orbits. Assume M D = is any quotient manifold of a strictly convex (with C 1 boundary) Hilbert geometry .; d /, with a discrete subgroup of Aut./. We use the same notation on and M , on H and HM . The orbit of a point w 2 H is the set ' w D f' t .w/; t 2 Rg. The orbit ' w of w 2 HM is closed if there exists T > 0 such that ' T .w/ D w. The smallest T > 0 which satisfies this equality is the length of the orbit. We also say that w is a periodic point and that T is its period. If G is a group, a non-identity element g of G will be called primitive if there is no h 2 G and k > 2 such that g D hk . The same will be said of the conjugacy class of g in G. For example, if Œg is a free homotopy class in M , that is, the conjugacy class of an element g in the group 1 .M /, Œg is primitive if “it does not make more than one loop”. A closed orbit on HM lifts to an orbit on H which projects down in onto an oriented line .x x C /, x ; x C 2 @, which is left invariant by some non-identity hyperbolic elements of . Among such elements, which are all hyperbolic of axis .x x C /, only one is primitive and has x C as attractive fixed point and x as repulsive fixed point. In this way, we can associate to each closed orbit a primitive hyperbolic element of . Conversely, to such an element is associated the oriented line .x x C /, which yields a closed orbit of the geodesic flow. Two such elements will define the same closed orbit if they are conjugate. Therefore we have the Proposition 2.7 (see [11], Proposition 5.1). Let .; d / be a strictly convex Hilbert geometry with C 1 boundary and M D = a compact quotient manifold, where is a discrete subgroup of Aut./. Closed orbits of the geodesic flow on HM are in bijection with conjugacy classes of primitive hyperbolic elements of . A closed orbit on HM gives by projection on M a closed geodesic, provided with an orientation. If g is a primitive hyperbolic element of , the orbits defined by g and its inverse g 1 project down on the same closed geodesic in M , the closed orbits differ only by the direction in which they run along the geodesic. A direct computation shows that the length of the associated orbit is 12 log n0 , where 0 and n are the moduli of the biggest and smallest eigenvalues of g.

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3 Differential objects in Finsler and Hilbert geometries Given a Riemannian manifold, its geodesic flow benefits a lot from the rich geometric structure of the manifold, and this can be extended to regular Finsler metrics. For less regular Hilbert geometries, this is much more delicate; only part of the objects can be constructed: these defects can be seen as an explanation for some rigidity results we will present later on.

3.1 The Hilbert 1-form For a regular Finsler metric, the geodesic flow is the Reeb flow of a 1-form A called the Hilbert 1-form. This is Proposition 3.2 below. The Hilbert 1-form is defined via the vertical derivative of the Finsler metric: Definition 3.1. Let F be a C 1 Finsler metric on a manifold M . Denote by p W T M ! M and r W T M Xf0g ! HM the canonical bundle projections. The vertical derivative of F is the 1-form dv F on T M defined by F .x; C "dp.Z// F .x; / ; (3.1) "!0 " where .x; / 2 T M and Z 2 T.x;/ T M . It descends by homogeneity on HM to give the Hilbert 1-form A of F , that is, dv F .x; /.Z/ D lim

A D r dv F: The definition of A is made possible because the 1-form dv F depends only Pon the direction Œ: it is invariant under the flow generated by the vector field D D i @@i . Let W HM ! M denote the bundle projection. Since d .X.x; Œ// 2 Œ and F .d .X.x; Œ/// D 1, we can write A.Z/ D lim

"!0

F .x; d .X "Z// 1 : "

Proposition 3.2. Let F be a regular Finsler metric on a manifold M and A its Hilbert 1-form. The generator X of the geodesic flow of F is the unique solution of A.X/ D 1I

dA.X; / D 0:

Moreover, the geodesic flow preserves the volume form A ^ dAn1 that we call the Liouville volume or Liouville measure. There is another way of seeing A as we explain in the next section. We learned this from T. Barthelmé, and more on this question can be found in his PhD thesis [5]; Arnold’s book [3] is still a good reference for the classical facts on Hamiltonian dynamics.

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3.2 The Legendre transform 3.2.1 Definition for regular metrics. We can see the geodesic flow of a regular Finsler metric as a Hamiltonian flow by using the Legendre transform. Definition 3.3. Let F be a regular Finsler metric on a manifold M . The Legendre transform LF W T M ! T M is defined by the formula LF .v/.u/ D

1 d ˇˇ F 2 .x; v C t u/: 2 dt t D0

(3.2)

For a Riemannian metric, the Legendre transform is linear: for a vector v 2 Tx M , the Legendre transform of v is the dual 1-form defined by v: LF .v/.u/ D hv; ui;

u 2 Tx M:

For a regular Finsler metric, we can see the Legendre transform geometrically in the following way. Let Bx .r/ and Sx .r/ be the metric ball and sphere of radius r > 0 of the Finsler norm F .x; :/ on Tx M . The Legendre transform LF .v/ of a vector v 2 Sx .r/ is then the 1-form such that LF .v/.v/ D F .v/2 I

ker LF .v/ D Tv Sx .r/:

3.2.2 Finsler cometrics Definition 3.4. A Finsler cometric is a function F W T M ! Œ0; C1/ such that: • F .x; ˛/ D F .x; ˛/, .x; ˛/ 2 T M , > 0; • F .x; ˛ C ˇ/ 6 F .x; ˛/ C F .x; ˇ/, x 2 M , ˛; ˇ 2 Tx M . We will say that a Finsler cometric F is regular if F is C 2 and the boundary of its unit coballs have positive definite Hessian. Its Legendre transform LF W T M ! T M is defined by LF .v/.u/ D

1 d ˇˇ .F /2 .x; v C t u/: 2 dt tD0

(3.3)

To a Finsler metric F is naturally associated a Finsler cometric F ; this is the usual dual norm, defined as F .x; ˛/ D maxf˛.v/; v 2 Sx .1/g: If F is regular, the Finsler cometric F is also regular. The Legendre transform is then a C 1 -diffeomorphism from T M to T M which sends F to F : for all v 2 v, F .LF .v// D F .v/. Its inverse is the Legendre transform LF of F . LF is trivially homogeneous of degree 1, that is, LF .v/ D LF .v/, so it defines a map lF from HM to H M D T M X f0g=RC that we still call the Legendre transform. Its inverse is the map lF .

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3.2.3 The Hilbert 1-form from the Hamiltonian point of view. The cotangent symplectic form ! being given space T M is canonically a symplectic manifold, theP by the exterior derivative of the Liouville form L D i dxi . The geodesic flow on T M is the Hamiltonian flow of the Hamiltonian function H.x; ˛/ D 12 .F .x; ˛//2 : this is the flow generated by the vector field XH such that dH.Z/ D !.XH ; Z/;

Z 2 T T M:

Let S M D fu 2 T M; F .u/ D 1g denote the unit sphere bundle of F . The restriction of the projection T M ! H M to S M , is a diffeomorphism that we denote by pF W S M ! H M . Through this map, we can define on H M the 1-form pF L, the 2-form pF ! and the vector field pF XH , which are the pushforward under pF of the restrictions of L; ! and XH to S M . The 2-form pF ! is invariant by the flow of pF XH , and the Liouville measure of this flow is the invariant volume given by pF L ^ .pF !/n1 . Proposition 3.5. Let F be a regular Finsler metric on a manifold M . The Hilbert form A, the 2-form dA and the Liouville volume A ^ dAn1 are the respective pullbacks by lF of the Liouville form pF L, the symplectic form pF !, and the volume pF L ^ .pF !/n1 . The Legendre transform lF also conjugates the flow of pF XH on H M and the geodesic flow of F on HM . 3.2.4 General Finsler metrics. For a general Finsler metric, the previous constructions have in general no meaning. Only the Finsler cometric F is well defined. Using the geometrical construction, the Legendre transform could also be seen as a multiplevalued function: for example, to a unit vector v 2 Tx M where the unit sphere Sx .1/ is not C 1 , the Legendre transform would associate the set of linear forms ˛ such that ˛.v/ D 1 and whose kernel is one of the supporting hyperplanes of Sx .1/ at v. If the Finsler metric is C 1 with strictly convex unit balls, the unit balls of the Finsler cometric F are also strictly convex with C 1 boundary. The Legendre transforms LF and lF are then well defined homeomorphisms through the geometrical construction, but they are no more of class C 1 . Thus, even if the differential forms L, ! and the vector field XH are well defined on T M , as well as their projections by (the now C 1 -map) pF on H M , we cannot pull these objects back to HM by the Legendre transform. However, I do not know if we can define a geodesic flow in this context.

3.3 Connection and parallel transport Given a manifold M , there are many ways to identify two distinct tangent spaces Tx M and Ty M . If M is the affine space Rn , then the most natural way is to identify the tangent spaces to the space itself. For a general manifold, such an identification is called a linear connection: any path c from x to y gives an identification, that is, a linear isomorphism, between Tx M and Ty M via parallel transport.

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When M is a Riemannian manifold, we would like the linear isomorphisms to be actual isometries between the tangent spaces. In other words, we want the Riemannian metric to be parallel. The Riemannian or Levi-Civita connection is the unique linear connection without torsion for which the metric is parallel. This construction cannot extend to the regular Finsler context. As we already said, the good geometrical space for a Finsler metric is the tangent bundle T M and not the manifold M . A “Finsler” connection should then be an object defined on T M , which would extend the Levi-Civita connection, in the sense that the latter should be recovered “by projection” in the case of a Riemannian metric. However, depending on which properties we want the connection to have (such as no torsion, parallelism of the Sasaki metric), we will obtain different connections. A way of avoiding these problems was discovered by P. Foulon: if we restrict ourselves to transport along geodesics then we get a well-defined linear transport. The good space to look at is then HM , and the good object is the geodesic flow. In this context, we can define in an intrinsic and canonical way linear objects generalizing the Riemannian ones. Foulon’s contribution appeared in [25], but one can find a shortest version in English in [27]. Here, we just describe the conclusions, which apply in general for C 3 regular Finsler metrics. The vertical bundle of HM is the bundle V HM D ker d , where is the canonical bundle projection W HM ! M . The tangent bundle to HM admits then a horizontal-vertical decomposition as T HM D R X ˚ hF HM ˚ V HM:

(3.4)

(The vector field X is the generator of the geodesic flow.) Furthermore, there is a pseudo-complex structure on hF HM ˚ VHM which exchanges vertical and horizontal subspaces; that is, a bundle isomorphism J F W hF HM ˚ VHM ! hF HM ˚ VHM such that J F .VHM / D hF HM; J F .hF HM / D VHM

and

J F B J F D Id:

Associated to this decomposition is a “partial” covariant derivative D X which defines the differential of a vector field Z defined along an orbit of the geodesic flow in the direction of the flow. This is a differential operator of order 1, D X W THM ! THM , which commutes with J F , and satisfies D X .X / D 0. A vector field Z defined along an orbit of the flow is parallel if D X .Z/ D 0. This allows us to define the parallel transport of a vector Z.w/ 2 Tw HM along the orbit ' w of w: it is the unique parallel vector field Z defined on ' w whose value at w is Z.w/. If we fix a t 2 R, we will denote by T t .Z.w// WD Z.' t .w// the parallel transport of Z.w/ at time t . This yields a bundle isomorphism T t W THM ! THM which sends TwHM on T' t .w/ HM .

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Since D X commutes with J F , the parallel transport T t also commutes with J F and, since and X is parallel, it preserves the decomposition THM D RX ˚hF HM ˚ VHM . The projection d W THM ! T M induces an isomorphism between the space R X.w/ ˚ hF HM.w/ and T.w/ M for each w 2 HM , so we can define a parallel transport along geodesics on M . Let c W R ! M be a geodesic and x D c.0/, w D Œc.0/ P 2 HM . If u 2 Tx M , we consider its lift U.w/ D d 1 .u/ to the space R X.w/ ˚ hF HM.w/, and define the parallel transport of u along the geodesic c by Tct .u/ D d .T t .U.w///;

t 2 R:

This gives a linear isomorphism T t between the tangent spaces Tx M and Tc.t / M .

3.4 Curvature and Jacobi fields The Jacobi operator RF is defined by RF .X/ D 0;

RF .Y / D pvF .ŒX; J F .Y //;

RF commutes with J F ;

where pvF .Z/ denotes the projection on the vertical subspace VHM of the vector Z 2 THM , with respect to the horizontal-vertical decomposition (3.4). In the case X is the geodesic flow of a Riemannian metric g on M , the Jacobi operator allows to recover the curvature tensor Rg of g: for u; v 2 Tx M X f0g, we have Rg .u; v/u D

d .RF V .x; Œu// ; g.u; u/

where V .x; Œu/ is the unique vector in R X.x; Œu/ ˚ hF HM.x; Œu/ such that d .V .x; Œu// D v. Definition 3.6. We will say that a regular Finsler manifold .M; F / is negatively curved if the Jacobi operator RF is negative definite. A Jacobi field is a vector field J on HM which satisfies D X D X J C RF J D 0: Jacobi fields form a 2.2n1/-dimensional vector space, which is J F -invariant. In the Riemannian case, one recovers the usual Jacobi fields by projection on the base. Jacobi fields on HM do not contain more information than the ones on M , the projection only rubs out the J F -symmetry between horizontal and vertical subspaces, as for the tensor curvature.

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3.5 Metrics on H M For a Riemannian metric g on M , there is a canonical associated Riemannian metric on HM called the Sasaki metric. To define it, we use the Levi-Civita connection THM D R X ˚ VHM ˚ hg HM and do the following construction. • The decomposition THM D R X ˚ VHM ˚ hg HM is orthogonal for the Sasaki metric. • Each subspace R X.u/ ˚ hgu HM is isomorphic to the tangent space T.u/ M of M via the projection ; the quadratic form on R X.u/ ˚ hg HM.u/ is defined as the pullback of g.u/ by . • Vertical and horizontal subspaces are identified by the complex structure J g ; the quadratic form on Vu HM is the push-forward by J g of the quadratic form on hgu T M . By construction, the Sasaki metric is invariant by the pseudo-complex structure, and vertical and horizontal subspaces are isometric. There are different ways of extending this definition to the Finsler context. Usually, Finsler geometers consider the following generalization for regular Finsler metrics. The Sasaki metric on HM associated to a regular Finsler metric F on M is the Riemannian metric g F defined by F .Y; Y 0 / D Hess.x;u/ .F 2 /.Y; Y 0 /; g.x;Œu/

Y; Y 0 2 VHM;

where u is the unit vector in Œu, and g F .X; X/ D 1;

g F .h; h0 / D g F .J F .h/; J F .h0 //;

h; h0 2 hF HM:

Equivalently, the formula on VHM can be replaced by g F .Y; Y 0 / D dA.ŒX; Y ; Y 0 /;

Y; Y 0 2 VHM;

where A is the Hilbert 1-form introduced above. Another way would be the following, but this would yield a Finsler metric on HM : we let, for Z D aX C Y C h, 1=2 : kZk D jaj2 C .F .d h//2 C .F .d J F Y //2 When the metric is Riemannian, we recover the Sasaki metric. When the metric is Finsler, this metric is only a Finsler metric (except for surfaces), whose regularity is one less than the original one. The complex structure is still an isometry between vertical and horizontal subspaces. In fact, I do not think the metric k k was already considered before. However, the first construction does not make sense for non-regular Finsler metrics and that is the reason why we consider the second one. In particular, we will use this second metric to study non-regular Hilbert geometries.

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3.6 The case of Hilbert geometries The above constructions work in general for at least C 3 regular Finsler metrics. It is however possible to exploit the flatness of Hilbert geometries to extend this formalism to the case where the convex set is strictly convex with C 1 boundary. Choose an affine chart adapted to the properly convex set RP n and fix a Euclidean metric on it. Denote by X e W H ! H the generator of the Euclidean geodesic flow. The crucial thing is the link X D mX e which exists between the geodesic flow of the Hilbert and the Euclidean metrics and the fact that m and its first derivatives are smooth in the direction of the flow(s). We can find details in [20]. It allows us to use freely Foulon’s dynamical formalism, that is, the objects defined in Sections 3.3 and 3.4. For example, we can prove the following proposition, which is immediate for regular metrics by using the differential dA of the Hilbert form: Proposition 3.7 ([20]). Let be a strictly convex subset of RP n with C 1 boundary and A the Hilbert form of the Hilbert metric F on . Then (i) ker A D VH ˚ hF H ; (ii) A is invariant under the geodesic flow. A general result linking RX and RX and a direct computation prove that such Hilbert geometries have constant curvature 1: e

Proposition 3.8. Let be a strictly convex subset of RP n with C 1 boundary. The Jacobi endomorphism of the Hilbert geometry .; d / is RF D Id on hF H C VH . However, the interpretation of such a result is not the same as in Riemannian geometry. We should not forget that the good geometrical space in Finsler geometry is the tangent bundle and not the manifold. Perhaps, the main consequence of this fact (at least in the context of this chapter) is an explicit computation of Jacobi fields: Lemma 3.9. Let .; d / be a Hilbert geometry defined by a strictly convex set with C 1 boundary. Let w 2 H , and Jw 2 VH C hF H . Then the Jacobi field J along the orbit of w such that J.w/ D Jw and D X J.w/ D ˙Jw is given by J.' t w/ D e ˙t T t Jw : 0 This describes the behaviour of all Jacobi fields since for any Zw ; Zw 2 TH , we can write C C Zw ; Zw D X.w/ C Zw

0 C Zw D 0 X.w/ C Zw Zw ;

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with C D Zw

0 Zw C Zw ; 2

Zw D

0 Zw Zw : 2

Hence the Jacobi field J along the orbit of w such that J.w/ D Zw and D X J.w/ D 0 will be given by Zw C J.' t w/ D .0 t C /X.' t w/ C e t T t Zw C e t T t Zw :

4 Stable and unstable bundles and manifolds We assume in this section that the convex set which defines the Hilbert geometry is strictly convex with C 1 boundary. The geodesic flow ' t W H ! H is then a C 1 flow. We want to study the (spatially infinitesimal) behaviour of the geodesic flow on the manifold H equipped with the Finsler metric k k introduced in Section 3.5. We denote by dH the metric induced by k k on H . Even if we will only consider compact quotients in the other sections, we do not make here any assumption on the existence of a quotient manifold, aiming to provide objects and results that could be useful for more general quotients.

4.1 Busemann functions and horospheres The Busemann function based at 2 @ is defined by b .x; y/ D lim d .x; p/ d .y; p/: p!

In some sense, it measures the (signed) distance from x to y in as seen from the point 2 @. A particular expression for b is given by b .x; y/ D lim d .x; .t // t; t!C1

where is the geodesic leaving y at t D 0 to . When is fixed, then b is a surjective map from onto R. When x and y are fixed, then b: .x; y/ W @ ! R is bounded by a constant C D C.x; y/. The horosphere passing through x 2 and based at 2 @ is the set H .x/ D fy 2 ; b .x; y/ D 0g: H .x/ is also the limit when p tends to of the metric spheres S.p; d .p; x// about p passing through x. In some sense, the points on H .x/ are those which are as far from as x is. The (open) horoball H .x/ defined by x 2 and based at 2 @ is the “interior” of the horosphere H .x/, that is, the set H .x/ D fy 2 ; b .x; y/ > 0g:

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It is easy to see that horospheres have the same kind of regularity as the boundary of . A consequence of (the proof of) Proposition 3.7 is the following Corollary 4.1. Let w D .x; Œ/ 2 H , w ˙ D ' ˙1 .w/ 2 @ and the unit vector in Œ. The projection d .VH C hF H / is the tangent space at x to both Hw C .x/ and Hw .x/ and the tangent space at to the unit sphere of F .x; / in Tx .

4.2 Stable and unstable bundles and manifolds Recall the following notions: Definition 4.2. The stable set of w 2 H is the set of points v 2 H such that lim t !C1 d.' t w; ' t v/ D 0. The unstable set is the set of points v 2 H such that lim t !1 d.' t w; ' t v/ D 0. Knowing a bit of hyperbolic geometry, it is not difficult to “find” what should be the stable and unstable sets of a point w D .x; Œ/ 2 H . Because d > d B , we see that the extreme point of the orbit of a point v 2 W s .w/ must coincide with the one for w, that is, v C D w C . Moreover, the C 1 regularity of the boundary @ implies that the horosphere Hw C ..w// about w C through .w/ satisfies Hw C ..w// D fv 2 H ; d ..' t .w//; .' t .v// D 0g : Thus, the stable set of w has to be a subset of the set W .w/ defined as the set of points v such that v C D w C and whose projection .v/ is on the horosphere through w about w C (see Figure 3): ˚ W .w/ D v 2 H j v C D w C ; .v/ 2 Hw C ..w// : Similarly, the unstable set of w has to be a subset of the set ˚ W C .w/ D v 2 H j v D w ; .v/ 2 Hw ..w// : Both sets W .w/ and W C .w/ are C 1 submanifolds of H . The sets W .w/, w 2 H , foliate H , and similarly the sets W C .w/; in general, these foliations are only C 0 . We will refer to them as the and C foliations. We immediately see that both foliations are invariant under the geodesic flow: W .' t .w// D ' t .W .w//

and

W C .' t .w// D ' t .W C .w//:

The rest of this section is dedicated to the following Theorem 4.3. Let w 2 H . The sets W .w/ and W C .w/ are the stable and unstable sets of w. The unstable and stable tangent bundles are given by E u D fY C J F .Y /; Y 2 VH g

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and E s D fY J F .Y /; Y 2 VH g D J F .E u /:

W − (w)

w−

x

ξ

w+

W + (w)

Figure 3. The submanifolds W C .w/ and W .w/, w D .x; Œ/.

4.2.1 A temporary Finsler metric. Let E and E C be the tangent bundles to the and C foliations. These bundles define a ' t -invariant decomposition of the tangent bundle T HM D R X ˚ E ˚ E C : Corollary 4.1 implies that E ˚ E C D VH C hF H . We define a temporary Finsler metric k k˙ by 1=2 : kZk˙ D jaj2 C 4.F .d Z C /2 C F .d Z /2 / It is not difficult to see that W C .w/ and W .w/ are the stable and unstable sets of w for this metric: this is what Corollary 4.5 below asserts. We first need to make a little computation. To simplify this computation, we will use the projective nature of our objects and choose a good affine chart and a good Euclidean metric on it. Let w D .x; Œ/ 2 H . A good chart at w is an affine chart where the intersection Tw C @ \ Tw @ is contained in the hyperplane at infinity, and a Euclidean structure on it so that the line .xw C / is orthogonal to Tw C @ and Tw @ (see Figure 4). Lemma 4.4. Let w 2 H , Z 2 E .w/ and fix a good chart at w. Set x D .w/, x t D ' t .w/, z D .Z /, z t D d d' t .Z /. We have jx t w C j jzj jx t w C j t ˙ C kd' .Z /k D 2F .z t / D : jxw C j jx t z tC j jx t z t j

Chapter 7. The geodesic flow of Finsler and Hilbert geometries

Tw− ∂Ω

181

Tw+ ∂Ω

x

w−

ξ

w+

Figure 4. A good chart at w D .x; Œ/.

Proof. We have kd' t .Z /k˙ D 2F .z t / by definition of the metric k k˙ . Now, by definition of F , we get jz t j 1 1 F .z t / D ; C jx t z t j 2 jx t z tC j where z tC and z t are the intersection points of the line fx t C z t ; 2 Rg and @ (see Figure 5). Consider the map h t W y 2 Hw C .x/ 7! y t D ' t .y; Œyw C / D .yw C / \ Hw C .x t /: We see that z t is given by z t D dh t .z/ D

jx t w C j z: jxw C j

This gives the result. We have a similar result for Z C 2 E C .w/. If Z C 2 E C .w/, with the same notation, we have jx t w j jzj jx t w j t C ˙ kd' .Z /k D F .z t / D : (4.1) C 2jxw j jx t z tC j jx t z t j The strict convexity of the convex set and the C 1 regularity of its boundary now yield the following Corollary 4.5. Let Z 2 E , Z C 2 E C . The map t 7! kd' t Z k˙ is a strictly decreasing bijection from R onto .0; C1/, and the map t 7! kd' t Z C k˙ is a strictly increasing bijection from R onto .0; C1/.

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zt+

z w

−

ξ x

zt xt

w+

zt− Figure 5. Contraction of stable vectors.

In particular, W .w/ and W C .w/ are the stable and unstable sets of w for the metric k k˙ . We will prove in the sequel that this metric actually coincides with the metric k k. 4.2.2 Identification of stable and unstable bundles for the metric k k. For a Riemannian or regular Finsler manifold M of variable negative curvature, the construction of stable and unstable manifolds is achieved through the understanding of Jacobi fields, via the following fact: for any Zw 2 Tw HM , the vector field Z defined along the orbit of w by Z.' t w/ D d' t Zw is a Jacobi field. This observation stays valid in Hilbert geometry and, as we have seen, the behaviour of Jacobi fields is easy to understand via parallel transport. In this way, we can identify the stable and unstable bundles in terms of the differential objects of Section 3. They are given by E u D fY C J F .Y /; Y 2 VHM g;

E s D fY J F .Y /; Y 2 VHM g D J F .E u /;

as asserts the next Proposition 4.6. Let Z s 2 E s , Z u 2 E u . The map t 7! kd' t Z s k is a strictly decreasing bijection from R onto .0; C1/, and the map t 7! kd' t Z u k is a strictly increasing bijection from R onto .0; C1/. To prove this proposition, we need some preparatory observations. We have THM D RX ˚E u ˚E s and this decomposition is ' t -invariant. As said above, for any Zw 2 Tw HM , the vector field Z defined along ' w by Z.' t w/ D

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u u d' t Zw is a Jacobi field. Furthermore, if Zw 2 E u .w/, we have D X .Z u /.w/ D Zw , and Proposition 3.9 implies that

d' t .Z u / D e t T t Z u :

(4.2)

s s If Zw 2 E s .w/, we have D X .Z s /.w/ D Zw and

d' t .Z s / D e t T t Z s :

(4.3)

To understand the behaviour of kd' t .Z u /k and kd' t .Z s /k, Z u 2 E u , Z s 2 E s , we have to understand the behaviour of kT t Z s k and kT t Z u k. Here is the big difference with the regular Finsler or Riemannian case. In the latter cases, the parallel transport preserves the Sasaki metric. For the Hilbert geometries under consideration, the parallel transport does not preserve the metric k k. In fact, we could prove the Proposition 4.7. Let .; d / be a Hilbert geometry defined by a strictly convex set with C 1 boundary. The following propositions are equivalent: • the boundary of the convex set is C 2 with positive definite Hessian; • for any w 2 H , there exists C > 0 such that C 1 kZk 6 kT t Zk 6 C kZk;

Z 2 Tw H :

To understand the parallel transport, we compare it with the parallel transport of the Euclidean structure. Fix an affine chart and a Euclidean metric j j on it, in which appears as a bounded open convex set. Denote by X e W H ! TH the generator of the Euclidean geodesic flow. Recall that X D mX e . Let w 2 HM and pick a vertical vector Y .w/ 2 Vw HM . Denote by Y and Y e its parallel transports with respect to X and X e along ' w. Let h D J F .Y / and e he D J X .Y e / be the corresponding parallel transports of h.w/ D J F .Y .w// and e he .w/ D J X .Y e .w// along ' w. The main result is the following Lemma 4.8. Along the orbit ' w, we have

m.w/ Y D m

1=2

Ye

and h D LY m X e C .m.w/m/1=2 he

m.w/ LX e m Y e : m

Now, we can complete a Proof of Proposition 4.6. Consider the vector Z.w/ D Y .w/ C h.w/ D Y .w/ J F .Y .w// 2 E s .w/. We use the notation of Proposition 4.4 and its proof. In a good chart at w, LY m D 0 along the orbit ' w, hence kT t Z.w/k D F .d .T t h.w/// D .m.w/m.' t .w///1=2 F .d .he .' t .w///;

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where he is as above. From Corollary 4.1 and the fact that hF H CVH D E s CE u , the vector d .T t .h.w/// is in Tx t H' t .w/ . The Euclidean parallel transport preserves the Euclidean metric so one has jd .he .' t .w//j D jd .he .w//j D jd .h.w//j. Keeping the same notation (see Figure 6), these two observations give t

t

kT Z.w/k D .m.w/m.' .w///

1=2 jd .h.w//j

C 1=2

jx t w j D C.w/ jx t z tC j

2

1 1 C C jx t z t j jx t z t j

jx t w C j1=2 ; C jx t z t j

for some constant C.w/ > 0. This equality is similar to the one in Lemma 4.4. The strict convexity of the convex set and the C 1 regularity of its boundary conclude the proof. zt+

dπ(h(w)) ω−

x

dπ(T t h(w)) xt

ω+

zt− Figure 6. Action of the parallel transport.

4.2.3 Both constructions coincide. We will now conclude the proof of Theorem 4.3, through the Proposition 4.9. Let .; d / a Hilbert geometry defined by a strictly convex set with C 1 boundary. We have E s D E , E u D E C and k k D k k˙ . I do not know a simple proof of this fact. If the geometry is divisible, then the following lemma concludes:

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Lemma 4.10. If k k and k k˙ are bi-Lipschitz equivalent on H , then E s D E , E u D E C and k k D k k˙ . Proof. Pick a vector Z 2 E , decompose it with respect to E s ˚ E u , and use Corollary 4.5 and Proposition 4.6 to conclude. Otherwise, we can do as follows. For related material concerning Benzécri’s theorem and Gromov-hyperbolic Hilbert geometries, Marquis’ contribution [46]. Proof of Proposition 4.9. Let X D f.; x/; x 2 g and X 0 D f.; x/ 2 X; is strictly convex with C 1 boundaryg: For ı > 0, let X ı D f.; x/ 2 X; the Hilbert geometry .; d / is ı hyperbolicg: Finally, let Xh D

[

Xı

ı>0

be the set of Gromov-hyperbolic Hilbert geometries (see Section 2.3.4 for the definition of Gromov-hyperbolicity). The space X is equipped with the topology induced by the Gromov-Hausdorff distance on subsets of RP n on the -coordinate, and the topology of RP n for the x-coordinate. The subsets X 0 , X h , X ı inherit the induced topology. We have X ı X h X 0 for any ı > 0. The space X h contains all .; x/ such that .; d / is regular ([18]), and so X h is dense in X 0 and X . J.-P. Benzécri [12] proved that the action of the projective group PGL.n C 1; R/ on X is proper and cocompact. Y. Benoist [10] proved that X ı is closed in X, so the action of PGL.n C 1; R/ on X ı is also proper and cocompact. Consider the map f W .; x/ 2 X 0 ! R defined by

²

³

kZk˙ ; Z 2 Tw H ; .w/ D x : f .; x/ D max kZk This map is continuous, positive and PGL.n C 1; R/-invariant on X 0 . So, by the Benoist–Benzécri result, there exists Cı such that Cı1 6 f 6 Cı on X ı . Hence, for any Gromov-hyperbolic Hilbert geometry .; d /, k k and k k˙ are bi-Lipschitz equivalent and Lemma 4.10 implies that E s D E , E u D E C and k k D k k˙ . In other words, f 1 on the subset X h of X 0 . Since X h is dense in X 0 and f is continuous, we have f 1 on X 0 , which concludes the proof.

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5 Hyperbolicity and Lyapunov exponents In this section, we want to understand the asymptotic behaviour of the geodesic flow locally around a given orbit; in particular, we want to see when it is locally hyperbolic. Since these are local considerations, we work in the universal cover and on H . Throughout this section, the Hilbert geometry .; d / is assumed to be defined by a strictly convex set with C 1 boundary.

5.1 Hyperbolicity of an orbit The orbit ' w of the point w 2 H of the geodesic flow ends at the point w C on the boundary. A fundamental observation is that most of the local properties of the geodesic flow around the orbit ' w are given by the local shape of the boundary @ at the point w C . Recall the following notions. Definition 5.1. Let k 2 N; 0 < " 6 1 and ˇ > 2. Let f W U Rn ! Rk be defined on an open subset U . Denote by T k f its Taylor expansion up to order k, if defined. The function f is said to be • of class C kC" on U if f is C k on U and, for some constant C > 0, jf .x/ f .y/ T k f .x/.x y/j 6 C jx yjkC" ;

x; y 2 U I

• of class D kC" at x 2 U if f is k-times differentiable at x and, for some constant C > 0 and a neighborhood V of x, jf .x/ f .y/ T k f .x/.x y/j 6 C jx yjkC" ;

y 2VI

• ˇ-convex on U if f is C 1 on U and, for some constant C > 0, jf .x/ f .y/ T 1 f .x/.x y/j > C jx yjˇ ;

x; y 2 U:

• ˇ-convex at x 2 U if f is differentiable at x and, for some constant C > 0 and a neighborhood V of x, jf .x/ f .y/ T 1 f .x/.x y/j > C jx yjˇ ;

y 2 V:

To study the behaviour around the orbit ' w of the point w 2 H , we look at the action of the differential d' t : we pick a vector Z 2 Tw H and look at the asymptotic behaviour of the function t 7! kd' t Zk. This function is constant for Z 2 R X. Corollary 4.5 states that, for Z 2 E s .w/ (resp. Z 2 E u .w/), kd' t Zk decreases to 0 (resp. increases to C1). To go further, we want to see when the contraction/expansion on stable and unstable subspaces are exponential.

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Definition 5.2. A point w 2 H or its orbit ' w is said to be hyperbolic if sup

lim sup

1 log kd' t Z s k < 0; t

inf

lim inf

1 log kd' t Z u k > 0: t

Z s 2E s .w/ t!C1

and Z u 2E u .w/

t!C1

From Equalities (4.2) and (4.3), we see that lim sup

1 1 log kd' t Z s k D 1 C lim sup log kT t Z s k; t t!C1 t

lim inf

1 1 log kd' t Z u k D 1 C lim inf log kT t Z u k; t!C1 t t

t !C1

Z s 2 E s .w/;

and t !C1

Z u 2 E u .w/:

Moreover, from the fact that E u D J F .E s / and from the J F -invariance of the norm, we see that w is hyperbolic if and only if 1 < lim inf

t!C1

1 1 log kT t Z s k 6 lim sup log kT t Z s k < 1; t t!C1 t

Z s 2 E s .w/:

In terms of parallel transport on , w is hyperbolic if and only if 1 < lim inf

t !C1

1 1 log F .Twt v/ 6 lim sup log F .Twt v/ < 1; t t!C1 t

v 2 T.w/ Hw C ..w//:

Define .w/ N D

sup

lim sup

1 log kd' t Z u k t

inf

lim inf

1 log kd' t Z u k: t

Z u 2E u .w/ t!C1

and .w/ D

Z u 2E u .w/ t!C1

These two numbers control the exponential rate of expansion on E u along the orbit ' w. Using formula (4.1), we can prove the Proposition 5.3. An orbit ' w is hyperbolic, with exponents 0 < N < 2 if and only if the boundary @ is of class C 1C" and ˇ-convex at the point w C for all 1 < 1 C " < 2N and ˇ > 2 .

5.2 Regular orbits and Lyapunov exponents The exponential rate of contraction/expansion can depend on the vector that we consider.

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Definition 5.4. A point w 2 H or its orbit ' w is said to be forward weakly regular if for any Z 2 Tw H , the quantity 1t log kd' t Zk admits a limit .Z/ when t goes to C1. It is said to be weakly regular if this quantity has the same limit when t goes to 1. From previous considerations, to see if a point w is forward weakly regular, we only have to check that the limit exists for all Z 2 E s .w/ or that .v/ D lim t !C1 1t log F .Twt v/ exists for all v 2 T.w/ Hw C ..w//. Let w 2 H be a forward weakly regular point. We call the number .v/ the parallel Lyapunov exponent of the vector v. These numbers f .v/; v 2 T.w/ Hw C ..w//g can take only a finite number 1 .w/ < < p .w/ of values, which are called the parallel Lyapunov exponents of w (or its orbit). There is then a decomposition T.w/ Hw C ..w// D E1 ˚ ˚ Ep called Lyapunov decomposition, such that, for any vector vi 2 Ei X f0g, 1 log kTwt .vi /k D i .w/: t!C1 t I made a detailed study of (parallel) Lyapunov exponents in [21]. I specify Proposition 5.3 by making a link between the Lyapunov exponents of w and some regularity properties of the boundary @ at w C . Finally, if we want to look at how the flow transforms volumes along a given orbit ' w, we can look at the quantity det Twt . lim

Definition 5.5. A point w 2 H is said to be regular if w is weakly regular, with parallel Lyapunov exponents 1 .w/ < < p .w/ and Lyapunov decomposition T.w/ Hw C ..w// D E1 ˚ ˚ Ep , and if p

X 1 log det Twt D dim Ei i .w/ DW .w/: t!˙1 t lim

iD1

Remark that in this definition, the determinant is not computed with respect to a Riemannian metric but with respect to a Finsler metric. This depends on the volume we associate to the Finsler metric. Since all natural volumes are equivalent, let us say that the determinant is computed with respect to the Busemann–Hausdorff volume Vol . An important fact is that the Lyapunov exponents of a periodic orbit of the geodesic flow of a quotient manifold M D = can be explicitly computed. As we have seen (Proposition 2.7), such an orbit corresponds to the conjugacy class of a hyperbolic element g 2 , and we get the following result, which involves the eigenvalues of g: Proposition 5.6. Let g be a periodic orbit of the geodesic flow of the manifold M D = , corresponding to a hyperbolic element g 2 . Denote by 0 > 1 > > p > pC1 the moduli of the eigenvalues of g. Then

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• g is regular and has no zero Lyapunov exponent; • the Lyapunov exponents . i .g// of the parallel transport along g are given by

i .g/ D 1 C 2

log 0 log i ; log 0 log pC1

i D 1 : : : pI

• the sum of the parallel Lyapunov exponents is given by

.g/ D .n C 1/

log 0 C log pC1 : log 0 log pC1

6 Compact quotients We now want to consider global properties of the geodesic flow of a compact quotient manifold M D =, in the case the convex set is strictly convex (and then with C 1 boundary).

6.1 The Anosov property We begin with the following definition. Definition 6.1. Let ' t W M ! M be a C 1 flow on a compact Finsler manifold .M; kk/. The flow ' t is an Anosov flow it there exists a ' t -invariant decomposition T HM D R X ˚ F s ˚ F u ; called the Anosov decomposition, and constants C; ˛ > 0 such that for any t > 0, kd' t .Z s /k 6 C e ˛t kZ s k; Z s 2 F s ; kd' t .Z u //k 6 C e ˛t kZ u k; Z u 2 F u : This property is named after D. V. Anosov. In [2], he proved that the geodesic flow of a negatively curved compact Riemannian manifold satisfies this property and used it to prove its ergodicity relative to its invariant volume (see Section 7.1). Theorem 6.2 (D. V. Anosov [2], P. Eberlein [23], P. Foulon [26]). Let F be a regular Finsler metric of negative curvature on a compact manifold M . The geodesic flow of F on HM is an Anosov flow. The proof of this result is based on a study of Jacobi fields, whose behaviour can be understood under the curvature assumption. As we have seen, the same can be done in Hilbert geometry, and it is even easier since the curvature is constant: we get actual equalities relating parallel transport and the action of the flow.

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Theorem 6.3 (Y. Benoist, [11]). Let M D = be a compact manifold. The geodesic flow of the Hilbert metric is an Anosov flow on HM , with invariant decomposition THM D R X ˚ E s ˚ E u : Moreover, the geodesic flow is topologically mixing: for any open subsets A and B of HM , there is a time T 2 R such that for all t > T , ' t .A/ \ B 6D ;. The Anosov property is a direct consequence of Corollary 4.5 together with compactness. We give a short proof of this fact below. For the topological mixing property, we refer to Section 5.3 of Benoist’s paper, or to [19] where the proof is extended to some noncompact quotients. The action of the group on the boundary @ is minimal, and this allows us to see that the set f.xgC ; xg /; g 2 g is dense in @@, that is, periodic orbits of the flow are dense in HM . Topological transitivity2 follows from this last result. Topological mixing then comes from the non-arithmeticity of the length spectrum, which is a consequence of Theorem 2.6 through the fact that the length of closed orbits is given by the eigenvalues of the elements of the group (see Section 2.4.2). Proof of the Anosov property. Choose T > 0. The set E1 D fZ 2 E s j kZk D 1g of unit stable vectors is compact, and the function Z 2 E1 7! kd' T .Z/k is continuous, hence attains its maximum for some vector ZM . Lemma 4.5 tells us that a WD kd' T .ZM /k < 1, so that, for all Z 2 E s , kd' T .Z/k 6 akZk: Finally, for t > 0, and setting n D Œt=T , we get kd' t .Z/k 6 an kd' tn Zk 6 DT an kZk 6

DT log at e kZk; at n

˚ t .Z/k ; Z 2 E s; 0 6 t 6 T . with DT D max kd'kZk This gives the upper bound for stable vectors; the same works for unstable ones.

Notice that there is an important difference with the negatively curved Riemannian case. Here, we do not really know what occurs for small t : we know that the norm of a stable vector decreases but we have no control on the rate of decreasing, unlike in the Riemannian case where even infinitesimal exponential rates are controlled by the bounds on the curvature. 2A flow ' t is topologically transitive if for any open subsets A and B of HM , there is a time T 2 R such that ' T .A/ \ B 6D ;

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6.2 Regularity of the boundary As a corollary of Proposition 5.3, we get Corollary 6.4. Let M D = be a compact quotient manifold. Then the boundary @ of is C 1C" and ˇ-convex for all 1 < 1 C " < ˛./ WD

2 2 ; DW ˇ./ ˇ .HM N / .HM /

where N .HM / D .HM N / D sup .w/; w2HM

inf .w/:

w2HM

In particular, the geodesic flow is C 1C" for some " > 0. From symmetry arguments, we can see that 1 1 C D 1: ˛./ ˇ./ Define ˛./ as the biggest 1 < ˛ D 1 C " 6 2 such that @ is C 1C" at all fixed points of the elements of . Obviously, we have ˛./ ˛./ and O. Guichard gave a geometrical proof of the following: Theorem 6.5 (O. Guichard [30]). Let M D = be a compact quotient manifold. We have ˛./ D ˛./. A dynamical proof had been given by U. Hamenstädt in a more general context. She proved the following which implies the previous result. Theorem 6.6 (U. Hamenstädt [31]). Let M D = be a compact quotient manifold. Let .Per/ N D supf .w/; N w 2 HM periodicg: We have .Per/ N D .HM N /.

6.3 Geometrical rigidity The algebraic nature of locally symmetric spaces confer them very specific properties that often turn to be characteristic. Some of these properties directly involve the dynamics of the geodesic flow. For example, let us recall this beautiful result. Theorem 6.7 (Y. Benoist–P. Foulon–F. Labourie [6]). Let .M; g/ be a compact Riemannian (or reversible regular Finsler) manifold of negative curvature. If the Anosov decomposition is C 1 then .M; g/ is locally symmetric.

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Actually, the authors proved a more general result about contact Anosov flows. In particular, they covered the case of a non-reversible Finsler metric: If the Anosov decomposition is C 1 , then there exists a closed 1-form ˛ on M such that F 0 D F C ˛ is a locally symmetric Riemannian metric on M . A similar result, which is easier to prove, allows to characterize the hyperbolic space among all strictly convex divisible Hilbert geometries. Theorem 6.8 (Y. Benoist [11]). Let .; d / be a strictly convex divisible Hilbert geometry. The following are equivalent: (1) the convex set is an ellipsoid; (2) the boundary @ is C 1C" for all 0 6 " < 1; (3) the Anosov decomposition is C " for all 0 6 " < 1. Proof. The implication .1/ ) .2/ is obvious. .2/ , .3/ comes from the description of stable and unstable bundles as E u D fY C J F .Y /; Y 2 VH g;

E s D fY J F .Y /; Y 2 VH g D J F .E u /

and the fact that the subbundle hF H and the map J F are C " if @ is C 1C" . The implication .2/ ) .1/ uses Corollary 6.4. If the boundary @ is C 1C" for all 1 6 ˛ < 2 this means that .HM N / D 1. In particular, the largest Lyapunov exponent of any periodic orbit is 1. By Proposition 5.6 this implies that 2

log 0 .g/ log i .g/ D1 log 0 .g/ log pC1 .g/

for all g 2 . By Theorem 2.6, this implies is not Zariski-dense in SL.n C 1; R/. Theorem 2.5 concludes that is an ellipsoid.

7 Invariant measures There are various ways of looking at a dynamical system. In this section, we want to look at geodesic flows from a measure-theoretic point of view. Let us first recall some basic definitions. We are given a flow ' t on a topological space X. • A Borel measure on X is invariant by the geodesic flow if ' t D , that is, for any Borel subset of X, we have .' t .A// D .A/. • An invariant measure is ergodic if any invariant Borel subset has zero or full measure. • An invariant probability measure is mixing if for any Borel sets A; B, we have lim t !C1 .' t .A/ \ B/ D .A/.B/; mixing implies ergodicity.

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Geodesic flows of compact negatively curved Riemannian or regular Finsler manifolds preserve lots of probability measures. The simplest are probably those defined by periodic orbits: if w is a periodic point of period Tw > 0, we can define an invariant probability measure by pushing forward by the application t 2 Œ0; Tw 7! ' t .w/ the Lebesgue measure of Œ0; Tw , and normalizing it. If we look at the geodesic flow from this measure-theoretic point of view, the system seems trivial: almost every point is periodic with the same period.

7.1 Absolutely continuous measures Let M be a manifold and the Riemannian measure of an arbitrary Riemannian metric on M . We say that a measure on M is smooth or is a volume if d D f d with f W M ! R everywhere positive and continuous; in the case f is only measurable and nonnegative, we say that is an absolutely continuous measure. Since a regular Finsler geodesic flow is in particular a Hamiltonian flow, it preserves the Liouville measure, which is given by A ^ dAn1 , where A denotes the Hilbert form of the metric (see Section 3.2.3). For a compact negatively curved manifold M , this measure is ergodic and so is the only invariant measure in its Lebesgue class.

7.2 Entropies 7.2.1 Topological entropy. A geodesic flow is a continuous action of the group R by diffeomorphisms. Its topological entropy measures the topological complexity of this action. Let us recall the definition for a flow ' t W X ! X on a compact metric space .X; d /. For t > 0, we define the distance d t on X by: d t .x; y/ D max d.' s .x/; ' s .y//; 06s6t

x; y 2 X:

For any " > 0 and t > 0, we consider coverings of X by open sets of diameter less than " for the metric d t . Let N.'; t; "/ be the minimal cardinality of such a covering. The topological entropy of the flow is then the (well-defined) quantity htop .' t / D lim lim sup 1t log N.'; t; "/ : "!0

t!1

It is important to remark that since the space is compact, the topological entropy does not depend on the metric d , but only on the topology defined by d on the space X : if we replace the metric by a topologically equivalent one, then we get the same number. 7.2.2 Measure-theoretical entropy. To every invariant probability measure m of the flow ' t is attached its measure-theoretical entropy hm .' t /, which measures the complexity of the flow from this measure point of view. We refer to Walters’ monograph for a complete description [54]. As could be expected, measures defined by periodic

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orbits have zero entropy whereas the Liouville volume has always positive entropy (in the negatively curved case). We will give an expression of the entropy of the Liouville measure in a forthcoming section. 7.2.3 Variational principle and measure of maximal entropy. A variational principle makes a link between measure-theoretical and topological entropies. This principle asserts that htop .' t / D sup h .' t /;

where the supremum is taken over all invariant probability measures. A measure which achieves the supremum is called a measure of maximal entropy. In some sense, such a measure is well adapted to describe the topological action. The following theorem gives existence and uniqueness of a measure of maximal entropy for hyperbolic dynamics. Theorem 7.1 (see for instance [35]). Let W be a compact manifold. A topologically transitive Anosov flow ' t W W ! W admits a unique measure of maximal entropy. As a consequence, the geodesic flow of a compact negatively curved manifold admits a unique measure of maximal entropy. It is usual to call it the Bowen–Margulis measure because R. Bowen and G. Margulis gave two independent constructions of it; we will denote it by BM . Both constructions are of particular interest. In his PhD thesis, Margulis [44], [45] constructed this measure as a product measure. Stable, unstable and orbit foliations provide local coordinate systems W s W u ."; "/: each point w has a neighborhood U in which any point is at the intersection of exactly one local stable leaf W s \U , one local unstable leaf W u \U and one local orbit ' w \ U . The measure is then described locally as a product BM D s u dt , where the measures s and u are measures on stable and unstable leaves, uniquely defined by the following transition property, involving topological entropy: ' t sw D e htop t s' t .w/ ;

' t uw D e htop t u' t .w/ ;

w 2 W:

In [16], [17], Bowen proved that for the geodesic flow of a compact hyperbolic manifold, closed orbits are uniformly distributed with respect to the Liouville measure, which in this case is also the measure of maximal entropy. Bowen’s construction extends to the case of a topologically transitive Anosov flow, and finally, we find that closed orbits are uniformly distributed with respect to a specific measure, which actually coincides with the one Margulis introduced.

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7.3 Harmonic measure and measure of maximal entropy We saw (Theorem 6.7) that locally symmetric spaces can be characterized among other Riemannian negatively curved compact manifolds by the regularity of their Anosov decomposition. We expect them to be also characterized by some property of their invariant measures. Of particular interest are the Liouville measure, the Bowen–Margulis measure and the harmonic measure. These three measures are naturally related to various aspects of the manifold: • the Liouville measure is directly defined through the local Euclidean structure defined by the Riemannian metric; • the Bowen–Margulis measure describes the distribution of closed orbits of the geodesic flow, hence it is related to the distribution of closed geodesics on the manifold; • the harmonic measure is related to the Brownian motion on the universal covering of the manifold, hence to the Laplace–Beltrami operator. For locally symmetric manifolds, such as hyperbolic manifolds, these three measures coincide and there are deep relations between the Riemannian structure, the distribution of closed geodesics and the Laplace–Beltrami operator. For non-locally symmetric manifolds, we expect the three measures to be distinct, and the previous links to fail. This has been confirmed for the harmonic measure: Theorem 7.2. Let M be a compact negatively curved Riemannian manifold. • The Bowen–Margulis measure coincides with the Liouville measure if and only if z is asymptotically harmonic, that is, horospheres have the universal covering M constant mean curvature. • The harmonic and Liouville measures coincide if and only if M is locally symmetric. The first point is due to Ledrappier [39]; he also proved in the same work that z is Liouville and harmonic measures coincide if and only if the universal covering M asymptotically harmonic. This last result is used by Besson, Courtois and Gallot [13] together with their minimal entropy rigidity theorem to deduce the second point. Concerning the Bowen–Margulis measure, the situation was clarified by A. Katok [36] for surfaces; Foulon [28] extended it to the more general case of contact Anosov flows. We give the following version: Theorem 7.3 (A. Katok [36], P. Foulon [28]). Let .M; F / be a compact regular Finsler surface of negative curvature. The Liouville measure coincide with the Bowen– Margulis measure if and only if M is a Riemannian manifold of constant curvature.

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7.4 Regular Finsler metrics For negatively curved regular Finsler metrics, comparing the Liouville measure A ^ dAn1 with the Bowen–Margulis one is still a relevant question that has not been considered except for Foulon’s contribution [28]. The extension to negatively curved regular Finsler metrics should not change the conjecture: if both measures coincide, the manifold should be Riemannian and locally symmetric. The case of the harmonic measure is more delicate, because there are various ways to extend the Laplace–Beltrami operator from the Riemannian to the Finsler world, namely because it lives on the manifold and not on its tangent bundle. There have been various attempts to define such an extension, but they have not been studied further. A more recent notion is T. Barthelmé’s definition [4] which is directly linked to the approach to Finsler geometry we have here. Barthelmé proved in his Ph.D. thesis [5] that the harmonic measure is well defined in the negatively curved case, but its properties have not been studied yet.

7.5 Hilbert geometries To my knowledge, there is no natural Brownian motion defined on a general Hilbert geometry. All proposed extensions of the Riemannian definition require the regularity of the metric; in particular, there is no natural way of defining a harmonic measure associated to the Hilbert metric of a convex projective compact manifold M D = (apart from the case of hyperbolic manifolds). Let M D = be a compact manifold, with .; d / a strictly convex Hilbert geometry, and a discrete subgroup of Aut./. We have seen that the geodesic flow on HM is a topologically mixing Anosov flow. In particular, according to Theorem 7.1, it admits a unique measure of maximal entropy BM , which is ergodic and mixing. The following general rigidity result implies in particular that this measure is absolutely continuous if and only if M is Riemannian hyperbolic, that is, is an ellipsoid. Theorem 7.4 (Y. Benoist [11]). Let M D = be a compact quotient manifold, with .; d / a strictly convex Hilbert geometry, and a discrete subgroup of Aut./. If is not an ellipsoid, the geodesic flow on HM admits no absolutely continuous invariant measure. Proof. A. Livšic [41], [42] proved that any absolutely continuous measure had to be smooth. In particular, for any periodic point w 2 HM of period T , the change of variable formulas implies that det dw ' T D 1: This implies in our context (see Section 5) that, for any periodic point w 2 HM of period T , det TwT D 1:

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In the notation of Section 5.2, this implies that, for any g 2 , we have .g/ D 0. According to Proposition 5.6, this is equivalent to a condition on the eigenvalues of the element of , which cannot be satisfied if is Zariski-dense in SL.n C 1; R/ via Theorem 2.6. Theorem 2.5 tells us that is an ellipsoid. As we already noticed in Section 3.2.4, the last result can be seen as a consequence of the lack of regularity of the Legendre transform. In particular, this raises the question of characterizing the pullback of the Liouville volume on H M by the Legendre transform. Finally, it seems that there is no counterpart neither for the harmonic measure nor for the Liouville measure in Hilbert geometry. Nevertheless, we will see in the next section that the Liouville measure is also characterized by another property than its absolute continuity, and this allows us to define an extension (in reality two) of the Liouville measure.

8 Entropies 8.1 Volume entropy The volume entropy of a Hilbert geometry measures the exponential growth of volume of balls. It is defined as 1 hvol D lim sup Vol .B.o; R// R R!C1 where o is an arbitrary point in . The Hilbert geometry defined by the simplex has zero volume entropy while the volume entropy of the .n 1/-dimensional hyperbolic space is n 1. We conjecture that those are the extremal cases: Conjecture. Let .; d / be a Hilbert geometry. Then its volume entropy is less than n 1. The result has been proved in dimension 2 by G. Berck, A. Bernig and C. Vernicos. Recently, Vernicos gave a proof in dimension 3. In higher dimensions, this is known to be true for polytopes, which have zero entropy, and for convex sets whose boundary is C 1C1 , for which hvol D n 1. For divisible convex sets, we can use a dynamical approach using the following Theorem 8.1 (A. Manning [43], see also [20]). Let .; d / be a Hilbert geometry and M D = a compact quotient manifold with G < Isom.; d /. The volume entropy of .; d / equals the topological entropy of the geodesic flow of M .

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8.2 Topological entropy Theorem 8.2. Assume M D = is compact, with strictly convex. The topological entropy of the geodesic flow satisfies htop 6 n 1, with equality if and only if is an ellipsoid. Sketch of the proof. Ruelle’s inequality [49] implies that Z

dBM : htop D hBM 6 n 1 C HM

Oseledec’s Theorem [48] says that the set ƒ HM of regular points has full BM measure. Furthermore, this set is invariant under the flip map W HM ! HM defined by .x; Œ/ D .x; Œ/; and we have B D on ƒ. To get the inequality, we just have to remark that, since F is reversible, conjugates t ' and ' t , hence BM is the measure of maximal entropy of both flows and R BM D BM . We conclude that HM dBM D 0. For the equality case, recall that F. Ledrappier and L.-S.Young proved that equality in Ruelle’s inequality Z h 6

C d

occurs if and only if the measure has absolutely continuous conditional measures on unstable manifolds. But the diffeomorphism sends stable manifolds to unstable ones. Since preserves BM , the Lebesgue class of its conditional measures on stable and unstable manifolds coincide. In particular, there is equality in Ruelle’s inequality for BM if and only if BM is itself absolutely continuous. By Theorem 7.4, this implies is an ellipsoid. The last two theorems imply that Conjecture 8.1 is true for divisible strictly convex Hilbert geometries. The numerous examples of such geometries also provide numerous examples whose volume entropy is positive but strictly smaller than n 1.

8.3 Variations of entropy Given a compact manifold M of dimension d , consider the moduli space ˇ.M / of marked convex projective structures on M . Such a structure can be described by a pair .dev; / consisting of z ! RP n which is a diffeomorphism onto a convex set • a developing map dev W M ; • a faithful morphism W 1 .M / ! PSL.n C 1; R/ called the holonomy map with respect to which dev is 1 .M /-equivariant; its image D .1 .M // divides with quotient = diffeomorphic to M .

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The space ˇ.M / is endowed with the topology given by uniform convergence on compact subsets on the first coordinate and the compact-open topology on the second. Proposition 8.3. Let M be a compact manifold of dimension n. The entropy function hvol W ˇ.M / ! .0; n 1 is continuous. Proof. Consider a deformation . ; dev /; 2 Œ1; 1 of a given structure . 0 ; dev0 /. These structures provide Finsler metrics F on the abstract manifold M . These metrics vary continuously with in the following sense: lim

sup

!0 TM Xf0g

F D 1: F0

For let T 1 M be the unit tangent bundle for F0 . Since T 1 M is compact and 7! dev is continuous, lim sup jF F0 j D 0: !0 T 1 M

Moreover minT 1 M F0 > 0, hence lim sup j

!0 T 1 M

F 1j D 0: F0

Homogeneity gives the result, that is, there exist reals C > 1 such that lim!0 C D 1 and F 6 C : C1 6 sup TM Xf0g F0 z z Denote by dQ the associated distances on R M . Let x; y 2 M , and c be the geodesic from x to y for the metric dQ , such that Fz .c0 .t // dt D dQ .x; y/. Then R R Fz .c0 .t// dt Fz .c00 .t // dt dQ .x; y/ 1 6 C : 6 6 C 6 R R Fz0 .c 0 .t// dt Fz0 .c 0 .t // dt dQ0 .x; y/ 0

z, Thus for any x; y 2 M C1 6

dQ .x; y/ 6 C : dQ0 .x; y/

From that we clearly get Bz .x; R/ Bz0 .x; C R/. Hence 1 cardfg 2 1 .M /; gx 2 Bz .x; R/g R R!1 1 6 lim sup cardfg 2 1 .M /; gx 2 Bz0 .x; C R/g R!1 R

hvol . ; dev / D lim sup

D C hvol . 0 ; dev0 /: Similarly,

C1 hvol . 0 ; dev0 /

6 hvol . ; dev /. This gives the continuity.

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For a 2-dimensional manifold M of genus g D 1, the convex structure is necessarily given by a triangle, whose entropy is zero. If g > 2, the space ˇ.M / has been well described by W. Goldman [29] as a manifold diffeomorphic to R16g16 . X. Nie studied in [47] a special kind of projective structures given by simplicial Coxeter groups, and among other things he showed the following Theorem 8.4 (X. Nie [47]). Let M be a compact 2-dimensional manifold. The entropy function hvol W ˇ.M / ! .0; 1 is a surjective continuous map.

8.4 Sinai–Ruelle–Bowen measures Consider a flow ' t on a compact manifold W preserving a smooth probability measure . Birkhoff’s ergodic theorem asserts that there is set A M of full -measure on which the quantity Z 1 t g.' t .x// dt t 0 converges to some g.x/ N as t goes to C1 for any integrable Borel function g. This limit function x ! g.x/ N is integrable and invariant under the flow.R In particular, in the case is ergodic, gN is (-almost everywhere) constant equal to g d . In general, a flow ' t on a compact manifold W does not preserve a smooth measure but it seems legitimate to ask about the behaviour of a random orbit, chosen randomly with respect to some Lebesgue probability measure . In particular, does there exist a subset of full -measure on which the Birkhoff averages Z 1 t g.' t .x// dt t 0 converge for all continuous functions? If this is the case and the limit is constant (-almost everywhere), we can then define a measure C by Z Z 1 t C g d D lim g.' t .x// dt: t!C1 t 0 Such a measure is called a physical measure for the flow ' t since it describes the asymptotic distribution of Lebesgue-almost every orbit. Theorem 8.5 (R. Bowen–D. Ruelle [15]). Any C 1C" Anosov flow on a compact manifold M admits a unique physical measure. Bowen and Ruelle constructed this measure as the equilibrium measure of the potential d ˇ t f C D ˇ tD0 log det d'E u: dt

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That is, C is the unique measure which maximizes the quantity Z P .f C ; / D h C f C d among all invariant probability measures. In particular, from general results, C is ergodic. Furthermore, the pressure P .f C / D sup P .f C ; / is zero and we have Z h D f C d: This implies that C achieves the equality in Ruelle’s inequality and is the unique measure to do so. In particular, from the Ledrappier–Young theorem [38], C is the only invariant probably measure to have absolutely continuous conditional measures on unstable manifolds. Recall that the flip map W HM ! HM is defined by .x; Œ/ D .x; Œ/. The measure WD C is also invariant by ' t and is in fact the physical measure of ' t . In other words, the Birkhoff averages Z 1 0 g.' t .x// dt jt j t R converge to g d when t goes to 1. Similarly, we find that is the equilibrium measure of the potential d ˇ t f D ˇ tD0 log det d'E s; dt and that it has absolutely continuous conditional measures on stable manifolds. Finally, both measures C and coincide if and only if one of them is absolutely continuous. All of this applies to the geodesic flow of a compact quotient of a strictly convex Hilbert geometry, which is C 1C" from Corollary 6.4. Corollary 8.6. Let M D = be a compact quotient manifold of a strictly convex Hilbert geometry. Unless is an ellipsoid, the three measures BM , C and are mutually singular. As a direct application of the fact that the measure C achieves the equality in Ruelle’s inequality, we can give a lower bound on its entropy, hence on topological entropy: Proposition 8.7 ([20]). Let .; d / be a strictly convex divisible Hilbert geometry where is not an ellipsoid. Assume @ is ˇ-convex for some ˇ 2 .2; C1/. Then htop >

2 .n 1/: ˇ

Another application is given in the next section.

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8.5 Curvature of the boundary of a divisible convex set Let us begin with an old theorem of A. D. Alexandrov [1] about convex functions: Theorem 8.8. Let f W U Rn 7! R be a convex function defined on a convex 2 open set U of Rn . The Hessian matrix Hess.f / D @@i @fj ij exists Lebesgue almost everywhere in U . Let be a bounded convex set of the Euclidean space Rn . It is then possible to compute the Hessian of its boundary at Lebesgue almost every point x 2 @. We will call a D 2 point a point x where this is possible. The Hessian is a positive symmetric bilinear form on the tangent space Tx @. It represents the curvature of the boundary at x. To be degenerate means that the curvature of the boundary is zero in some tangent direction. The Hessian is a Euclidean notion, but its degeneracy is not. Namely, if is a properly convex open set of RP n and x a point of @, we can choose an affine chart centered at x and a metric on it and compute the Hessian of @ at x; its degeneracy does not depend on the choice of the affine chart and the metric. We can measure the vanishing of the curvature of @ in the following way. Fix a smooth measure on the boundary of the dual convex set , and call its pull-back to @. Then can be seen as a measure of the curvature of @. It can be decomposed as D ac C sing ; where ac is an absolutely continuous measure and sing is singular with respect to any Lebesgue measure on @. For example, if @ is not differentiable at some point x then will have an atom at x. The support of ac is the closure of the set of D 2 points with nondegenerate Hessian. Though is convex, it may happen that ac D 0, that is, is singular with respect to some (hence any) smooth measure on @. This is equivalent to the fact that the Hessian is degenerate at almost all D 2 point of @. We then say that the curvature of the boundary is supported on a set of zero Lebesgue measure. The following lemma gives a criterion for this to happen, due to Benzécri [12]: Lemma 8.9. Let Xn denote the set of properly convex open sets of RP n , and pick 2 Xn . If there exists a D 2 point x 2 @ with nondegenerate Hessian, then the closure of the orbit PGL.n C 1; R/ in Xn contains an ellipsoid. Proof. Choose an affine chart and a Euclidean metric on it such that appears as a bounded convex open set of Rn . Let x be a point of @ with nondegenerate Hessian. Let E be the osculating ball of @ at x. It defines a hyperbolic geometry .E; dE /. Pick a point y 2 @E distinct from x, and choose a hyperbolic isometry g of E whose attracting fixed point is y and the repulsive one is x. Now, since @E and @ are tangent up to order 2, it is not difficult to see that g n converges to E when n goes to C1. This proves the statement.

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As a consequence, we get the following Theorem 8.10. Let .; d / be a divisible Hilbert geometry, and assume is not an ellipsoid. Then any D 2 point has degenerate Hessian. In particular, the curvature of @ is supported on a subset of zero Lebesgue measure. Proof. This is a consequence of the last proposition and of the following fact: if is divisible then the orbit of under PGL.n C 1; R/ is closed in Xn (for a proof of this fact, see Proposition 9.18 in Marquis’ contribution [46]). The boundary of a divisible convex set has thus a quite mysterious geometry. In the case of a strictly convex divisible set, we can be more precise using the fact that the unstable conditional measures of C are absolutely continuous: Proposition 8.11 ([21]). Let .; d / be a strictly convex divisible Hilbert geometry. There exists " > 0 such that, for Lebesgue almost every point x 2 @, there exists a 2-dimensional subspace Px intersecting and containing x such that the boundary of the 2-dimensional convex set \ Px is D 2C" at x. In particular, if RP 2 , there exists " > 0 for which the boundary @ is D 2C" at Lebesgue almost every point. Acknowledgement. This work is partially supported by CONICYT (Chile), FONDECYT project NB 3120071.

References [1]

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[10] Y. Benoist, Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. Inst. Hautes Études Sci. 97 (2003), 181–237. [11] Y. Benoist, Convexes divisibles 1. In Algebraic groups and arithmetic, Tata Inst. Fund. Res. Stud. 17, Narosa Publishing House, New Delhi 2004, 339–374. [12] J.-P. Benzécri, Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88 (1960), 229–332. [13] G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5 (1995), no. 5, 731–799. [14] A. Bosché, Symmetric cones, the Hilbert and Thompson metrics. Preprint, arXiv:1207.3214. [15] R. Bowen and D. Ruelle, The ergodic theory of axiom A flows. Invent. Math. 29 (1975), 181–202. [16] R. Bowen, The equidistribution of closed geodesics. Amer. J. Math 94 (1972), 413–423. [17] R. Bowen, Periodic orbits for hyperbolic flows. Amer. J. Math 94 (1972), 1–30. [18] B. Colbois and P. Verovic, Hilbert geometry for strictly convex domains. Geom. Dedicata 105 (2004), 29–42. [19] M. Crampon and L. Marquis, Le flot géodésique des quotients geometriquement finis des géométries de Hilbert. Preprint, arXiv:1105.6275v1. [20] M. Crampon, Entropies of strictly convex projective manifolds. J. Mod. Dyn. 3 (2009), no. 4, 511–547. [21] M. Crampon, Lyapunov exponents in Hilbert geometry. Preprint, arXiv:1204.5005v1. [22] P. de la Harpe, On Hilbert’s metric for simplices. In Geometric group theory (Graham A. Niblo et al., eds.,) Vol. 1, London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge 1993, 97–119. [23] P. Eberlein, When is a geodesic flow of Anosov type? I, II. J. Differential Geom. 8 (1973), 437–463; ibid. 565–577, 1973. [24] J. Faraut and A. Korányi, Analysis on symmetric cones. Oxford Math. Monogr., Oxford Science Publications, New York 1994. [25] P. Foulon, Géométrie des équations différentielles du second ordre. Ann. Inst. Henri Poincaré 45 (1986), 1–28. [26] P. Foulon, Estimation de l’entropie des systèmes lagrangiens sans points conjugués. With an appendix “About Finsler geometry”. Ann. Inst. H. Poincaré Phys. Théor. 57 (1992), no. 2, 117–146. [27] P. Foulon, Locally symmetric Finsler spaces in negative curvature. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 10, 1127–1132. [28] P. Foulon, Entropy rigidity of Anosov flows in dimension three. Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1101–1112. [29] W. M. Goldman, Convex real projective structures on compact surfaces. J. Differential Geom. 31 (1990), 791–845.

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[30] O. Guichard, Régularité des convexes divisibles. Ergodic Theory Dynam. Systems 25 (2005), no. 6, 1857–1880. [31] U. Hamenstädt, Anosov flows which are uniformly expanding at periodic points. Ergodic Theory Dynam. Systems 14 (1994), 299–304. [32] D. Johnson and J. J. Millson, Deformation spaces associated to compact hyperbolic manifolds. In Discrete groups in geometry and analysis, 1984), Progr. Math. 67, Birkhäuser Boston 1987, 48–106. [33] V. G. Kac and E. B. Vinberg, Quasi-homogeneous cones. Mat. Zametki 1 (1967), 347–354. [34] M. Kapovich, Convex projective structures on Gromov-Thurston manifolds. Geom. Topol. 11 (2007), 1777–1830. [35] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Cambridge University Press, Cambridge 1995. [36] A. Katok, Four applications of conformal equivalence to geometry and dynamics. Ergodic Theory Dynam. Systems 8 (Charles Conley Memorial Issue) (1988), 139–152. [37] M. Koecher, The Minnesota notes on Jordan algebras and their applications. Lecture Notes in Math. 1710, Springer-Verlag, Berlin 1999. [38] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ann. of Math. 122 (1985), 509–574. [39] F. Ledrappier, Harmonic measures and Bowen-Margulis measures. Israel J. Math. 71 (1990), no. 3, 275–287. [40] B. Lemmens and C. Walsh, Isometries of polyhedral Hilbert geometries. J. Topol. Anal. 3 (2011), , no. 2, 213–241. [41] A. Livšic, Certain properties of the homology of Y-systems. Mat. Zametki 10 (1971), 555–564. [42] A. Livšic, Cohomology of dynamical systems. Math. USSR Izvestija 6 (1972), 1278–1301. [43] A. Manning, Topological entropy for geodesic flows. Ann. of Math. 110 (1979), 567–573. [44] G. Margulis, Certain measures associated with Y-flows on compact manifolds. Functional Anal. Appl. 4 (1969), 55–67. [45] G. Margulis, On some aspects of the theory of Anosov systems. Springer Monogr. Math., Springer-Verlag, Berlin, 2004. [46] L. Marquis, Around groups in Hilbert geometry. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 207–261. [47] X. Nie, On the Hilbert geometry of simplicial Tits sets. Preprint, arXiv:1111.1288v1. [48] V. I. Osedelec, A multiplicative ergodic theorem. Trans. Moscow Math. Soc. 19 (1968), 197–231. [49] D. Ruelle, An inequality for the entropy of differentiable maps. Bol. Soc. Bras. de Mat. 9 (1978), 83–87. [50] É. Socié-Méthou, Comportements asymptotiques et rigidité en géométrie de Hilbert. PhD thesis, Université de Strasbourg, 2000; available at http://www-irma.u-strasbg.fr/annexes/publications/pdf/00044.pdf.

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[51] É. Socié-Méthou, Caractérisation des ellipsoïdes par leurs groupes d’automorphismes. Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 4, 537–548. [52] E. B. Vinberg, The theory of homogeneous convex cones. Trudy Moskov. Mat. Ob˘sc˘ . (12) (1963), 303–358. [53] E. B. Vinberg, The structure of the group of automorphisms of a homogeneous convex cone. Trans. Math. Mosc. Soc. (13) (1965), 63–93. [54] P. Walters, An introduction to ergodic theory. Grad. Texts in Math. 79, Springer-Verlag, New York 1982.

Chapter 8

Around groups in Hilbert geometry Ludovic Marquis

Contents 1 2

3

4

5

6

7

Generalities on the group Aut./ . . . . . . . . . . . . . . . . . . The automorphisms of a properly convex open set . . . . . . . . . 2.1 The first lemma . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The matrix point of view . . . . . . . . . . . . . . . . . . . . 2.3 The dynamical point of view . . . . . . . . . . . . . . . . . . 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of “large” groups acting on properly convex open subsets 3.1 Homogeneous properly convex open set . . . . . . . . . . . . 3.2 Symmetric properly convex open set . . . . . . . . . . . . . 3.3 Spherical representations of semi-simple Lie groups . . . . . 3.4 Schottky groups . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Strictly convex divisible and quasi-divisible convex sets . . . 3.6 Non strictly convex divisible and quasi-divisible convex sets . Convex hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 An alternative construction . . . . . . . . . . . . . . . . . . . Zariski closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Definitions of proximality . . . . . . . . . . . . . . . . . . . 5.2 Positive proximality . . . . . . . . . . . . . . . . . . . . . . 5.3 Cocompact and finite-covolume case . . . . . . . . . . . . . 5.4 A sketch of proof when is strictly convex . . . . . . . . . . Gromov-hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The first step . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The duality step . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The key lemma . . . . . . . . . . . . . . . . . . . . . . . . . Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 A naturality statement . . . . . . . . . . . . . . . . . . . . . 7.2 A sketch of the proof of closedness . . . . . . . . . . . . . . 7.3 The openness . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 A description of the topology for the surface . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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211 212 212 213 214 220 222 223 224 224 225 226 229 231 231 231 233 233 233 234 235 235 237 237 239 239 240 240 241 242 243

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7.5 Description of the topology for 3-orbifolds . . . . . . 8 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 For strongly convex bodies . . . . . . . . . . . . . . . 8.2 For round convex bodies . . . . . . . . . . . . . . . . 8.3 For any convex bodies . . . . . . . . . . . . . . . . . 9 Benzécri’s theorem . . . . . . . . . . . . . . . . . . . . . . 9.1 The proof . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Natural things are equivalent . . . . . . . . . . . . . . 9.3 Two not-two-lines applications of Benzécri’s theorem 10 The isometry group of a properly convex open set . . . . . . 10.1 The questions . . . . . . . . . . . . . . . . . . . . . . 10.2 The knowledge . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hilbert geometries were introduced by Hilbert as examples of geodesic metric spaces where straight lines are geodesics. We will completely forget this story. We will take Hilbert geometry as a very simple recipe for metric spaces with several different flavours. Hilbert geometries are Finsler manifold, so it is not easy to say that they are non-positively curved, but we hope that at the end of this text the reader will have noticed some flavours of non-positively curved manifolds and will start to see them as “damaged non-positively curved manifold”. The most interesting examples of Hilbert geometries in the context of geometric group theory are called, following Vey in [88], divisible convex sets. These are those properly convex open subsets of the real projective space P d D P d .R/ such that there exists a discrete subgroup of the group PGLd C1 .R/ of projective transformations which preserves and such that the quotient = is compact. In 2006, Benoist wrote a survey [12] of divisible convex sets and in 2010, Quint wrote a survey [75] of the work of Benoist on divisible convex sets. Thus, we will not concentrate on divisible convex sets since the survey of Benoist does this job very well, even if we consider divisible and quasi-divisible convex sets1 as the most important class of convex sets. We want to describe the groups that appear in Hilbert geometry without restriction and also how groups can be used for the purpose of Hilbert geometry. The first two sections of this survey are very elementary. Their goal is to make the reader familiar with the possible automorphisms of a convex set from a matrix point of view and a dynamical point of view. The third section presents existence results on convex sets with a “large” group of symmetries. This section presents the examples that motivated this study. The reader in quest of motivation should skip Section 2 to go straight to the third section. 1 Those

for which the quotient = is of finite volume rather than compact.

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The remaining seven sections are roughly independent. They support the claim that geometric group theory mixed with Hilbert geometry meet (and need) at least: differential geometry, convex affine geometry, real algebraic group theory, metric geometry, moduli spaces, hyperbolic geometry, symmetric spaces, Hadamard manifolds and geometry on manifolds. We have tried to give sketchy proofs in these sections. We do not report on the links with Coxeter group theory and partial differential equations.

Context We will work with the real projective space P d .R/ D P d of dimension d (i.e. the space of lines of the real vector space Rd C1 ). An affine chart of P d is the complement of a projective hyperplane. Every affine chart carries a natural structure of an affine space. A convex subset of the projective space P d is a subset of P d which is either P d or is included in an affine chart A and is convex in this affine chart in the usual sense. A convex subset C of P d is properly convex when there exists an affine chart such that C is bounded in it or, equivalently, when C does not contain any affine line. Our playground will be a properly convex open set, also called a convex body. We will always denote a properly convex open subset of P d by the letter with a subscript if necessary. On a properly convex open set, one can define the Hilbert distance. Given x ¤ y 2 , let p, q be the intersection points of the line .xy/ with the boundary @ of in such a way that x is between p and y and y between x and q (see Figure 1). pC

q

v x

p

y

p

Figure 1. Hilbert distance.

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We set d .x; y/ D

1 jpyj jqxj 1 log Œp W x W y W q D log and d .x; x/ D 0; 2 2 jpxj jqyj

where the quantity Œp W x W y W q is the cross-ratio of the four points p, x, y, q, and x of . j j is any Euclidean norm on any affine chart A containing the closure 2 The cross-ratio is a projective notion. Thus it is clear that d does not depend on A or on the choice of the Euclidean norm on A. We also note that the distance d is invariant by the group of projective automorphisms preserving . The metric space .; d / is geodesic 3 , proper 4 and the topology induced by the Hilbert distance and the projective space coincide. But this space is not uniquely geodesic 5 in general (see Proposition 10.3). This distance d is called the Hilbert distance and has the good taste of coming from a Finsler metric on defined by a very simple formula. Letˇ x be a point in d ˇ d .x; x C t v/ and v a vector in the tangent space Tx of at x; the quantity dt t D0 is homogeneous of degree one in v, therefore it defines a Finsler metric F .x; v/ on . Moreover, if we choose an affine chart A containing and any Euclidean norm j j on A, we get (see Figure 1): 1 d ˇˇ jvj 1 C F .x; v/ D ˇ d .x; x C t v/ D dt tD0 2 jxp j jxp C j The Finsler structure gives rise to a measure on which is absolutely continuous with respect to the Lebesgue measure, called the Busemann volume. To define it, choose an affine chart A containing , a Euclidean norm j j on A and let Leb be the Lebesgue measure on A normalised by the fact that the volume of the unit cube is 1. The Busemann volume 6 of a Borel set A is then defined by the following formula: Z !d .A/ D d Leb.x/ Tang .1// A Leb.Bx Tang

where !d is the Lebesgue volume of the unit ball of A for the metric j j and Bx is the unit ball of the tangent space Tx of at x for the metric F .x; /.

.1/

2 In fact, one should notice that if we allow the symbol 1 in our computation, then we can define d on any affine chart containing . 3A metric space is geodesic when for any two points, there exists a geodesic joining them. 4A metric space is proper when the closed balls are compact, and so a proper space is complete. 5A geodesic metric space is uniquely geodesic when the geodesic between any two points is unique. 6 There is another notion of volume which is often used in Finsler geometry, the so-called Holmes–Thompson volume HT , defined by the following formula:

Z HT .A/ D CoTang

A

CoTang

Leb.Bx !d

.1//

d Leb.x/

where Bx .1// is the unit ball of the cotangent space of at x for the dual norm induced by .Tx ; F .x; //.

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A charming fact about Hilbert geometries is that they are comparable between themselves. Indeed, if 1 2 , then we can compare the distance, the balls, etc. of those two properly convex open sets. The moral statement is the following: The Hilbert distance is decreasing with . The balls (in and in the tangent space) are increasing with . The Busemann volume is decreasing with . The precise statement is the following: Comparision Theorem. Let 1 , 2 be two properly open convex subset such that 1 2 . Then the following holds. • For every x; y 2 1 , d1 .x; y/ 6 d2 .x; y/. • For every x 2 1 , for every v 2 Tx 1 D Tx 2 , F1 .x; v/ 6 F2 .x; v/. • For any Borel set A of 1 , 2 .A/ 6 1 .A/. We will see that regularity conditions of @ have a crucial impact on the geometry of .; d /. A properly convex open set is strictly convex if there does not exist any nontrivial segment in its boundary @. A point p 2 @ is of class C 1 if the hypersurface @ is differentiable at p; since is convex this is equivalent to the uniqueness of a supporting hyperplane7 at p of . A properly convex open set has C 1 -boundary if every point of @ is of class C 1 . We will see that these two properties are dual to each other (see Subsection 2.3). A properly convex set which verifies both properties is called round. Round convex sets have a hyperbolic behaviour. Non-round convex sets have common properties with “the geometry of a normed plane”. An analogy should be made with symmetric spaces. We recall that rank-one symmetric spaces are Gromov-hyperbolic, and that higher rank symmetric spaces contain Euclidean planes. Acknowledgements. The author thanks the anonymous referee for his useful comments and Athanase Papadopoulos for his careful and patient review. The author also wants to thanks Constantin Vernicos and Mickaël Crampon for all the discussions they had around Hilbert geometry. The author is supported by the French program ANR Finsler.

1 Generalities on the group Aut./ The main goal of this chapter is to study the group ˚ Coll˙ ./ D 2 PGLd C1 .R/ j ./ D where is a properly convex open subset of P d . The following fact is very basic and also very useful. 7A

hyperplane H is a supporting hyperplane at p 2 @ when H \ D ¿ and p 2 H .

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Proposition 1.1. The action of the group Coll˙ ./ on is by isometries with respect to the Hilbert distance. Consequently, the action of Coll˙ ./ on is proper, so Coll˙ ./ is a closed subgroup of PGLd C1 .R/ and therefore is a Lie group. One may want to introduce the group Isom./ of isometries of .; d / for the Hilbert metric. This group is rather mysterious. We postpone some remarks on the general knowledge on this group to the end of this chapter. Before going to a “matrix study” of the elements of Coll˙ ./, let us make a remark which allows us to see Coll˙ ./ as a subgroup of the group SL˙ d C1 .R/ of d C1 linear transformations of R with determinant 1 or 1. We denote by P W Rd C1 X f0g ! P d the canonical projection. Let be a properly convex open subset of P d . The cone above is one of the two connected components C of P 1 ./.8 We introduce the group ˚ Aut ˙ ./ D 2 SL˙ d C1 .R/ j .C / D C rather than Coll˙ ./. The canonical morphism W SL˙ d C1 .R/ ! PGLd C1 .R/ is onto with kernel equal to f˙1g. But, the restriction of W Aut˙ ./ ! Coll˙ ./ is an isomorphism since elements of Aut ˙ ./ preserve C . With this in mind, we will now concentrate on Aut./ which is the group of linear automorphisms of Rd C1 preserving C and having determinant one: ˚ Aut./ D 2 SLd C1 .R/ j .C / D C : Sometimes it is very useful to look at the two-fold cover Sd of P d . A projective way to define Sd is to consider it as the space of half-lines of Rd C1 . The group of d projective automorphisms of Sd is the group SL˙ d C1 .R/. One advantage of S is that the definition of convexity is neater in Sd than in P d .

2 The automorphisms of a properly convex open set 2.1 The first lemma Lemma 2.1. Every compact subgroup of Aut./ fixes a point of . Proof. The proof of the lemma relies on the construction of a “center of mass” for every bounded subset of . This construction relies on the Vinberg convex hypersurface, see Lemma 4.2.

8 This

is a little abusive since there are two such cones, but nothing depends on this choice.

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2.2 The matrix point of view Let be an element of SLd C1 .R/. We denote by 1 . / > 2 . / > > d C1 . / the moduli of the eigenvalues of listed in decreasing order with multiplicity. The 1 spectral radius C is by definition 1 ./. We also define D d C1 . / D C1 . C C An element is semi-proximal if or is an eigenvalue of . An element is positively semi-proximal if C is an eigenvalue of . An element is proximal if 1 ./ > 2 . /. This implies that is semi-proximal. An element is positively proximal if is positively semi-proximal and proximal. Since, det./ D 1, we remark that if C D 1 then D 1 also, and i . / D 1 for all i D 1; : : : ; d C 1. An element is bi-“something” if and 1 are “something”. The main use will be for biproximal elements and positively biproximal elements. Let k D R or C. An element is k-semi-simple if in a suitable k-basis is diagonal. An element is S1 -semi-simple 9 if it is C-semi-simple and all its eigenvalue are on the unit circle. An element is unipotent if . 1/d C1 D 0. The power of a unipotent element is the smallest integer k such that . 1/k D 0. For every element 2 SLd C1 .R/, there exists a unique triple consisting of an R-semi-simple element h , an S1 -semi-simple element e and a unipotent element u such that D h e u and these three elements commute with each other (see for example §4.3 of the book [71] of Witte Morris or [72], Chapter 3, Part 2). Let 2 SLd C1 .R/ and be an eigenvalue of . The power of is the size of the maximal Jordan block of with eigenvalue . In other terms, the power of is the multiplicity of in the minimal polynomial of . Of course, when is unipotent the power of 1 is exactly what we previously called the power of . Proposition 2.2. Suppose the element 2 SLd C1 .R/ preserves a properly convex open set . Then is positively bi-semi-proximal. Moreover, if C D 1, then its power is odd. Finally, the projective traces of the eigenspaces ker. C / and ker. / x meet the convex set . Proof. First, we have to prove that is positively semi-proximal. This is exactly the content of Lemma 3.2 of [9]. We give a rough proof. Consider the subspace V of Rd C1 whose complexification VC is the sum of the eigenspaces corresponding to the eigenvalues of modulus D C . If x 2 , then a computation using the Jordan form of shows that any accumulation point of the x \ P .V /. sequence n .x/ belongs to x \ P .V / which is a properly convex open Consider the relative interior 0 of subset of a vector space V 0 and consider the restriction 0 of to V 0 . The element 0 is semi-simple and all its eigenvalues have the same modulus . Therefore, the element 9 Usually, we say that is hyperbolic when is R-semi-simple, and we say that is elliptic when is S1 -semi-simple, but we shall not use this terminology because it will create a conflict with the next subsection.

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00 D 0 = is S1 -semi-simple and preserves 0 . Since 00 is S1 -semi-simple, the group generated by 00 is relatively compact. Lemma 2.1 shows that 1 is an eigenvalue of 00 and that P .ker. 00 1// meet 0 . The “bi” statement is obvious. Finally, we have to show that the Jordan multiplicity of in is odd, when D 1. This is almost contained in Lemma 2.3 of [10]. One has to study the action of Aut./ on Sd , the 2-fold covering of P d . Denote by k the integer minfl j . /l D 0g and take any point x 2 which is not in ker. /k1 . A computation using a Jordan form of shows that lim n .x/ D lim n .x/

n!C1

if k is even, and

n!1

lim n .x/ D lim n .x/

n!C1

n!1

if k is odd. Therefore preserves a properly convex open set if and only if k is odd.

2.3 The dynamical point of view Definition 2.3. Let 2 Aut./. The translation length of on is the quantity

. / D inf d .x; .x//. We will say that is x2

(1) elliptic if ./ D 0 and the infimum is achieved; (2) parabolic if ./ D 0 and the infimum is not achieved; (3) hyperbolic if ./ > 0 and the infimum is achieved; (4) quasi-hyperbolic if ./ > 0 and the infimum is not achieved. A complete classification of automorphisms of properly convex open sets was given in dimension 2 in [19] and [65]. There is a classification under the hypothesis that the convex set is round in [32]. Finally, one can find a classification in the general context in [29]. Our exposition is inspired by the last reference. The faces of a properly convex open set. We present the notion of faces of a convex body. For more details and a different point of view, the reader can consult the book [17], Chapter 1, §1.5. Let be a properly convex open subset of P d . We introduce the following equivx x y when the segment Œx; y can be extended beyond x and alence relation on : x the closure of an open y. The equivalence classes of are called open faces of ; x face is a face of . The support of a face or of an open face is the smallest projective space containing it. The dimension of a face is the dimension of its support. The behaviour of open faces and faces with respect to closure and relative interior is very nice, thanks to convexity. More precisely, the interior of a face F in its support (i.e. its relative interior) is equal to the unique open face f such that fN D F . Finally,

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x then f is a properly convex open set one should remark that if f is an open face of in its support. The notion of face is important to describe the dynamics, more precisely to describe the set of attractive and repulsive fixed points of an automorphism. We give a family of examples. Consider the convex subset of Rd formed by the points with strictly positive coordinates. This is a properly convex open subset of P d . The diagonal matrix 2 SLd C1 .R/ given by .1 ; : : : ; d C1 / with i > 0, …i D 1 and 1 6 2 6 6 d C1 preserves . If 1 > 2 and d > d C1 , then has one attractive fixed point and one repulsive fixed point corresponding to the eigenlines of 1 and d C1 . Now, if 1 D 2 > 3 and d > d C1 , then has a set of attractive fixed points and one repulsive fixed point. More precisely, the set of attractive fixed points of is a segment included in @; it is even a face of . We can build a lot of examples with various sizes of attractive or repulsive fixed points with this convex set . Each time the attractive set and the repulsive set will be faces of . This is a general fact. Horospheres in Hilbert geometry The round case. Horospheres are very important metric objects in the study of the geometry of metric spaces. But horospheres are not easy to define in a general Hilbert geometry. We begin by a definition in the context of round Hilbert geometry. Let be a round convex subset of P d . First, we define the Busemann function at a point p 2 @. Let x; y 2 . We set ˇp .x; y/ D lim d .y; z/ d .x; z/: z!p

x and To make the computation, compute first in any affine chart containing then send p to infinity. Take the supporting hyperplane H of at p and do your computation in the affine chart A D P d X H . You get: jx qx j 1 ˇp .x; y/ D log 2 jy qy j where qx (resp. qy ) is the point p; x/ \ @ (resp. p; y/ \ @). We recall that the horosphere passing trough x is the set of points y of such that ˇp .x; y/ D 0. Therefore, in the affine chart A, the horospheres are just the translates of the set @ X fpg in the direction given by the line p 2 P d which are in (see Figure 2). Hence, we showed that in a round convex set the horospheres are round convex sets and have the same regularity as @.

The general case: algebraic horospheres. Walsh gave a complete description of the horofunction boundary of a Hilbert geometry in [92]. We will not go that far in the study of horospheres, we will content ourselves with simpler objects.

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p X fpg x

qx

Figure 2. Algebraic horosphere for the ellipsoid.

The convergence of Busemann functions is no more true if the convex set is not round. Cooper, Long and Tillmann introduced an alternative definition in [29]. This time a horosphere will not be defined at a point but at a point p together with a supporting hyperplane H of at p. The algebraic horospheres based at .p; H / of a properly convex open set are the translates of @ X H in the affine chart P d X H in the direction10 p 2 P d which are in (see Figure 3).

p x qx

Figure 3. Algebraic horosphere for the triangle.

10 We

insist on the fact that p is a point of P d which is not in A; therefore p is also a direction in A.

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Hence, we get two transverse foliations of , the one given by the straight line ending at p and the one given by the algebraic horosphere based at .p; H /. We need one more notion before stating and proving the classification. Duality. We present the notion of duality for convex bodies. For more details and a different point of view, the reader can consult the books [17], [81]. A convex cone of a real vector space is said to be sharp when it does not contain any affine line. Hence, there is a correspondence between open sharp convex cones of Rd C1 and properly convex open subsets of P d via the natural projection. If C is a convex cone in a vector space E, then we define C D ff 2 E j for all x 2 Cx X f0g; f .x/ > 0g: We first remark that C is a convex open cone of E . Secondly, C is non-empty if and only if C is sharp and C is sharp if and only if C has non-empty interior. This operation defines an involution between the set of all sharp convex cones of E and the set of all sharp convex cones of E . If is a properly convex open subset of P .E/ then we define the dual of /. This defines an involution between the set of all properly convex by D P .C open subsets of P .E/ and the set of all properly convex open subsets of P .E /. One can remark that is a space of hyperplanes of P d , hence is also a space of affine charts since there is a correspondence between hyperplanes and affine charts. Namely, can be identified with the space of affine charts A containing the closure x of or with the space of hyperplanes H such that H \ x D ¿. An important proposition about duality for convex subsets is the following: Proposition 2.4. Let be a properly convex open set. Then is strictly convex if and only if @ is C 1 . We can explain this proposition in dimension 2. If @ is not C 1 at a point p 2 @, then the space of lines containing p and not intersecting is a segment in the boundary of , showing that is not strictly convex. We consider the following set: X D f.; x/ j is a properly convex open subset of P d and x 2 g endowed with the Hausdorff topology. The group PGLd C1 .R/ acts naturally on X . We denote by P d the projective space P ..Rd C1 / /. We also consider its dual, X D f.; x/ j is a properly convex open subset of P d and x 2 g: The following theorem needs some tools, so we postpone its proof to Subsection 9.1.2 (Lemma 9.3) of this chapter. We think that this statement is natural and we hope the reader will feel the same.

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Lemma 2.5. There is a continuous bijection ? W X ! X which associates to a pair .; x/ 2 X a pair . ; x ? / 2 X where is the dual of , and this map is PGLd C1 .R/-equivariant. We are now able to give a dynamical description of the automorphisms. The classification in the general context Proposition 2.6. Let 2 Aut./. The following are equivalent: (1) is elliptic; (2) fixes a point of ; (3) is S1 -semi-simple. Proof. 1/ () 2/ This is the definition. 2/ ) 3/ Since fixes a point x, also fixes the hyperplane x ? dual to x with respect to (cf. Lemma 2.5). Therefore, is a linear transformation of the affine chart A D P d X x ? centered at x which preserves the convex , hence preserves the John ellipsoid11 of (centered at x in A), so is conjugate to an element in SOd . The conclusion follows. 3/ ) 1/ The closure G of the group generated by is compact, therefore every orbit of is bounded, hence is elliptic by Lemma 2.1. Given a point p 2 @ and a supporting hyperplane H at p of @, one can define the group Aut.; H; p/ of automorphisms of which preserve p and H . This group acts on the set of algebraic horosphere based at .p; H / of . Since different algebraic horospheres correspond by translation in the direction p in the chart P d X H , we get a morphism h W Aut.; H; p/ ! R that measures horosphere displacement. The proofs of two following lemmas are left to the reader. Lemma 2.7. Let 2 Aut.; H; p/. If C D 1, then h. / D 0 and preserves all the algebraic horospheres based at .H; p/ of . If C > 1, then h. / ¤ 0. Lemma 2.8 (McMullen, Theorem 2.1 of [68]). Let 2 Aut./. The translation length satisfies 1 log max. C ; 1= ; C = / 6 . / 6 log max. C ; 1= /: 2 In particular, C D 1 if and only if is elliptic or parabolic. x of a properly convex open set When a subset A of P d is included in the closure , we can define its convex hull in , i.e. the smallest convex set of containing A 11 We recall that the John ellipsoid of a bounded convex of a real vector space is the unique ellipsoid with maximal volume included in and with the same center of mass.

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in its closure. We will denote this set by Conv.A/. An element 2 Aut./ is planar when is R-semi-simple and has exactly two eigenvalues. C In that case, the function 1 d .x; .x// is constant on and equal to 2 log = . Proposition 2.9. Let 2 Aut./. The following are equivalent: (1) is hyperbolic or quasi-hyperbolic. (2) C > 1. x which are fixed and the action (3) There exist two disjoint faces F C and F of of on the properly convex set axe D Conv.F [ F C / is planar. In the last case ./ D 12 log. C = /. Moreover, is quasi-hyperbolic if and only if axe @; otherwise is hyperbolic. Proof. Lemma 2.8 shows that 1/ ) 2/. For 2/ ) 3/ and 1/, since C > 1 we have < 1. So consider the projective C x spaces E C D P .ker. C // and E D P .ker. //, and let us set F C D \E x \ E . Since the action of on is proper, the convex sets F C and F and F D are included in @. Moreover, a computation using the Jordan form of shows that for any point x 2 the limit n x as n tend to C1 (resp. 1) belongs to F C (resp. x Finally the restriction of to the F ), therefore F ; F C are non-empty faces of . C convex hull axe D Conv.F [ F / is clearly planar, and so for every x 2 axe we get d .x; .x// D daxe .x; .x// D 12 log. C = /. Hence, . / 6 12 log. C = /, but . / > 12 log. C = / thanks to Lemma 2.8. 3/ ) 2/ Since the action of on axe is planar we get C > 1. Proposition 2.10. Let 2 Aut./. The following are equivalent: (1) is parabolic. (2) C D 1 and is not elliptic. (3) h D 1 and the power of u is odd and > 3. x and there is only one maximal face (4) fixes every point of a face F @ of fixed by . (5) preserves an algebraic horosphere and is not elliptic. Proof. 1/ ) 2/ is a consequence of Lemma 2.8. 2/ ) 3/ is a consequence of Proposition 2.2. For 3/ ) 4/, denote by k the power of u and consider the subspace E D P .Im.u 1/k1 / of P d . From the Jordan form of , we get that every point of X P .ker.u 1/k1 / accumulates on a point of F D @ \ E; hence F is a face x Moreover, every point of F must be fixed by since it is fixed by h , and of . and h commutes. Finally, such an F is unique since otherwise the action of on Conv.F [ F 0 / would be planar and C > 1.

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4/ ) 5/ Take any point p in the relative interior of F . One can find a supporting hyperplane H fixed by that contains F . Since F is unique we have C D 1, and one concludes with Lemma 2.8. 5/ ) 1/ Lemma 2.7 shows that C D 1. Since is not elliptic, Lemma 2.8 shows that is parabolic. The classification in the strictly convex case or C 1 boundary case Proposition 2.11. Suppose the element 2 SLd C1 .R/ preserves a properly convex open set . If is strictly convex or has C 1 boundary then is not quasi-hyperbolic. If is hyperbolic then is positively proximal and if is parabolic then its Jordan block of maximal size is unique 12 . Proof. First, assume that is strictly convex. If is quasi-hyperbolic, then by Proposition 2.9, one gets that axe @, hence is not strictly-convex. Now, if is hyperbolic and not proximal then it is an exercise to check that @ must contain a non-trivial segment. Analogously, if is parabolic and its Jordan block of maximal size is not unique, then @ must contain a non-trivial segment. Finally, if has a C 1 boundary then the dual is strictly convex (Proposition 2.4) and is preserved by t 1 .

2.4 Examples 2.4.1 Ellipsoid. Consider the quadratic form q.x/ D x12 C Cxd2 xd2 C1 on Rd C1 . The projective trace Ed of the cone of timelike vectors Cd C1 D fx 2 Rd C1 j q.x/ < 0g is a properly convex open set. In a well chosen chart, Ed is a Euclidean ball. We call any image of Ed by an element of SLd C1 .R/ an ellipsoid. The ellipsoid is the leading example of a round Hilbert geometry. The group Aut.Ed / is the group SOd;1 .R/13 and .Ed ; dEd / is isometric to the real hyperbolic space of dimension d . In fact, Ed is the Beltrami–Cayley–Klein projective model of the hyperbolic space. The elements of SOd;1 .R/ can be of three types: elliptic, hyperbolic or parabolic. The stabilizer of any point of Ed is a maximal compact subgroup of SOd;1 .R/, and it acts transitively on @Ed . The stabilizer P of any point p of @Ed is an amenable subgroup of SOd;1 .R/ that acts transitively on Ed . Moreover, P splits as a semi-direct product of a compact group isomorphic to SOd 1 and a solvable subgroup S which acts simply transitively on Ed . And S also splits as a semi-direct product S D U ÌRC where RC is the stabilizer of p and any point q ¤ p, q 2 @Ed . It is a group composed uniquely of hyperbolic elements, and U is a unipotent subgroup composed only of parabolic elements. It is isomorphic to Rd 1 and it acts simply transitively on each 12 We

mean by this that has only one Jordan block of size its power. precisely it is the identity component of SOd;1 .R/.

13 More

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horosphere of Ed . Finally, one can show that the Lie algebra u of U is conjugate to the following Lie algebra of sld C1 .R/: 8 0 19 ˇ 0 0 u1 ud 1 > ˆ ˇ > ˆ > ˆ ˇ B C > ˆ 0 0 0 u ˆ 1 ˇ B C> = < C : d 1 ˇ B : :: : .u1 ; : : : ; ud 1 / 2 R ˇ B C 0 : C> : ˇ B ˆ > ˆ ˇ @ ˆ 0 ud 1 A> > ˆ ˇ > ˆ ; : 0 The case of the ellipsoid is not only the illuminating example of the world of round convex sets, its properties are also the main tool for studying groups acting on round convex sets. 2.4.2 Cone over an ellipsoid. The cone of timelike vectors Cd C1 D fx 2 Rd C1 j q.x/ < 0g is a sharp convex cone of Rd C1 , so it is also a properly convex open subset of P d C1 . It is not strictly convex nor with C 1 boundary and its group of automorphisms is isomorphic to SOd;1 .R/ RC . One should remark that the group Aut.Cd C1 / is “affine” since every automorphism preserves the support of the ellipsoidal face of Cd C1 , hence the affine chart Rd C1 . This properly convex open set gives several counter-examples to statements about strictly convex open sets. 2.4.3 Simplex. Consider an open simplex Sd in Rd . This is a polyhedron (and so a properly convex open subset of P ) whose group of automorphisms is the semi-direct product of the group Dd of diagonal matrices with positive entries and determinant one and the alternate group on the d C 1 vertices of Sd . The group Dd is isomorphic to Rd and it acts simply transitively on Sd . One should remark that Aut.Sd / does not preserve any affine chart of P d (i.e. it is not affine). This very simple properly convex open set is more interesting than it looks. One can remark that since every collineation of P d fixing d C 2 points in generic position is trivial and every polyhedron of P d which is not a simplex has d C2 vertices in generic position, we get that the automorphism group of every polyhedron which is not a simplex is finite. Therefore the only polyhedron (as a properly convex open set) which can be of interest for the geometric group theorist is the simplex. 2.4.4 The symmetric space of SLm .R/. Consider the convex cone of symmetric positive definite matrices of Mm .R/. It is a sharp convex cone in the vector space of CC symmetric matrices. We consider its trace Sm on P d where d D .m1/.mC2/ . This 2 is a properly convex open set which is not strictly convex nor with C 1 boundary, if m > 3.

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CC The automorphism group of Sm is SLm .R/ via the representation m .g/ M D gM tg. This representation offers plenty of examples of various conjugacy classes of matrices acting on a convex set. The interested reader can find a very nice description of S3CC and its relation to the triangle S2 in [35].

3 Examples of “large” groups acting on properly convex open subsets Before giving the examples, we need a nice context and for that a notion of indecomposability. This is the goal of the next lines. Let us begin by a remark. Given a properly convex open subset of the projective space P .E/, the cone C above is a properly convex open subset of the projective space P .E ˚ R/. We will say that a properly convex open set is a convex cone if there exists an affine chart such that is a cone in this affine chart. The following definition is very natural: a sharp convex cone C of a vector space E is decomposable if we can find a decomposition E D E1 ˚ E2 of E such that this decomposition induces a decomposition of C (i.e. Ci D Ei \ C and C D C1 C2 ). A sharp convex cone is indecomposable if it is not decomposable. We apply this definition to properly convex open sets. A properly convex open set is indecomposable if the cone C above is indecomposable. This definition suggests a definition of a product of two properly convex open sets which is not the Cartesian product. Namely, given two properly convex open sets 1 and 2 of the projective space P .E1 / and P .E2 /, we define a new properly convex open set 1 ˝ 2 of the projective space P .E1 E2 / by the following formula: if Ci is the cone above i then 1 ˝ 2 D P .C1 C2 /. It is important to note that if i is of dimension di then 1 ˝ 2 is of dimension d1 C d2 C 1. For example if the i are two segments then 1 ˝ 2 is a tetrahedron. Here is a more pragmatic way to see this product. Take two properly convex sets !i of a projective space P .E/ of disjoint support. The !i are not open but we assume that they are open in their supports. Then there exists an affine chart containing both !i and the convex hull, and such an affine chart14 of !1 [ !2 is isomorphic to !1 ˝ !2 . Let us finish by an explanation of why the Cartesian product is not a good product for convex sets from our point of view. The Cartesian product is an affine notion and not a projective notion, namely the resulting convex set depends on the affine chart containing the convex sets. For example if the i are segments (which is a projective notion) then in an affine chart i can be a half-line or a segment. If they are both half-lines then the Cartesian product is a triangle. But if they are both segments then the Cartesian product is a square which is not projectively equivalent to a triangle. 14 Of

course, this does not depend of the choice of the affine chart containing the convex sets.

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We get that a properly convex open set is either indecomposable or decomposable. We stress that a cone C can be written C D ˝ where is just one point, which is a properly convex open set of S0 . One can easily see that 1 ˝ 2 is never strictly convex nor with C 1 boundary, except if 1 ˝ 2 is a segment, and that the automorphism group of 1 ˝ 2 is by a canonical isomorphism the group Aut.1 / Aut.2 / R and that “the group R” can be written in a good basis as a diagonal matrix group with diagonal entries: .; : : : ; ; ; : : : ; / „ ƒ‚ … „ ƒ‚ … d1 C1

with d1 C1 d2 C1 D 1:

d2 C1

3.1 Homogeneous properly convex open set A properly convex open set is said to be homogeneous when its automorphism group acts transitively on it. Homogeneous convex sets have been classified in the sixties in [90], [89] by Vinberg. Rothaus gave an alternative construction of all homogeneous convex sets in [78]. If 1 and 2 are two homogeneous properly convex open sets, then the product 1 ˝ 2 is also homogeneous. In fact, one should also remark that the cone above 1 is a homogeneous properly convex open set with automorphism group G1 R. We will not spend time on homogeneous properly convex open sets because they do not give new examples of divisible or quasi-divisible convex sets. This is due to the following corollary of the classification: Proposition 3.1. A homogeneous properly convex open set is quasi-divisible if and only if it is symmetric. Moreover, it admits a compact and a finite-volume non-compact quotient. A properly convex open set is symmetric15 when it is homogeneous and self-dual16 . Proof. If is symmetric, then G D Aut./ is semi-simple and acts transitively and properly on . Theorem 3.2 shows that there exists a uniform lattice17 and a non-uniform lattice 0 of G, therefore admits a compact and a finite-volume non-compact quotient. If is quasi-divisible, then G D Aut./ admits a lattice, but a locally compact group which admits a lattice has to be unimodular (i.e., its Haar measure is rightinvariant and left-invariant). Therefore, the group G is unimodular. The classification of Vinberg shows that the only case where is homogeneous with Aut./ unimodular is when is symmetric. 15 This is equivalent to the fact that for every point x 2 there exists an isometry of .; d / which fixes x and whose differential at x is Id . 16 That is, and are isomorphic as properly convex open set. 17A lattice in a locally compact group G is a discrete subgroup such that the quotient G= is of finite volume for the induced Haar measure. A lattice is uniform when the quotient is compact.

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Theorem 3.2 (Borel–Harish-Chandra). A semi-simple Lie group admits a uniform lattice and a non-uniform lattice.

3.2 Symmetric properly convex open set Symmetric properly convex open sets have been classified by Koecher in the sixties using the classification of Jordan algebras ([53], [36], [89]). The classification of indecomposable symmetric properly convex open sets is given by the two following theorems: Theorem 3.3. Let be a symmetric properly convex open subset of P d which is strictly convex. Then is an ellipsoid. In other words, is the symmetric space associated to SOd;1 .R/. For d D 1, we warn the reader that the usual name for an ellipsoid of dimension 1 is a segment. Theorem 3.4. Let be an indecomposable symmetric properly convex open subset of P d which is not strictly convex. Then is the symmetric space associated to SLm .K/ where K D R; C; H and m > 3 or to the exceptional Lie group E6.26/ . We can give an explicit description of these symmetric properly convex open sets. For example, the symmetric properly convex open set associated to SLm .R/ is the projective trace of the cone of positive definite symmetric matrices of size m m. For the other non-exceptional ones, just take the Hermitian or quaternionic Hermitian matrices. We remark that such convex sets do not exist in all dimensions and that the smallC2/ D est one is of dimension 5. The real ones are of dimension DR .d / D .d 1/.d 2 5; 9; 14; 20; 27; 35; : : : , the complex ones are of real dimension DC .d / D d 2 1 D 8; 15; 24; 35; 48; 63; : : : , the quaternionic ones DH .d / D .2d C 1/.d 1/ D 14; 27; 44; 65; 90; 119; : : : , the exceptional one of dimension 26.

3.3 Spherical representations of semi-simple Lie groups In [91], Vinberg characterised the representation of a semi-simple real Lie group that preserves a properly convex open set. We use the classical notation. Let K be a maximal compact subgroup of G and let P be a minimal parabolic subgroup of G. The characterisation is the following: Theorem 3.5. Let W G ! SL.V / be an irreducible representation of a semi-simple group G. Then the following are equivalent: (1) The group .G/ preserves a properly convex open subset of P .V /.

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(2) The vector space V K of vectors fixed by K is not zero. (3) The group P preserves a half-line. Furthermore, if one of the previous assertions is true, then dim.V K / D 1 and there exist two properly convex open sets min and max such that for any properly convex open set preserved by we have min max . Such a representation is called a spherical representation. The properly convex open set min is the convex hull of the limit set of G and max is the dual convex to min .

3.4 Schottky groups The following subsection relies on the machinery of representation of real Lie groups. It can be skipped without consequence for the understanding of the rest of the chapter. Let G be a semi-simple real linear connected Lie group and an irreducible representation of G on a real vector space V of finite dimension. One can ask the following question: When does there exist a Zariski-dense subgroup of G such that ./ preserves a properly convex open subset of P .V /? Benoist gives an answer to this question in [5]. To present this theorem we need to introduce some vocabulary. Let G be a linear algebraic semi-simple Lie group and W G ! V an irreducible representation. We suppose that G; and V are defined over the field R. Let B be a Borel subgroup of G containing a maximal torus T of G. Using T , one can decompose the representation into weight spaces V , 2 X .T / D Hom.T; C /. Let …. / be the set of weights of the representation , i.e. …. / D f 2 X .T / j V ¤ 0g. Take any order on X .T / such that the roots of G are positive on B. There is a unique maximal weight 0 (called the highest weight of ) for this order and the corresponding weight space V 0 is a line. This line is the unique line of V stabilized by B. We also need some “real” tools. The maximal torus T contains a maximal R-split torus S and the Borel subgroup B is contained in a minimal parabolic R-subgroup P of G. Using S, one can decompose the representation into weight spaces V , 2 X .S / D Hom.S; R /. Let …R . / be the set of restricted weights of the representation , i.e. …R . / D f 2 X .S / j V ¤ 0g. The order on X .T / induces an order on X .S/ such that the roots of G are positive on P . There is a unique maximal restricted weight 0 (called the highest restricted weight of ) for this order and the corresponding weight space V 0 has no reason to be a line. The representation is proximal when dim.V 0 / D 1. We denote by PG the lattice of restricted weights associated to P . We have the following equivalences due to Abels, Margulis and So˘ıfer in [1]: (1) The representation of G is proximal. (2) The group .GR / contains a proximal element.

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(3) Every element in the interior of a Weyl chamber of S is mapped to a proximal element. (4) V 0 is stabilized by P . Furthermore, if one of the previous assertions is true, then the subset of GR of all elements g 2 G such that .g/ is proximal is Zariski-dense in G. A representation is orthogonal (resp. symplectic) when G preserves a nondegenerate symmetric (resp. antisymmetric) bilinear form on V . We say that two irreducible representations , 0 are equal mod 2 when the difference of their restricted highest weight is in 2PG . Theorem 3.6 (Benoist [5]). Let G be a semi-simple real linear connected Lie group with finite center and an irreducible representation of G on a real vector space V of finite dimension. (1) G contains a Zariski-dense subgroup which preserves a properly convex open subset of P .V / if and only if is proximal and not equal mod 2 to an irreducible proximal symplectic representation. (2) Every Zariski-dense subgroup 0 of G contains a Zariski-dense subgroup which preserves a properly convex open subset of P .V / if and only if is proximal and equal mod 2 to an irreducible proximal orthogonal representation. Remark 3.7. All subgroups constructed by Benoist for this theorem are Schottky groups, so in particular they are discrete free groups. We give three examples. The interested reader can find more examples in the article by Benoist. The canonical representation of SLd C1 .R/ on Rd C1 satisfies (1) if and only if d > 2 and it never satisfies (2). The canonical representation of SOp;q .R/ on RpCq satisfies (1) and (2). The canonical representation of SLm .C/ on C m never satisfies (1) nor (2).

3.5 Strictly convex divisible and quasi-divisible convex sets A properly convex open set is divisible (resp. quasi-divisible) if the automorphism group of contains a discrete subgroup such that the quotient = is compact (resp. of finite volume). Figures 4 and 5 were made by Xin Nie, and they show tilings of 2-dimensional open convex bodies obtained by triangular Coxeter groups. In the first example, each tile is compact. In the second example, each tile is of finite volume but not compact. 3.5.1 Existence of non-trivial examples Theorem 3.8 (Folklore). In every dimension d > 2, there exists a divisible (resp. quasi-divisible but not divisible) convex set which is strictly convex and is not an ellipsoid.

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Figure 4. Three divisible convex sets divided by the triangular Coxeter group .3; 3; 7/.

Let us say a few words about the history of divisible and quasi-divisible convex sets: The fact that the ellipsoid is divisible can be found at least in small dimensions in the work of Poincaré. The general case is due to Borel and Harish-Chandra (Theorem 3.2). The fact that the only strictly convex open set which is homogeneous and divisible is the ellipsoid is a consequence of the work of Vinberg and Koecher as explained in Sections 3.2 and 3.1. Kac and Vinberg showed using triangular Coxeter groups in [48] that in dimension 2 there exist divisible convex sets which are not homogeneous, but they did not show that their examples are strictly convex (this is a consequence of a theorem of Benzécri [14] (Theorem 3.14) or the article [55] of Kuiper). Johnson and Millson described in [47] a deformation18 of the projective structure of “classical arithmetic” hyperbolic manifolds. A previous theorem of Koszul in [54] (Theorem 7.7) showed that the deformed manifold is actually a convex projective manifold if the deformation is small enough. Finally, Benoist showed that the convex sets given by this deformation are strictly convex [8] (Theorem 6.1) and that in fact the deformed structure is always convex even in the case of a big deformation [9] (Theorem 7.1). In [67], the author shows the theorem in dimension 2 by describing explicitly the moduli space of convex projective structures of finite volume on a surface. This explicit description is a small extension of Goldman’s parametrisation in the compact 18 This

deformation, introduced by Thurston in [84] for quasi-fuchsian groups, is called bending.

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Figure 5. Three quasi-divisible convex sets quasi-divided by the triangular Coxeter group .3; 3; 1/.

case [40]. In [64], the author shows the theorem in any dimension, using a “bending construction”, but in that case, the convexity of the projective structure is obtained “by hand”, i.e. without a theorem like Koszul’s theorem. Theorem 3.8 needs a security statement: Proposition 3.9. If is an indecomposable properly convex open set which has a compact and a finite-volume non-compact quotient, then is a symmetric properly convex open set. Proof. This proposition is a consequence of Theorem 5.3 which states that either is symmetric or Aut./ is Zariski-dense in SLd C1 .R/. Therefore, either is symmetric or Aut./ is discrete, since a Zariski-dense subgroup of a quasi-simple Lie group is either discrete or dense. Now, the case where Aut./ is discrete is excluded since the quotient of by Aut./ would have to be compact and non-compact at the same time. 3.5.2 A non-existence result on exotic examples in small dimensions Proposition 3.10. Every group dividing (resp. quasi-dividing) a strictly convex properly convex open set of dimension 2 or 3 is a uniform lattice (resp. a lattice) of SO2;1 .R/ or SO3;1 .R/.

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Proof. The articles [32] and [29] show that such a quotient = is the interior of a compact manifold. Moreover, such a manifold is aspherical since its universal cover is homeomorphic to Rd . In dimension 2, = is homeomorphic to a compact surface with a finite number of punctures and this surface has negative Euler characteristic since is strictly convex; therefore is a lattice of SO2;1 .R/. In dimension 3, we distinguish the case of a compact quotient from the case of a non-compact quotient. Suppose = is of finite volume and non-compact, then = is the interior of an aspherical atoroidal compact 3-manifold with non-empty boundary, hence = is a Haken manifold and Thurston’s hyperbolization Theorem of Haken manifolds implies that is a non-uniform lattice of SO3;1 .R/. Finally, if = is compact then = is an aspherical atoroidal compact 3-manifold and does not contain Z2 since is Gromov-hyperbolic (Theorem 6.1). Hence, Perelman’s theorem on Thurston’s geometrization conjecture shows that is a uniform lattice of SO3;1 .R/. 3.5.3 An existence result of exotic examples in higher dimensions Theorem 3.11 (Benoist (d D 4) [11], Kapovich (d > 4) [49]). In every dimension d > 4, there exists a strictly convex divisible convex set which is not quasi-isometric to the hyperbolic space. We just point out that the examples of Benoist are obtained using a Coxeter group, and that Kapovich constructed a convex projective structure on some “Gromov– Thurston manifold”. We recall that Gromov–Thurston manifolds are sequences of manifolds Mn of dimension d > 4 such that none of them carries a Riemannian metric of constant curvature 1 but all of them carry a Riemannian metric of variable negative curvature such that the pinching constant converges to zero as n goes to infinity ([44]). The following question is then natural and open: Open question 1. Does there exist in every dimension d > 4 a strictly convex quasidivisible convex set which is not quasi-isometric to the hyperbolic space and which is not divisible?

3.6 Non strictly convex divisible and quasi-divisible convex sets 3.6.1 Intermission: Vey’s theorem. The following theorem describes the splitting of a divisible convex set into indecomposable pieces. Theorem 3.12 (Vey [88]). Let be a discrete subgroup of SLd C1 .R/ that divides a properly convex open set . There exists a subgroup 0 of of finite index such that the following holds.

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(1) There exists a decomposition V1 ˚ ˚ Vr of Rd C1 such that 0 preserves this decomposition and for each i D 1; : : : ; r, the group i0 D Stab 0 .Vi / acts strongly irreducibly on Vi . x are faces of . We denote by !1 ; : : : ; !r the corre(2) The convex sets P .Vi / \ sponding open faces. We have D !1 ˝ ˝ !r . (3) Each !i is an indecomposable divisible convex set divided by i0 . Corollary 3.13. Let be a discrete subgroup of SLd C1 .R/ that divides a properly convex open set . Then is indecomposable if and only if is strongly irreducible. This theorem leads to the following open question: Open question 2. Does Vey’s theorem hold if quasi-divides ? 3.6.2 A non-existence theorem of non-trivial examples in small dimensions Theorem 3.14 (Kuiper [55] or Benzécri [14] (divisible), Marquis [65] (quasi-divisible)). In dimension 2, the only non-strictly convex divisible properly convex open set is the triangle, and there does not exist any quasi-divisible non-divisible non-strictly convex properly convex open subset of dimension 2. Corollary 3.15. In dimension 2, every indecomposable quasi-divisible convex open set is strictly convex with C 1 boundary. 3.6.3 An existence result Theorem 3.16 (Benoist (3 6 d 6 7, divisible) [10], Marquis (d D 3, quasi-divisible) [66]). In every dimension 3 6 d 6 7 (resp. d D 3), there exists an indecomposable divisible (resp. quasi-divisible non-divisible) convex set which is not strictly-convex nor with C 1 boundary. We stress the fact that Benoist describes very precisely the structure of every divisible convex set in dimension 3 in [10]. In particular, he shows that every segment in the boundary of is in fact in a unique triangle T such that @T @ and such that every triangle in the boundary is stabilized by a virtually Z2 -subgroup of . Moreover, every Z2 -subgroup of stabilizes such a triangle. Finally the projection of T on the quotient = is a Klein bottle or a torus, giving a geometric version of the Jaco–Shalen–Johannson decomposition of the quotient. 3.6.4 A question in higher dimensions Open question 3. Does there exist in every dimension d > 4 an indecomposable divisible (resp. a quasi-divisible not divisible) convex set which is not-strictly convex?

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4 Convex hypersurfaces A very important tool in the study of groups acting on Hilbert geometries is the notion of a convex hypersurface. Informally, it consists in building the analogue of the hyperboloid associated to an ellipsoid for any properly convex open set.

4.1 The theorem Let X D f.; x/ j is a properly convex open subset of P d and x 2 g endowed with the Hausdorff topology. The group PGLd C1 .R/ acts naturally on X . Theorem 4.1 (Vinberg [89]). Let be a properly convex open subset of P d , and C the cone above . There exists a map D W ! C which defines a strictly convex analytic embedded hypersurface of Rd C1 which is asymptotic to C and this map is Aut./-equivariant. In fact, one can define this map as X ! Rd C1 so that it becomes PGLd C1 .R/-equivariant. A convex hypersurface of Rd C1 is an open subset of the boundary of a convex set of . A convex hypersurface † is asymptotic to an open convex cone C containing R † if every affine half-line contained in the cone C intersects the hypersurface (see Figure 6). d C1

† †

Figure 6. An asymptotic and a non-asymptotic convex hypersurface.

R Proof. Consider the map ' W C ! RC given by '.x/ D C e f .x/ df . We leave it as an exercise that the level set of this map gives the intended hypersurface.

4.2 Consequences Existence of centers of mass Lemma 4.2. Let B be a bounded part of a properly convex open set. There exists a point xB 2 Conv.B/ such that for every 2 Aut./, if .B/ D B then .xB / D xB .

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Proof. Denote by D W ! † the convex hypersurface given by Theorem 4.1. Consider C D Conv.D.B//. It is a bounded convex part of Rd C1 . The action of Aut./ on Rd C1 is linear, so the projection on of the center of mass of C is the point xB we are looking for. Existence of an invariant Riemannian metric Lemma 4.3. Let be a properly convex open set. There exists on a Riemannian metric which is Aut./-invariant. In fact, one can define this metric from X so that it becomes PGLd C1 .R/-equivariant. Proof. Since the hypersurface of Theorem 4.1 is strictly convex, its Hessian is definite positive at every point, therefore it defines an invariant analytic Riemannian metric.

Existence of a convex locally finite fundamental domain Lemma 4.4. Let be a properly convex open set. There is a map H from to the space of hyperplanes intersecting such that for every x; y 2 the hyperplane H.x; y/ separates x from y and H is Aut./-equivariant, and such that if yn ! p 2 @ then H.x; yn / tends to a supporting hyperplane at p. Proof. Consider the affine tangent hyperplanes Hx and Hy at the point D.x/ and D.y/ of the convex hypersurface † D D./. Set H.x; y/ D P .Vect.Hx \ Hy //. It is an exercise to check that H.x; y/ does the job (see Figure 7).

†

H.x; y/

y

x

O Figure 7. Bisector.

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The existence of convex locally finite fundamental domains for actions of discrete groups is very useful. For the hyperbolic space, the existence of fundamental domains is due to Dirichlet. A direct application of the Dirichlet techniques does not work in Hilbert geometry since the bisector of two points is no longer a hyperplane. Indeed, in the case of Hilbert geometry the two connected components given by a bisector have no reason to be convex. Nevertheless, thanks to Lemma 4.4, Lee shows: Theorem 4.5 (Lee [57] or [58]). Let be a discrete subgroup of SLd C1 .R/ acting on a properly convex open set . There exists a locally finite 19 convex fundamental domain for the action of on . The method of Lee relies on the existence and geometry of affine spheres (cf. next paragraph). The reader can also find a proof which relies only on Vinberg’s hypersurface in [65].

4.3 An alternative construction There is an another construction of similar convex hypersurfaces. Theorem 4.6 (Cheng–Yau, Calabi–Nirenberg). Let be a properly convex open 0 W ! C which defines a strictly convex subset of P d . There exists a map D 20 embedded hyperbolic affine sphere of Rd C1 which is asymptotic to C and this map is Aut./-equivariant. In fact, one can define this map from X to Rd C1 so that it becomes PGLd C1 .R/-equivariant. The main application of this theorem is Theorem 7.9.

5 Zariski closure 5.1 Definitions of proximality A subgroup of SLd C1 .R/ is proximal if it contains a proximal element. The action of a group on P d is proximal if for every x; y 2 P d , there exists a 2 such that .x/ and .y/ are arbitrarily close. We give the definition of a proximal representation of a semi-simple real Lie group in Subsection 3.4. 19A fundamental domain is locally finite when each compact subset of intersects only a finite number of translates of the fundamental domain. 20 We refer to the survey [62] of Loftin for a definition of an affine sphere and for references, see also Chapter 11 of this volume [52].

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These three definitions of proximality have the following link. When is strongly irreducible, the action of on P d is proximal if and only if is proximal. Moreover, if G is the identity component of the Zariski closure of , then the representation W G ! SLd C1 .R/ is proximal if and only if G is proximal if and only if is proximal.

5.2 Positive proximality The following theorem characterizes irreducible subgroups of SLd C1 .R/ which preserve a properly convex open set. A proximal group is positively proximal when every proximal element is positively proximal. Theorem 5.1 (Benoist [5]). A strongly irreducible subgroup of SLd C1 .R/ preserves a properly convex open set if and only if is positively biproximal. If G is the identity component of the Zariski closure of then G has no reason to be positively biproximal. The group G is positively biproximal if and only if the representation W ! SLd C1 .R/ is spherical (cf. Section 3.3). This phenomenon is not exceptional: just take any divisible convex set which is not an ellipsoid. Any group dividing it is Zariski dense in SLd C1 .R/ by Theorem 5.3 below and SLd C1 .R/ is not positively proximal. Finally, we stress on the following fact about positively proximal groups. Theorem 5.2. Suppose is a strongly irreducible group which preserves a properly convex open set. Then there exists a unique closed -invariant minimal subset ƒ of P d ; hence there exist two properly convex open sets min and max preserved by such that for every properly convex open set preserved by , we have min max . Therefore, if we start with a strongly irreducible group preserving a properly convex open set, by taking its Zariski closure we get a reductive 21 group G. We then get an irreducible representation W G ! SLd C1 .R/ which is proximal. Since we assume that 6 SLd C1 .R/, we get that G is in fact semi-simple 22 . The irreducible representations of a semi-simple group are completely classified. The next question is: what can we say about this representation? Theorem 3.6 gives a complete answer to this question. But we can say more in the case of a finite-covolume action. linear Lie group is reductive when it does not contain any non-trivial normal unipotent subgroup. since G is reductive, we just need to show that the center of G is discrete. Take any element g in the center of G; g has to preserve all the eigenspaces of all the proximal elements of G, hence g is a homothety, so g D ˙1 since G 6 SLd C1 .R/. 21A

22 Indeed,

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5.3 Cocompact and finite-covolume case The following theorem of Benoist completely describes the Zariski closure of a group dividing a properly convex open set. Theorem 5.3 (Benoist [6]). Suppose that divides an indecomposable properly convex open set which is not homogeneous. Then is Zariski-dense. The following question is open: Open question 4. Suppose that quasi-divides an indecomposable convex set which is not homogeneous. Is Zariski-dense? The following theorem answers Question 4 for quasi-divisible convex sets in the strictly convex case. Theorem 5.4 (Crampon–Marquis [32]). Suppose that quasi-divides a strictly convex open set which is not an ellipsoid. Then is Zariski-dense. For results of this kind in the context of geometrically finite actions, the reader is invited to read [32].

5.4 A sketch of proof when is strictly convex We only sketch the proof when the convex set is strictly convex. The actual techniques for the non-strictly convex case are different but the hypothesis of strict convexity already gives a nice feeling of the proof. Lemma 5.5 (Vey [88]). Let be a discrete subgroup of SLd C1 .R/ that divides a properly convex open set . Then, for every x 2 , the convex hull of the orbit of x is , i.e. Conv. x/ D . In particular, if is strictly convex, then ƒ D @. Proof. The action is cocompact so there exists a number R0 > 0 such that given any point x 2 , the projection of the ball Bx .R0 / of radius R0 is = ; in other words, the ball Bx .R0 / meets every orbit. Let p be an extremal point of @. We are going to show that p 2 x, and this will prove the first part of the lemma. Suppose that p … x. Then there exists a neighbourhood U of p such that U \ x D ¿; but since p is extremal, the set U \ contains balls of .; d / of arbitrary size, contradicting the first paragraph. Since @ is closed and -invariant we always have ƒ @. Now if is strictly convex, suppose that ƒ ¨ @. Then since is strictly convex we get Conv.ƒ / ¨ ; in particular the convex hull of the orbit of any point of Conv.ƒ / is not all of , contradicting the first part of this lemma. Hence, ƒ D @ when is strictly convex.

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Proof of 5.3 in the case where is strictly convex. Let G be the identity component of the Zariski closure of . Since is strictly convex, ƒ D @ by Lemma 5.5, is strongly irreducible by Theorem 3.12 and positively proximal by Theorem 5.1; hence the group G is semi-simple and the representation W G ! SLd C1 .R/ is irreducible and proximal. Consider the limit set ƒG . The limit set is an orbit of G because the action of G on P d is by projective transformations, hence every orbit is open in its closure. Hence, the limit set is the unique closed orbit of G acting on P d . It is the orbit of the line of highest restricted weight. Hence, ƒG ƒ D @. There are two possibilities, ƒG D @ or ƒG D P d . In the first case, the maximal compact subgroup K of G acts also transitively on ƒG D @. But K fixes a point x of and a point x ? of since K is compact, hence K preserves the John ellipsoid E of centered at x in the affine chart P d X x ? . As K acts transitively on @, we get D E. In the second case, the lemma below shows that either G D SLd C1 .R/ or G D Sp2d .R/ and preserves a symplectic form. But a group which preserves a properly convex open set cannot preserve a symplectic form (Corollary 3.5 of [5]). Lemma 5.6 (Benoist, Lemma 3.9 of [5]). Let G be a linear semi-simple connected Lie group and . ; V / a faithful irreducible and proximal representation. The action of G on P .V / is transitive if and only if (1) G D SLd C1 .R/ and V D Rd C1 with d > 1, or (2) G D Sp2d .R/ and V D R2d with d > 2. If one reads carefully the proof, one should remark that we only used the hypothesis “ is strictly convex” to get ƒ D @. So, in fact we have shown that: Theorem 5.7. Let be a discrete group of SLd C1 .R/ that preserves a properly convex open set . If acts minimally on @, then is Zariski-dense or is an ellipsoid. This leads to the following question: Open question 5. Let be a discrete group of SLd C1 .R/ that divides (or quasidivides) an indecomposable properly convex open set which is not homogeneous. Does act minimally on @? If the answer to this question is yes, then we get an alternative proof of Theorem 5.3. We note that Benoist answers the last question in dimension 3 in [10]. Namely, he shows that the action of any group dividing an indecomposable properly convex open set is minimal on @.

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6 Gromov-hyperbolicity The notion of Gromov-hyperbolicity is a very powerful tool in geometric group theory and metric geometry. The goal of this section is to catch the link between Gromovhyperbolicity and roundness of convex bodies. A proper geodesic metric space X is Gromov-hyperbolic when there exists a number ı such that given any three points x; y; z 2 X, and given any geodesics Œx; y, Œy; z and Œz; x, the geodesic Œx; y is included in the ı-neighbourhood of Œy; z [ Œz; x. A group of finite type is Gromov-hyperbolic if its Cayley graph with respect to one of its finite generating sets 23 is Gromov-hyperbolic for the word metric. Theorem 6.1 (Benoist [8]). Let be a divisible convex set divided by a group . Then the following are equivalent: (1) The metric space .; d / is Gromov-hyperbolic. (2) The convex set is strictly convex. (3) The boundary of is C 1 . (4) The group is Gromov-hyperbolic. Note that a similar statement is true by [29] in the case of quasi-divisible convex sets. There is also a statement of this kind in [22] for non-compact quotients. And, finally, there is a weaker statement of this kind for geometrically finite actions in [32]. We will not review these results. We shall present a rough proof of this theorem.

6.1 The first step Proposition 6.2 (Benoist [8], Karlsson–Noskov [50]). Let be a properly convex open set. If the metric space .; d / is Gromov-hyperbolic, then the convex set is strictly convex with C 1 boundary. Proof. Suppose is not strictly convex and take a maximal non trivial segment s @. Choose a plane … containing s and intersecting , choose a sequence of points xn and yn in \ … converging to the different endpoints of s, and, finally, take any point z 2 \ …. It is an exercise to show that sup u2Œxn ;yn

d .u; Œxn ; z [ Œz; yn / ! 1;

which shows that is not Gromov-hyperbolic (see Figure 8)

23 This

property does not depend on the choice of the generating set.

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z

u

xn

yn

s

Figure 8. Strict convexity proof.

Suppose that @ is not C 1 but Gromov-hyperbolic. Then there exist a point z 2 @, a plane … 3 z such that \ … ¤ ¿, a triangle T of … containing \ … such that z is a vertex of T and such that the two segments of T issuing from z are tangent to \ … at z (see Figure 9).

T

x

0 n

y

xn y n

n

z

Figure 9. C 1 proof.

Now, choose two points x and y in \ … and two sequences xn and yn on Œx; zŒ and Œy; zŒ converging to z. Consider the triangle n of vertices xn , yn and z. We want to show that supu 2 Œxn ; yn d .u; Œxn ; z [ Œz; yn / ! 1. This needs some attention. One should remark that the Comparison Theorem (p. 211) shows that

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for any u 2 Œxn ; yn , we have dT .u; Œxn ; z [ Œz; yn / 6 d .u; Œxn ; z [ Œz; yn /. But the sequence dT .u; Œxn ; z [ Œz; yn / is eventually constant and strictly positive. In particular, the sequence dT .u; Œxn ; z [ Œz; yn / does go to 1, unfortunately. Nevertheless, one must remember that must be strictly convex thanks to the previous paragraph. Hence, we can find a triangle 0n whose sides tend to the side of T . Such a triangle contradicts Gromov-hyperbolicity.

6.2 The duality step We recall the following proposition. Proposition 6.3. Let be a properly convex open set. Then is strictly convex if and only if @ is C 1 . The following proposition is very simple once you know the notion of virtual cohomological dimension. The cohomological dimension of a torsion-free group of finite type is an integer d such that if acts properly on Rd then d > d and the quotient Rd = is compact if and only if d D d . The virtual cohomological dimension of a virtually torsion-free 24 group is the cohomological dimension of any of its torsion-free finite-index subgroup. A reference is [82]. Proposition 6.4. Let be a discrete group of SLd C1 .R/ acting on a properly convex open set . Then the action of on is cocompact if and only if the action of on is cocompact. We grab the opportunity to mention the following open question: Open question 6. Let be a discrete group of SLd C1 .R/ acting on a properly convex open set . Is it true that the action of on is of finite covolume if and only if the action of on is of finite covolume? It is known that the answer is yes in dimension 2 from [65] and also yes in any dimension if you assume that the convex set is strictly convex or with C 1 boundary by [29] and [32].

6.3 The key lemma Theorem 6.5. Suppose that is divisible and that is strictly convex. Then is Gromov-hyperbolic. 24 We

recall that Selberg’s lemma shows that every finite type subgroup of GLm .C/ is virtually torsion-free.

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This theorem is the heart of the proof of Theorem 6.1. We postpone its proof until Subsection 9.3.2 where we will show a more general theorem (Corollary 9.15). Proof of Theorem 6.1 assuming Theorem 6.5. 1/ ) 2/ and 1/ ) 3/ are the content of Proposition 6.2. 2/ ) 1/ is the content of Theorem 6.5. The equivalence 1/ () 4/ is a consequence of the fact that since divides , with the word metric is quasi-isometric to .; d / and Gromov-hyperbolicity is invariant by quasi-isometry. Let us show that 3/ ) 4/ to finish the proof. The properly convex open set dual to is strictly convex, by Proposition 6.3, and the action of on is cocompact by Proposition 6.4. Therefore, by Theorem 6.5, the metric space . ; d / is Gromovhyperbolic. But the group acts by isometries and cocompactly on . ; d /, hence the group is Gromov-hyperbolic. Theorem 6.1 has a fascinating corollary: Corollary 6.6. Suppose a group divides two properly convex open sets and 0 . Then is strictly convex if and only if 0 is strictly convex. We also want to stress that Theorem 6.1 is the only way known by the author to show that a divisible convex set (which is not an ellipsoid) of dimension at least 3 is strictly convex. Hence, the actual proof of the existence of a strictly convex divisible convex set relies on this theorem. We shall see in Subsection 8.2 that Gromov-hyperbolicity also implies some regularity of the boundary of .

7 Moduli spaces 7.1 A naturality statement Let be a group of finite type and d > 2 an integer. The set of homomorphisms Hom.; SLd C1 .R// can be identified with a Zariski closed subspace of SLd C1 .R/N , where N is the number of generators of . We put on it the topology induced from Hom.; SLd C1 .R//. We denote by ˇ the subspace of representations of in SLd C1 .R/ which divide a non-empty properly convex open set of P d . The following striking theorem shows that convex projective structures are very natural. Theorem 7.1 (Koszul–Benoist). Suppose M is a compact manifold of dimension d and D 1 .M / does not contain an infinite nilpotent normal subgroup. Then the space ˇ is a union of connected components of the space Hom.; SLd C1 .R//.

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We will see in the next paragraph that Koszul showed the “open” part of this theorem in [54]. The “closed” part was proved by Choi and Goldman when d D 2 in [23], by Kim when d D 3 and is a uniform lattice of SO3;1 .R/ in [51] and, finally, Benoist showed the general case in [9] (which is the hard step of the proof). One can also find a version of this theorem in the finite volume context in dimension 2 in [67]. Remark 7.2. In [9] (Corollary 2.13), Benoist showed that if divides a properly convex open set, then the group does not contain any infinite nilpotent normal subgroup if and only if is strongly irreducible.

7.2 A sketch of the proof of closedness The proof of the fact that ˇ is closed is quite hard in the general case. In dimension 2, we can give a simpler proof due to Choi and Goldman [23]. The following proposition is a corollary of a classical theorem of Zassenhaus ([95]): Proposition 7.3. Let be a discrete group that does not contain an infinite nilpotent normal subgroup. Then any limit of a sequence of discrete and faithful representations n W ! SLd C1 .R/ is also discrete and faithful. One can find a proof of this proposition in [42] or [75]. This theorem explains the beginning of the story. Lemma 7.4 (Choi-Goldman [23]). Let be a discrete group such that does not admit any infinite nilpotent normal subgroup. Let .n ; n / be a sequence where n is a properly convex open subset of P d and n a sequence of discrete and faithful representations, n W ! SLd C1 .R/, such that n ./ is a subgroup of Aut.n /. Suppose that n ! 1 2 Hom.; SLd C1 .R//. If 1 is irreducible, then 1 preserves a properly convex open set. Proof. Endow the space of closed subsets of P d with the Hausdorff topology. The subspace of closed convex subsets of P d is closed for this topology, therefore it is compact. Consider an accumulation point K of the sequence .n /n2N in this space. The convex set K is preserved by 1 . We have to show that K has non-empty interior and is properly convex. To do that, consider the 2-fold cover Sd of P d and rather than taking an accumulation point K of .n /n2N in P d , take it in Sd and call it Q. We have three possibilities for Q: (1) Q has empty interior; (2) Q is not properly convex; (3) Q is properly convex and has a non-empty interior. Suppose Q has empty-interior. As Q is a convex subset of Sd , this implies that Q spans a non-trivial subspace of Rd C1 which is preserved by 1 . This is absurd since

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1 is irreducible. Suppose now that Q is not properly convex. Then Q \ Q ¤ ¿ spans a non-trivial subspace of Rd C1 . This is again absurd. Therefore Q is properly convex and has non-empty interior. In dimension 2, using Lemma 7.5, one can show easily that every accumulation point of ˇ is an irreducible representation, hence the preceding lemma shows that ˇ is closed. Unfortunately, this strategy does not work in high dimensions and the machinery to show closedness is highly more involved. Lemma 7.5. Every Zariski-dense subgroup of SL2 .R/ contains an element with a negative trace. Every element of SL3 .R/ preserving a properly convex open set is such that Tr./ > 3. Remark 7.6. The second assertion of this lemma is a trivial consequence of Proposition 2.2. The first part is a lemma of [23]. Proof of the closedness in dimension 2 assuming Lemma 7.5. We choose a sequence n 2 ˇ which converges to a representation . Suppose is not irreducible. Then up to conjugation and transposition, the image of restricted to Œ; is included in the special affine group of the canonical affine charts of RP 3 (this subgroup is of course isomorphic to SL2 .R/ Ì R2 ). Hence, Lemma 7.5 shows that there exists an element 2 such that Tr. 1 .// < 1. But Lemma 7.5 shows that for every element , we have Tr. n .// > 3. Hence, the representation is irreducible. Therefore by Lemma 7.4, the representation preserves a properly convex open set . The action of on is proper using the Hilbert metric and the quotient is compact since the cohomological dimension of is 2.

7.3 The openness Let M be a manifold. A projective manifold 25 is a manifold with a .P d ; PGLd C1 .R//structure 26 . A marked projective structure on M is a homeomorphism h W M ! M where M is a projective manifold. A marked projective structure is convex when the projective manifold M is convex, i.e., the quotient of a properly convex open set by a discrete subgroup of Aut./. Two marked projective structures h; h0 W M ! M; M0 are equivalent if there exists an isomorphism i of .P d ; PGLd C1 .R//-structures between M and M 0 and the homeomorphism h01 B i B h W M ! M is isotopic to the identity. 25All projective structures are assumed to be flat all along this chapter. Here is an alternative definition: two torsion-free connections on a manifold M are projectively equivalent if they have the same geodesics, up to parametrizations. A class of projectively equivalent connections defines a projective structure on M . A projective structure is flat if every point has a neighbourhood on which the projective structure is given by a flat torsion-free connection. 26 See [85], [43] and Chapter 11 of the present volume [52] for the definition of a .G; X /-structure.

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We denote by P .M / the space of marked projective structures on M and by ˇ.M / the subspace of convex projective structures. There is a natural topology on P .M /, see [43] for details. Theorem 7.7 (Kozsul [54]). Let M be a compact manifold. The subspace ˇ.M / is open in P .M /. One can find a proof of this theorem in [54], but also in a lecture of Benoist [4] or in a paper of Labourie [56]. In the last reference, Labourie did not state the theorem but his Theorem 3.2.1 implies Theorem 7.7. The formalism of Labourie is the following: a projective structure on a manifold M is equivalent to the data of a torsion-free connexion r and a symmetric 2-tensor h satisfying a compatibility condition. Namely, let r h be a connexion on the vector bundle TM R ! M given by the formula: rXh .Z; / D rX Z C X; LX ./ C h.Z; X / where X, Z are vector fields on M , is a real-valued function on M and LX is the derivation of along X. We say that the pair .r; h/ is good if r preserves a volume and r h is flat. Labourie shows that every good pair defines a projective structure and that every projective structure defines a good pair. Finally, Labourie shows that a projective structure is convex if and only if the symmetric 2-tensor h of the good pair is definite positive. Hence, he shows that being convex is an open condition.

7.4 A description of the topology for the surface Let † be a compact surface with a finite number of punctures. We denote by ˇ.†/ and Hyp.†/ the moduli spaces of marked convex projective structures and hyperbolic structures on †, and by ˇf .†/ and Hypf .†/ the finite-volume ones (for the Busemann volume). In dimension 2, we can give three descriptions of the moduli space ˇ.†/. The first one comes from Fenchel–Nielsen coordinates, the second one comes from Fock– Goncharov coordinates on higher Teichmüller space and finally the third one was initiated by Labourie and Loftin independently. Theorem 7.8 (Goldman). Suppose that † is a compact surface with negative Euler characteristic . Then the space ˇ.†/ is a ball of dimension 8. Goldman shows this theorem for compact surfaces in [40]27 by extending Fenchel– Nielsen coordinates on Teichmüller space. Choi and Goldman extend this theorem to compact 2-orbifold in [24]. Finally, the author of this chapter extends Goldman’s theorem to the case of finite-volume surfaces in [67]. 27 This

is by the way a very nice door to the world of convex projective geometry.

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At a different time by completely different methods, Fock–Goncharov find a system of coordinates for higher Teichmüller space described in [37]. Since the space ˇ.†/ is the second simplest higher Teichmüller space (after Teichmüller space itself), their results give another system of coordinates on ˇ.†/. The situation of SL3 .R/ is much simpler than the situation of a general semi-simple Lie group, and there is a specific article dealing with Fock–Goncharov coordinates in the context of ˇ.†/, that the reader will be happy to read: [38]. Note that the Fock–Goncharov coordinates are nice enough to describe very simply and efficiently ˇf .†/ in ˇ.†/. This leads us to the last system of coordinates: Theorem 7.9 (Labourie [56] or Loftin [61] (compact case), Benoist–Hulin (finite volume case [13])). There exists a fibration ˇf .†/ ! Hypf .†/ which is equivariant with respect to the mapping class group of †.

7.5 Description of the topology for 3-orbifolds 7.5.1 Description of the topology for the Coxeter polyhedron. There is an another context where we can describe the topology of ˇ . It is the case of certain Coxeter groups in dimension 3. Take a polyhedron G , label each of its edges e by a number e 20; 2 and consider the space ˇG of marked 28 polyhedra P of P 3 with a reflection s across each face s such that for every two faces s, t of P sharing an edge e D s \ t , the product of the two reflections s and t is conjugate to a rotation of angle e . We define the quantity d.G / D e C 3 where e C is the number of edges of G not labelled 2 . We need an assumption on the shape of P to get a theorem. Roughly speaking, the assumption means that P can be obtained by induction from a tetrahedron by a very simple process. This process is called “écimer” in French. The English translation of this word is “to pollard” i.e. cutting the head of a tree. An ecimahedron is a polyhedron obtained from the tetrahedron by the following process: see Figure 10.

Figure 10. Ecimation or truncation.

28 This

means that you keep track of a numbering of the faces of P .

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A bad 3-circuit of G is a sequence of three faces r; s; t that intersect each other but such that the edges e D r \ s, f D s \ t and g D t \ s do not intersect and such that

e C f C g > and one of the angles e ; f ; g is equal to 2 . Theorem 7.10 (Marquis [66]). Suppose G is an ecimahedron without bad 3-circuits, which is not a prism and such that d.G / > 0. Then the space ˇG is a finite union of balls of dimension d.G /. Moreover, if G is a hyperbolic polyhedron, then ˇG is connected. The number of connected components of ˇG can be computed, but we refer the reader to [66] for a statement. The expression “G is a hyperbolic polyhedron” means that there exists a hyperbolic polyhedron which realizes G . Andreev’s theorem perfectly answers this question (see [2] or [77]). The assumption that G is not a prism is here just to simplify the statement. The case of the prism can be worked out very easily. We also mention that Choi shows in [21] that under an “ordonnability” condition, ˇ.G / is a manifold of dimension d.G /. Choi, Hodgson and Lee in [25] and Choi and Lee in [26] study the regularity of the points corresponding to the hyperbolic structure in ˇ.G /. 7.5.2 Description of the topology for 3-manifolds. Heusener and Porti show the “orthogonal” theorem to Theorem 7.10. Namely, in practice, Theorem 7.10 shows that one can find an infinite number of hyperbolic polyhedra such that their moduli space of convex projective structures is of dimension as big as you wish. Heusener and Porti show: Theorem 7.11 (Heusener and Porti [46]). There exist an infinite number of compact hyperbolic 3-manifolds such that ˇ.M / is a singleton. Cooper, Long and Thistlethwaite compute explicitly, using a numerical program, the dimension of ˇ.M / for M a compact hyperbolic manifold, using an explicit presentation and a list of 3-manifolds in [28]. It would be very interesting to find a topological obstruction to the existence of a deformation. And also a topological criterion for the smoothness of the hyperbolic point in ˇ.M /, as started by Choi, Hodgson and Lee. We also want to mention that very recently Ballas obtained in [3] a version of Theorem 7.11 in the context of finite-volume convex manifolds.

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8 Rigidity For us, a rigidity result is a theorem that says: “If a convex set have some regularity plus a big isometry group, then this convex has to belong to this list”. We give three precise statements. The interested reader can also read the chapter [45] of this Handbook.

8.1 For strongly convex bodies We say that a properly convex open set is strongly convex if its boundary is C 2 with positive Hessian. Colbois and Verovic show in [27] that a strongly convex body of P d endowed with its Hilbert metric is bi-lipschitz equivalent to the hyperbolic space of dimension d . Theorem 8.1 (Sasaki [79], Socié-Méthou [83]). Let be a strongly convex open set. If the group Aut./ is not compact, then is an ellipsoid. Sasaki proves this theorem in [79] in the case where the boundary is C 1 and d > 4. Podestà gives in [74] a detailed proof of Sasaki’s theorem with some refinements. Finally Socié-Methou gives in [83] the proof in full generality. In fact, Socié-Methou shows a more precise statement; she shows that if a properly convex open set admits an infinite-order automorphism which fixes a point p 2 @, then if @ admits an osculatory ellipsoid at p then is an ellipsoid. We remark that the technique of Sasaki-Podestà and Socié-Methou are completely different. The first one uses geometry of affine spheres and the second one uses only elementary techniques.

8.2 For round convex bodies The following theorem shows that the boundary of a quasi-divisible convex open set cannot be too regular unless it is an ellipsoid. Theorem 8.2 (Benoist (divisible) [8], Crampon–Marquis (quasi-divisible) [30]). Let be a quasi-divisible strictly convex open set. Then the following are equivalent: (1) The convex set is an ellipsoid. (2) The regularity of the boundary of is C 1C" for every 0 6 " < 1. (3) The boundary of is ˇ-convex for every ˇ > 2. Let ˛ > 0 and consider the map W Rd ! R, x 7! jxj˛ , where j j is the canonical Euclidean norm. The level set of defines a properly convex open subset E˛ of P d which is analytic outside the origin and infinity. A point p 2 @ is of class C 1C" (resp. ˇ-convex) if and only if one can find an image of E" (resp. Eˇ ) by a projective

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transformation inside (resp. outside) and such that the point origin of E" (resp. Eˇ ) is sent to p. We recall that a quasi-divisible strictly convex open set is Gromov-hyperbolic, therefore there exists an " > 0 and a ˇ > 2 such that the boundary of is C 1C" and ˇ-convex ([7]). The reader should find more details about this theorem in Chapter 7 of this volume [33].

8.3 For any convex bodies The following statement is just a reformulation of Theorem 5.3. We stress that this statement can be seen as a rigidity theorem. Theorem 8.3 (Benoist [6]). Let be a divisible convex open set. Then the following are equivalent: (1) The group Aut./ is not Zariski-dense. (2) The convex set is symmetric.

9 Benzécri’s theorem Let X D f.; x/ j is a properly convex open subset of P d and x 2 g. The group PGLd C1 .R/ acts naturally on X . The following theorem is fundamental in the study of Hilbert geometry. Theorem 9.1 (Benzécri’s Theorem). The action of PGLd C1 .R/ on X is proper and cocompact. Roughly speaking, even if the action of PGLd C1 .R/ on X is not homogeneous, the quotient space is compact; therefore it means that we can find some homogeneity in the local geometric properties of Hilbert geometry. Precise examples of this rough statement can be found in Subsection 9.2 of the present text.

9.1 The proof To prove Theorem 9.1, we will need two lemmas. The first one is very classical, the second one is less classical. 9.1.1 John’s ellipsoid Lemma 9.2. Given a bounded convex subset of the affine space Rd with center of mass at the origin, there exists a unique ellipsoid E with center of mass at the origin which is included in and of maximal volume. Moreover, we have E d E.

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9.1.2 Duality. The following lemma may sound strange to the reader not used to jump from the projective world to the affine world and vice versa; nevertheless, once we understand it, the lemma should sound right; but its proof needs some analytic tools. Lemma 9.3. Given a properly convex open set of P d and a point x 2 , there exists a unique hyperplane x ? 2 such that x is the center of mass of viewed in the affine chart P d X x ? . Moreover, the map X ! .P d / defined by .; x/ 7! x ? is continuous, PGLd C1 .R/-equivariant and the map ! given by x 7! x ? is analytic. We first give a Proof of Benzécri’s Theorem assuming Lemmas 9.2 and 9.3. We consider the space S D f.E; x/ j E is an ellipsoid of P d and x 2 Eg. The action of PGLd C1 .R/ on S is transitive with stabilizer PSOd .R/, a compact subgroup of PGLd C1 .R/, so S is a PGLd C1 .R/-homogeneous space on which PGLd C1 .R/ acts properly. We are going to define a fiber bundle of X over S with compact fiber which is PGLd C1 .R/-equivariant. Namely, the map ' W X ! S which associates to the pair .; x/ the pair .E; x/ where E is the John ellipsoid of viewed in the affine chart P d X x ? . The map ' is continuous and well-defined, thanks to Lemma 9.3. Lemma 9.2 shows that the fibers of this map are compact and this map is clearly equivariant. Since the action of PGLd C1 .R/ on S is proper and cocompact, it follows that the action of PGLd C1 .R/ on X is proper and cocompact. 9.1.3 A sketch of the proof of Lemma 9.3. In [89] Vinberg introduces a natural diffeomorphism between a sharp convex cone and its dual. Let C be a sharp convex cone of Rd C1 . The map ' W C ! C ; 7! Center of mass.C .d C 1//; where C .d C 1/ D fu 2 C j .u/ D d C 1g, is an equivariant diffeomorphism. For a proof of this statement, the reader can consult [89] but also [41]. We give a geometric presentation of this diffeomorphism but in practice an analytic presentation is needed to understand it correctly. Proof of Lemma 9.3. Let be a properly convex open set and x a point in . Consider C and C , the cone above and respectively, and the diffeomorphism ' W C ! C. Take a point u in C such that Œu D x. Let 2 C be the linear form defined by D '1 .u/. Then .u/ D d C 1 and u is the center of mass of C .d C 1/. Therefore, x is the center of mass of in the affine chart P d X 1 .0/. In other words, x ? D Œ 2 is the point we are looking for.

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9.2 Natural things are equivalent 9.2.1 Definitions. We denote by X the following space: X D f j is a properly convex open subset of P d g: A projective volume is a map from X which associates to a properly convex open set an absolutely continuous measure on with respect to Lebesgue measure. A projective metric is a map F from X which associates to a pair .; x/ 2 X a norm F .x/ on the tangent space Tx of at x. Each of these notions is said to be natural when it is invariant by PGLd C1 .R/ and continuous. The two basic examples are the Hilbert distance which gives rise to the Busemann volume and the Holmes–Thompson volume. We give a brief definition of both volumes in the introduction. We will denote by Hil any measure, distance, norm, etc. coming from the Hilbert distance. 9.2.2 Volume Proposition 9.4. In every dimension d , given any natural projective volume , there exist two constants 0 < ad < bd such that for any properly convex open set we have ad 6 Hil 6 bd . Proof. Given a properly convex open set , let us denote by f (resp. g ) the density of (resp. Hil ) with respect to Lebesgue measure. We get a map from X to RC f .x/ given by .; x/ 7! g .x/ . This map is continuous since and Hil are absolutely continuous. This map is also PGLd C1 .R/-invariant. So, Theorem 9.1 shows that this map attains its maximum and its minimum which are two strictly positive constants. The two following corollaries are now trivial. We just recall some definitions for the convenience of the reader. Given a projective volume , a properly convex open set is said to be -quasi-divisible if there exists a discrete subgroup of Aut./ such that .=/ < 1. Corollary 9.5. If and 0 are two natural projective volumes then any properly convex open set is -quasi-divisible if and only if it is 0 -quasi-divisible. This corollary justifies the usual definition of quasi-divisible convex set which uses Busemann volume. Given a projective volume , the -sup-volume entropy of is the quantity log..B.x; R/// ; R!1 R lim

Hil where x is any point 29 of and B.x; R/ is the ball of radius R of .; d /. 29 This

quantity does not depend on x.

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Corollary 9.6. If and 0 are two natural projective volumes then for any properly convex open set the -sup-volume entropy and the 0 -sup-volume entropy coincide. Of course, the same result is true if we change the supremum into an infimum. 9.2.3 Metric Proposition 9.7. In every dimension d , given any natural projective metric F there exist two constants 0 < ad < bd such that for any properly convex open set and for any point in x, we have ad F .x/ 6 FHil .x/ 6 bd F .x/. Proof. We denote by F Hil the Hilbert metric and we introduce a slightly bigger space than X . We take X ¾ D f.; x; v/ j .; x/ 2 X ; v 2 Tx and FHil .x/.v/ D 1g: The action of PGLd C1 .R/ on X ¾ is again proper and cocompact since X ¾ is a PGLd C1 .R/-equivariant fiber bundle over X with compact fiber. Now the following map is continuous, PGLd C1 .R/-invariant and takes strictly positive values: .; x; v/ 2 X ¾ 7! F .x/.v/. The conclusion is straightforward. 0 Corollary 9.8. If d and d are two distances coming from natural projective metrics 0 / are then for any properly convex open set , the metric spaces .; d / and .; d bi-Lipschitz equivalent through the identity map.

We remark that there is no reason for which the volume entropy of two bi-Lipschitz spaces is the same. We also remark that if a convex set is Gromov-hyperbolic for the Hilbert distance, it is Gromov-hyperbolic for all natural distances. 9.2.4 Curvature. It is hard to give a meaning to the sentence “Hilbert geometries are non-positively curved” since the Hilbert distance is a Finsler metric and not a Riemannian metric. A Hilbert geometry which is CAT(0) is an ellipsoid, see [34]. Lemma 4.2 allows to construct two natural Riemannian metrics on a Hilbert geometry, gvin and gaff . The first one is constructed using the Vinberg hypersurface (Theorem 4.1) and the second one thanks to the affine hypersurface (Theorem 4.6). Proposition 9.9. In any dimension, there exist two numbers 1 6 2 such that for any properly convex open set, any point x 2 and any plane … containing x, the sectional curvature of gvin and gaff at .x; …/ is between 1 and 2 . The triangle gives an example where the curvature is constant and equal to 0. The ellipsoid is an example where the curvature is constant equal to 1. It would have been good if 2 6 0. Unfortunately, Tsuji shows in [86] that there exists a properly convex open set , a point x and a plane … such that the

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sectional curvature of gaff at .x; …/ is strictly positive. Since the example of Tsuji is a homogeneous properly convex open set, we stress that gvin D gaff for this example. Hence, Hilbert geometry cannot be put in the world of non-positively curved manifolds using a Vinberg hypersurface or an affine sphere. Finally, we note that Calabi shows in [18] that the Ricci curvature of gaff is always non-positive.

9.3 Two not-two-lines applications of Benzécri’s theorem 9.3.1 The Zassenhaus–Kazhdan–Margulis lemma in Hilbert geometry. The use of the Margulis constant has proved to be very useful in the study of manifolds of non-positive curvature. The following lemma says that this tool is also available in Hilbert geometry. Theorem 9.10 (Choi [20], Crampon–Marquis [31], Cooper–Long–Tillmann [29]). In any dimension d , there exists a constant "d such that for every properly convex open set , for every point x 2 , for every real number 0 < " < "d and for every discrete subgroup of Aut./, the subgroup " .x/ generated by the set f 2 j d .x; x/ 6 "g is virtually nilpotent. The following lemma of Zassenhaus is the starting point: Lemma 9.11 (Zassenhaus [95]). Given a Lie group G, there exists a neighbourhood U of e such that for any discrete group of G, the subgroup U generated by the set f 2 j 2 Ug is nilpotent. First let us “virtually” prove the lemma in the case where is an ellipsoid E. Since the action of Aut.E/ on E is transitive and since the Margulis constant is a number depending only on the geometry of the space at the level of points, we just have to prove it for one point. We choose a point O 2 E and we have to show that if " is small enough then every discrete group of Aut.E/ generated by elements moving O at a distance less that " is virtually nilpotent. The Zassenhaus lemma gives us a neighbourhood U of e in Aut.E/ such that for any discrete subgroup of G, the subgroup U generated by f 2 j 2 Ug is nilpotent. The open set O D f 2 Aut.E/ j dE .O; .O// < "g is contained in the open set StabO U if " is small enough. What remain is that U is of finite index N in the group " generated by \ O. For this we have to show that N is less than the number of translates of U by StabO needed to cover entirely StabO . In the case of Hilbert geometry, we have to replace the fact that the action of Aut.E/ on E is transitive by Benzécri’s theorem 9.1. The proof of Benzécri’s theorem gives us an explicit and simple compact space D of X such that PGLd C1 .R/ D D X . Namely, choose any affine chart A, any point O 2 A, any scalar product on A, denote

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by B the unit ball of A and set: D D f.; O/ j O is the center of mass of in A and B dBg; where dB is the image of B by the homothety of ratio d and center at O in A. We need to link the Hilbert distance with the topology on PGLd C1 .R/. This is done by the following two lemmas: Lemma 9.12. For every " > 0, there exists a ı > 0 such that for every .; O/ 2 D and every element 2 Aut./, we have: d .O; .O// 6 " H) dPGLd C1 .R/ .1; / 6 ı: Lemma 9.13. For every " > 0, there exists a ı > 0 such that for every .; O/ 2 D and every element 2 Aut./, we have: d .O; O/ 6 ı H) dPGLd C1 .R/ .Stab .O/; / 6 ": The rest of the proof is like in hyperbolic geometry. The details can be found in [29] or [31]; the strategy is the same and can be traced back to the two-dimensional proof of [20]. 9.3.2 A characterisation of Gromov-hyperbolicity using the closure of orbits under PGLdC1 .R/. Recall that X is the space of properly convex open subsets of P d endowed with the Hausdorff topology. For every ı > 0, we denote by Xı the space of properly convex open subsets which are ı-Gromov-hyperbolic for the Hilbert distance. Theorem 9.14 (Benoist [7]). (1) The space Xı is closed in X . (2) Conversely, for every closed subset F of X which is PGLd C1 .R/-invariant and contains only strictly convex open sets, there exists a constant ı such that F Xı . Proof. We only sketch the proof. We start by showing the first point. Take a sequence n 2 Xı converging to a properly convex open set 1 . We have to show that 1 is ı-hyperbolic; we first show that it is strictly convex. Suppose 1 is not strictly convex. Then there exists a maximal non-trivial segment Œx1 ; y1 1 . Therefore, we can find (see the proof of Lemma 6.2) a sequence of triangles zn ; xn ; yn included in 1 whose “size” in 1 tends to infinity. If n is large enough, these triangles are in fact included in n and so their size in n should be less than ı. Passing to the limit we get a contradiction. To show that 1 is Gromov-hyperbolic, we do almost the same thing. Suppose it is not in Xı . Take a triangle of 1 whose size is more than ı. Since 1 is strictly convex, the side of such a triangle is a segment, and one can conclude like in the previous paragraph. Now for the second affirmation. Suppose such a ı does not exist. Then we can find a sequence of triangles Tn whose size goes to infinity in a sequence of properly

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convex open sets n . Precisely, this means that there exists a sequence xn , yn , zn , un of points of n such that un 2 Œxn ; yn and the quantity dn .un ; Œxn ; zn [ Œzn ; yn / tends to infinity. Using Benzécri’s theorem, we can assume that the sequence .n ; un / converges to .1 ; u1 /. Since F is closed, we get that 1 is strictly convex. One can also assume that the triangle xn , yn , zn converges to a triangle of 1 , which cannot be degenerate since its size is infinite. But the strict convexity of 1 contradicts the fact that the size of the limit triangle is infinite. Corollary 9.15. Let be a properly convex open subset of P d . The following properties are equivalent: (1) The metric space .; d / is Gromov-hyperbolic. (2) The closure of the orbit PGLd C1 .R/ in X contains only strictly convex properly convex open sets. (3) The closure of the orbit PGLd C1 .R/ in X contains only properly convex open sets with C 1 boundary. Proof. We just do 1/ () 2/. If the closure F of the orbit PGLd C1 .R/ in X contains only strictly convex properly convex open sets, since F is closed and PGLd C1 .R/-invariant, Theorem 9.14 shows that all the elements of F are in fact Gromov-hyperbolic, hence is Gromov-hyperbolic. Now, if is Gromov-hyperbolic, there is a ı > 0 such that 2 Xı . This space is a closed and PGLd C1 .R/-invariant subset of X, hence the closure of PGLd C1 .R/ is included in Xı . Hence Proposition 6.2 concludes the proof. This corollary has a number of nice corollaries. The first one is an easy consequence, using Proposition 6.3. Corollary 9.16. The metric space .; d / is Gromov-hyperbolic if and only if the space . ; d / is Gromov-hyperbolic. The second one is the conclusion of Theorem 6.1 that we have left so far. Corollary 9.17. A divisible convex set which is strictly convex is Gromov-hyperbolic. To show this corollary, one just needs to know the following proposition which is a direct consequence of Benzécri’s theorem. Proposition 9.18. Let be a divisible convex set. Then the orbit PGLd C1 .R/ is closed in X. Proof. Benzécri’s theorem shows that the action of PGLd C1 .R/ on X is proper, hence the orbit PGLd C1 .R/ .; x/ is closed for every point x 2 . Now, we remark that the orbit PGLd C1 .R/ is closed in X if and only if the union [ PGLd C1 .R/ .; x/ PGLd C1 .R/ .; / WD x2

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is closed in X . Now, since there is a group which divides , one can find a compact subset K of such that K D . Hence the conclusion follows from this computation: [[ PGLd C1 .R/ .; .K// PGLd C1 .R/ .; / D x2K 2

D

[

PGLd C1 .R/ .; K/:

x2K

10 The isometry group of a properly convex open set 10.1 The questions We basically know almost nothing about the group of isometries Isom./ of the metric space .; d /. Open question 7 (raised by de la Harpe in [35]). Is Isom./ a Lie group? Is Aut./ always a finite-index subgroup of Isom./? If yes, does this index admit a bound Nd depending only of the dimension? Does Nd D 2? By the Arzelá–Ascoli theorem, the group Isom./ is a locally compact group for the uniform convergence on compact subsets. We shall see that the answer to the first question is yes and it is already in the literature. The other questions are open and a positive answer would mean that the study of Aut./ or Isom./ is the same from a group-geometrical point of view. The first question is a corollary of a wide open conjecture: Conjecture (Hilbert–Smith). A locally compact group acting effectively on a connected manifold is a Lie group. This conjecture is known to be true in dimension d D 1; 2 and 3. For the proof in dimension 1 and 2, see [70], or Theorem 4.7 of [39] for a proof in dimension 1. The proof in dimension 3 is very recent and due to Pardon [73] (2011). The first article around this conjecture is due to Bochner and Montgomery [15], ˘ cepin who proved it for smooth actions. Using a theorem of Yang ([94]), Repov˘s and S˘ showed in [76] that the Hilbert–Smith conjecture is true when the action is by biLipschitz homeomorphism with respect to a Riemannian metric. There is also a proof of Maleshich in [63] for Hölder actions. For a quick survey on this conjecture, we recommend the article of Pardon [73]. The author of the present chapter is not an expert in this area and this paragraph does not claim to be an introduction to the Hilbert–Smith conjecture. ˘ cepin works for Finsler metrics since it uses only the The proof of Repov˘s and S˘ fact that the Hausdorff dimension of a manifold with respect to a Riemannian metric

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is the dimension of the manifold. We sketch the argument for the reader. We want to show that: ˘ cepin [76]). A locally compact group acting effectively Theorem 10.1 (Repov˘s and S˘ by Lipschitz homeomorphisms on a Finsler manifold is a Lie group. In particular, the group Isom.; d / is a Lie group. Proof. A classical and very useful reduction (see for example [59]) of the Hilbert– Smith conjecture shows that we only have to show that the group Zp of p-adic integers cannot act on M . Since every neighbourhood of the identity of Zp contains a copy of Zp , we can assume that Zp acts by Lipschitz homeomorphisms with bounded Lipschitz constant. Now, since Zp is compact, we can assume that this action is in fact by isometries by averaging the metric using the Haar measure of Zp . ˘ cepin. We have the following Now comes the key argument of Repov˘s and S˘ inequality where dim, dimH and dimZ mean respectively the topological dimension, the Hausdorff dimension and the homological dimension. dim.M / D dimH .M / > dimH M=Zp > dim M=Zp > dimZ M=Zp D dim.M / C 2 The first equality comes from the fact that the Hausdorff dimension of a Finsler manifold is equal to its topological dimension.30 The first inequality follows from the fact that since the action is by isometries, the quotient map is distance non-increasing. The second and third inequalities follow from the fact that we always have dimH .X / > dim.X / > dimZ .X/ when X is a locally compact Hausdorff metric space.31 The last equality is the main result of Yang in [94].

10.2 The knowledge Since in this context subgroups of finite-index are important, we think that the group Aut˙ ./ of automorphisms of of determinant ˙1 is more adapted to the situation. Proposition 10.2 (de la Harpe [35]). Let be a properly convex open set. Suppose that the metric space .; d / is uniquely geodesic. Then Aut˙ ./ D Isom./. Proof. If the only geodesics are the segments, this implies that the image of a segment by an isometry g is a segment. Therefore the fundamental theorem of projective geometry implies that g is a projective transformation. topological dimension of a compact topological space X is the smallest integer n such that every finite open cover A of X admits a finite open cover B of X which refines A in which no point of X is included in more than n C 1 elements of B. If no such minimal integer n exists, the space is said to be of infinite topological dimension. The topological dimension of a non-compact locally compact Hausdorff metric space is the supremum of the topological dimensions of its compact subspaces. 31 For the definition of the homological dimension we refer to [94]. For the inequalities, we refer to [80] for the first and [94] for the second. 30 The

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The following proposition is a nice criterion for uniqueness of geodesics .; d /. Proposition 10.3 (de la Harpe [35]). Let be a properly convex open set. Then the metric space .; d / is uniquely geodesic if and only if for every plane … intersecting , the boundary of the 2-dimensional convex set \ … contains at most one maximal segment. We stress the following corollary: Corollary 10.4. If is strictly convex, then Isom./ D Aut˙ ./. Theorem 10.5 (de la Harpe [35] for dimension 2, Lemmens–Walsh [60] for the general case). Suppose that is a simplex. Then Aut˙ ./ is of index two in Isom./. Proof. We give a rough proof in the case of a triangle . A nice picture will show you that the balls of the triangle are hexagons. Moreover, the group R2 acts simply transitively on the triangle . The induced map is an isometry from the normed vector space R2 with the norm given by the regular hexagon to with the Hilbert distance. Now, every isometry of a normed vector space is affine. So the group Isom./ has twelve connected components since the group Isom./=Isom0 ./ is isomorphic to the dihedral group of the regular hexagon, where Isom0 ./ is the identity component of Isom./. But the group Aut˙ ./ has six connected components and the group Aut ˙ ./=Aut˙0 ./ is isomorphic to the dihedral group of the triangle. Theorem 10.6 (Molnar [69] for the complex case, Bosché [16] for the general case). Suppose that is a non-strictly convex symmetric properly convex open set. Then Aut˙ ./ is of index two in Isom./. In the case where is the simplex given by the equations D fŒx1 W x2 W W xd C1 j xi > 0g; then an example of a non-linear automorphism is given by Œx1 W x2 W W xd C1 7! Œx11 W x21 W W xd1C1 : In the case where is a non-strictly convex symmetric space, then can be described as the projectivisation of a convex cone of symmetric definite positive matrices (or of positive Hermitian symmetric complex matrices, or the analogue with the quaternions or the octonions), and the non-linear automorphism is given by M 7!t M 1 (or analogous). In both cases, the non-linear automorphisms are polynomial automorphisms of the real projective space, the group Aut˙ ./ is normal in Isom./, and even better, Isom./ is a semi-direct product of Aut ˙ ./ with a cyclic group of order 2. Finally, Lemmens and Walsh showed the following theorem:

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Theorem 10.7 (Lemmens and Walsh [60]). Suppose that is a polyhedron which is not a simplex. Then Isom./ D Aut ˙ ./. Lemmens and Walsh conjecture that Isom./ ¤ Aut˙ ./ if and only if is a simplex or is symmetric but not an ellipsoid. For a survey on the isometry group of Hilbert geometry, see Chapter 5 of this volume [93].

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Chapter 9

Dynamics of Hilbert nonexpansive maps Anders Karlsson

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic numbers associated to semicontractions . . . . . . . . . . . Horofunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . Horofunctions and semicontractions . . . . . . . . . . . . . . . . 4.1 An argument of Beardon . . . . . . . . . . . . . . . . . . . 4.2 A general abstract result . . . . . . . . . . . . . . . . . . . 5 Horofunctions and the asymptotic geometry of Hilbert geometries 6 Some consequences for Hilbert metric semicontractions . . . . . 6.1 Karlsson–Noskov . . . . . . . . . . . . . . . . . . . . . . . 6.2 Monotone orbits . . . . . . . . . . . . . . . . . . . . . . . 6.3 Linear drift . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Two-dimensional geometries . . . . . . . . . . . . . . . . . 7 The theorems of Lins and Nussbaum . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

263 265 266 267 267 268 269 270 270 270 271 271 272 272

1 Introduction In a letter to Klein, published in Mathematische Annalen in 1895, Hilbert noted that a formula which Klein observed giving the projective model of the hyperbolic plane, provides a metric on any bounded convex domain. Hilbert’s concern was in the foundations of geometry, going back to the famous Euclid postulates, and his fourth problem asks about geometries where straight lines are geodesics. Today, with an extensive development of metric geometry, we consider these Hilbert geometries as beautiful concrete examples of metric spaces interpolating between (certain) Banach spaces and the real hyperbolic spaces. To a certain limited extent, the Hilbert metric could also be viewed as a simpler analogue of Kobayashi’s (pseudo-) metric in several variable complex analysis, such as the Teichmüller metric. It is then all the more remarkable that this metric discovered and discussed from a foundational geometric point of view (by Hilbert and Busemann) is a highly useful metric for other mathematical sciences, providing yet another example of research driven by purely theoretical curiosity leading to unexpected applications. This was

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probably first noticed, seemingly independently and simultaneously, by Birkhoff [5] and Samelson [27]. The Hilbert metric is defined via the cross-ratio from projective geometry. By virtue of being invariant, it is therefore not surprising that it is useful in the study of groups of projective automorphisms, see for example [26]. A philosophical explanation for the abundance of applications beyond group theory is that positivity (say of population sizes in biology, concentrations in chemistry or prices in economics) is ubiquitous and that many even nonlinear maps contract Hilbert distances. The contraction property leads to fixed point theorems which in turn are a main source of existence of solutions to equations. Recently, a very good book was published on this subject, nonlinear Perron–Frobenius theory, written by Lemmens and Nussbaum [17]. The following conjecture was made independently by Nussbaum and the author of this chapter about a decade ago: Conjecture. Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Then either there is a fixed point of f in C or the forward orbits f n .x/ accumulate at one unique closed face. In view of Nussbaum [23] we know the first part, namely if the orbits do not accumulate on the boundary of C , then indeed there is a fixed point. That the accumulation set does not depend much on which point x in C is iterated is not difficult to see, but what is not quite settled yet is whether the limit set necessarily belongs to the closure of only one face. In this brief text, I will focus on this issue and explain the known partial results. Taken together, they provide convincing support for the conjecture as they treat in particular two basic and opposite situations, the strictly convex case (Beardon) and the polyhedral case (Lins). A general result of Noskov and the author shows that the conjecture in general is at least almost true, see below. As Volker Metz originally indicated to me, in practice it is often hard to show that orbits are bounded (this is needed for establishing the existence of a fixed point), but arguing by contradiction and assuming that the orbits are unbounded, then if one can show the existence of a limiting face, this sometimes leads to the desired contradiction. There are several studies on the bounded orbit case, which is not considered here; for that we refer to [16], [15], [17], [18] and references therein. There is a wealth of applications of the Hilbert metric, see [23], [25], [17]; let me just add two papers, one on decay of correlation [21] and another on growth of groups [3]. A final remark is that from a certain point of view, the conjecture is somewhat reminiscent of the notorious invariant subspace problem in functional analysis, although surely much less difficult. Acknowledgements. I have at various times benefited from discussions on this general topic with Alan Beardon, Gena Noskov, Volker Metz, Thomas Foertsch, Roger Nussbaum, Brian Lins, Bruno Colbois, Constantin Vernicos, Enrico Le Donne, and Bas Lemmens. I would also like to thank Athanase Papadopoulos and Marc Troyanov for numerous improvements on the text. This work is supported in part by the Swiss NSF grant 200021 132528/1.

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2 Basic numbers associated to semicontractions Let .X; d / be a metric space. A semicontraction f W X ! X is a map such that d.f .x/; f .y// d.x; y/ for every x; y 2 X. Other possible terms for this are 1-Lipschitz, non-expanding or nonexpansive maps . Since f is a self-map we may study it by iterating it. If one orbit ff n .x/gn>0 is bounded, then every orbit is bounded, since d.f n .x/; f n .y// d.x; y/ for every n. Therefore, having bounded or unbounded orbits is a well-defined notion. We use the same terminology for subsets, semigroups or groups of semicontractions. For example, if a semigroup of semicontractions fixes a point, then trivially any orbit is bounded. The translation number of f is 1 d.x; f n .x//; n which exists (and is independent of x) by the following lemma: f D lim

n!1

Lemma 2.1. Let an be a subadditive sequence of real numbers, that is, anCm an C am . Then the following limit exists: 1 1 an D inf am 2 R [ f1g: n!1 n m>0 m lim

Proof. Given " > 0, pick M such that aM =M inf an =n +". Decompose n as kn M C rn , where 0 rn < M . Hence kn =n ! 1=M . Using the subadditivity and considering n big enough (n > N."// 1 1 1 1 am an akn M Crn .kn aM C arn / m>0 m n n n 1 1 aM C " inf am C 2": m>0 m M Since " is at our disposal, the lemma is proved. inf

Furthermore we define ıf .x/ D lim d.f n .x/; f nC1 .x//; n!1

which exists because it is the limit of a bounded monotone sequence. Finally, let Df D inf d.x; f .x// x2X

be the minimal displacement, sometimes also called the translation length. Since d.x; f n .x// d.x; f .x// C C d.f n1 .x/; f n .x// nd.x; f .x//

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we have 0 f Df ıf .x/: If f is an isometry, then ıf is constant along every orbit, which means that typically ıf .x/ > Df . Moreover, it might happen that 0 D f < Df , for example if f has finite order and X is an orbit. A metric space is proper if every closed and bounded subset is compact. A theorem of Calka [6] asserts that provided that X is proper, if an orbit is unbounded, then it actually escapes every bounded set, so that d.f n x; x/ ! 1 as n ! 1 for any x. This applies to the Hilbert metric case that we consider here. In this case this fact was independently discovered by Nussbaum [23]. For non-proper spaces this is not true in general.

3 Horofunctions Horospheres first appeared in noneuclidean geometry in the work of Lobachevsky. Horofunctions appear later in the work of Busemann. Consider the unit disk D in the complex plane. The metric with constant Gaussian curvature 1 is ds D

2 jdzj 1 jzj2

;

and in particular d.0; z/ D log

1 C jzj : 1 jzj

This metric is called the Poincaré metric and for a point 2 @D one has the horofunction j zj2 : h .z/ D log 1 jzj2 These appear explicitly or implicitly for example in the Poisson formula in complex analysis, Eisenstein series, and the Denjoy–Wolff theorem. An abstract definition was later introduced by Busemann who defines the Busemann function associated to a geodesic ray to be the function h .z/ D lim d..t /; z/ t: t!1

Note here that the limit indeed exists in any metric space, since the triangle inequality implies that the expression on the right is monotonically decreasing and bounded from below by d.z; .0//. The convergence is moreover uniform if X is proper as can be seen from a 3"-proof using the compactness of closed balls. Horoballs are sublevel sets of horofunctions h./ C . In Euclidean geometry horoballs are halfspaces and in the Poincaré disk model of the hyperbolic plane horoballs, or horodisks in this case, are Euclidean disks tangent to the boundary circle.

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There is a more general definition, perhaps first considered by Gromov around 1980. Namely, let X be a complete metric space and let C.X / denote the space of continuous real functions on X equipped with the topology of uniform convergence on bounded subsets. Fix a base point x0 2 X . Consider now the map ˆ W X ! C.X / defined by ˆ W z 7! d.z; / d.z; x0 /: A related map was considered by Kuratowski [14] and independently K. Kunugui [13] in the 1930s. We will sometimes denote by hz the function ˆ.z/. Note that every hz is 1-Lipschitz because jhz .x/ hz .y/j D jd.z; x/ d.z; x0 / d.z; y/ C d.z; x0 /j d.x; y/ which applied to y D x0 gives jhz .x/j d.x; x0 /: The map ˆ is a continuous injection. We denote the closure ˆ.X) by Xx H or X [ @H X and call it the horofunction compactification of X. The elements in @H X are called horofunctions. In some sense, horofunctions are to metric spaces what linear functionals (of norm 1) are to vector spaces. More on horofunction boundaries, and in particular on the horofunction boundary of Hilbert geometry, is contained in Chapter 5 of this volume [29], by Cormac Walsh.

4 Horofunctions and semicontractions It has long been observed, starting with the studies of Wolff and Denjoy on holomorphic self-maps, that the notion of horofunction is relevant for the dynamical behaviour of semicontractions. See §4.2 of Chapter 10 of this volume [18].

4.1 An argument of Beardon Here is a sketch of a nice argument due to Beardon [4]. Suppose that the semicontraction f of a proper metric space X can be approximated by uniform contractions fk , so that for every x one has fk .x/ ! f .x/. By the contraction mapping principle, every fk has a unique fixed point yk since X is complete. Now take a limit point y of this sequence in a compactification Xx of X . For simplicity of notation, we assume that yk ! y. If the limit point belongs to X , then it must be fixed by f . Indeed: f .y/ D lim fk .yk / D lim yk D y: k!1

k!1

If the limit point instead belongs to the associated ideal boundary @X D Xx n X , then we define the associated “horoballs”: z 2 Hfyk g .x/ if and only if z is the limit of

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points zk belonging to the closed balls centered at yk with radius d.x; yk /. These are invariant in the sense that f .Hfyk g .x// Hfyk g .x/, as is straightforward to verify ([4]). In the special case that the compactification is the horofunction compactification and the ideal boundary @H X, the horoballs are not dependent on the sequence, just the limit y and one has for any x 2 X , hy .f .x// D lim d.yk ; f .x// d.yk ; x0 / k!1

D lim d.yk ; fk .x// d.yk ; x0 / k!1

D lim d.fk .yk /; fk .x// d.yk ; x0 / k!1

lim d.yk ; x/ d.yk ; x0 / k!1

D hy .x/: This implies in particular: Theorem 4.1 ([4]). Let C be a bounded strictly convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Then either there is a fixed point of f in C or the forward orbits f n .x/ converge to a unique boundary point z 2 @C . In a more recent paper of Gaubert and Vigeral [8], an interesting sharpening (in some sense) of Beardon’s argument is used to establish a result inspired by the Collatz– Wielandt characterisation of the Perron root in linear algebra. Theorem 4.2 ([8]). Let f be a semicontraction of a complete metrically star-shaped hemi-metric space .X; d /. Then, Df D f D max inf h.x/ h.f .x//: x H x2X h2X

This is just the first part of the theorem; it has an interesting second part as well. For precise definitions we refer to the paper in question, but roughly speaking, the metric space is allowed to be asymmetric and should have a canonical choice of geodesics which have a Busemann non-positive curvature property. If .X; d / is not proper, one has to interpret the horofunctions appropriately. To a certain extent this theorem unifies Beardon’s result with the result in the next paragraph in the setting considered. Both Beardon and Gaubert–Vigeral results apply to semicontractions of the Hilbert metric.

4.2 A general abstract result Beardon’s argument depends on finding approximations to f . Here is a quite general version of the relationship between semicontractions and horofunctions, not using any approximating sequence:

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Proposition 4.3 ([9]). Let .X; d / be a proper metric space. If f is a semicontraction with translation number , then for any x 2 X there is a function h 2 Xx H such that h.f k .x// k for all k > 0. Proof. Given a sequence i & 0 we set bi .n/ D d.f n .x/; x/. i /n. Since these numbers are unbounded, we can find a subsequence such that bi .ni / > bi .m/ for any m < ni and by sequential compactness we may moreover assume that f ni .x/ ! h 2 Xx H , for some h. We then have, for any k 1, h.f k .x// D lim d.f k .x/; f ni .x// d.x; f ni .x// i!1

lim inf d.x; f ni k .x// d.x; f ni .x// i!1

lim inf bi .ni k/ C . i /.ni k/ bi .ni / . i /ni i!1

lim k C i k D k; i!1

as was to be shown.

5 Horofunctions and the asymptotic geometry of Hilbert geometries Being a convex domain in Euclidean space, C has a natural extrinsic boundary @C . One faces the challenge to compare it with the intrinsic metric boundary @H C . Two papers that consider this comparison are [12] and [28]. In the former, only the case of simplices is treated and a very precise description is obtained. In the latter, reference to the general situation is considered which is useful here. It is observed that the horoballs are convex sets, and this means in particular that any intersection of horoballs where the base points tend to the boundary of C can at most contain one closed face. Walsh also shows that any sequence convergent in the horofunction boundary also converges to a point in @C . We will make free use of these facts below. Notice that since any two orbits lie at a bounded distance from each other, it follows from simple well-known estimates on the Hilbert metric that the closed face, as predicted by the conjecture, must be independent of the point x being iterated. This is a consequence of the following more general statement: Proposition 5.1 ([10]). Let C be a bounded convex domain with the Hilbert metric d . Fix y 2 C and define the Gromov product: .x; x 0 / D

1 .d.x; y/ C d.x 0 ; y/ d.x; x 0 //: 2

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Assume that we have two sequences of points in C that converge to boundary points: N z, N xn ! xN 2 @C and zn ! zN 2 @C . If @C does not contain the line segment Œx; then there is a constant K D K.x; N zN / such that lim sup.xn ; zn / K: n!1

As remarked in the same paper, this is useful for the study of iteration of semicontractions f , see Theorem 6.1 below. The result has also found other applications, such as in [7]. To explain the above point about different orbits in detail, notice first that d.f n .x/; f n .z// d.x; z/ for all n > 0. Hence for any unbounded f one has 1 .d.f n .x/; y/ C d.f n .z/; y/ d.f n .x/; f n .z/// ! 1: 2 In view of the proposition, any subsequence of the iterates on x and the same for z must accumulate on the same closed faces. .f n .x/; f n .z// D

6 Some consequences for Hilbert metric semicontractions The previous results have a number of applications for 1-Lipschitz maps in Hilbert geometry.

6.1 Karlsson–Noskov In [10], a certain simple asymptotic geometric fact in terms of the Gromov product is established as recalled above. Now following [9] we select a subsequence ani of an WD d.f n .x/; x/ such that ani > ak for all k < ni . It follows that the Gromov product .f ni .x/; f k .x// tends to infinity. By selecting a converging subsequence of the ni ’s, we then conclude from Proposition 5.1 that: Theorem 6.1 ([10]). Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Then either there is a fixed point of f in C or there is a boundary point z 2 @C such that the line segment between z and any limit point of a forward iteration f n .x/ is contained in @C . In [11] the same result but in a more general setting is discussed.

6.2 Monotone orbits If the sequence d.x; f n .x// grows monotonically, then every convergent subsequence can play the role of ni as in the proof of Proposition 4.3. This means that in Theo-

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rem 6.1, the special property of z is held by any limit point, thus verifying the conjecture. Explicitly: Theorem 6.2. Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Suppose that for some x 2 X , d.x; f n .x// % 1 monotonically. Then there is a closed face that contains every accumulation of the iterations of f . The monotonicity can be weakened for this argument to go through with some smaller addition, although this may not suffice for the general case.

6.3 Linear drift If f is strictly positive, then by the general abstract result, Proposition 4.3, the orbit goes deeper and deeper inside the horoballs of a fixed horofunction. Since these are convex sets, their intersection is convex and must therefore be a closed face. The orbit can only accumulate here since the horoballs contain all orbit points: Theorem 6.3. Let C be a bounded convex domain in RN equipped with the Hilbert metric d . Let f W C ! C be a 1-Lipschitz map. Suppose that f > 0; then there is a closed face that contains every accumulation point of the iterations of f . It is interesting to note, as is done in [18], that the conjecture also holds in the quite opposite situation where lim inf k!1 d.f k .x/; f kC1 .x// D 0, as shown by Nussbaum in [25].

6.4 Two-dimensional geometries In this note we have described two methods: Beardon’s method with approximations and another one coming from [9] in terms of orbits. Unfortunately, there is at present no known relation between the associated boundary points these two arguments give. If they would give the same point, then the conjecture would follow. In dimension two, where the extrinsic geometry is limited, one can play out these two boundary points to conclude: Theorem 6.4. The Conjecture is true in dimension 2. Proof. By Karlsson–Noskov, there is a star containing all orbit points. At worst, this is made up of two adjacent line segments; suppose this is the case since otherwise we are done. Beardon’s boundary point must lie in one of these. But no matter in which, since the horoballs are invariant sets, we again get that since the orbit accumulates at all the faces in the star we get a contradiction, and there can only be one closed face

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containing all limit points. In the case Beardon’s point is the corner point, then this coincides with the point in [9]. So Beardon’s point is a limit point, but in this case the closed face is just that same point. This proves the conjecture.

7 The theorems of Lins and Nussbaum Lins showed in his thesis [20] that the conjecture is true for every polyhedral domain. This is particularly interesting since it seems to be the case most opposite to the one that Beardon handled, i.e. the strictly convex domains. The proof goes via an isometric embedding into a finite-dimensional Banach space. Lins and Nussbaum [19] showed that for projective linear maps, a stronger version of the conjecture is true: the orbits converge to a finite number of points. (The paper [7], which proves the convergence of geodesics rays, is somewhat analogous to this). This constitutes a beautiful addendum to the classical Perron–Frobenius theorem. Further established cases of the conjecture for specific maps of practical interest can be found in [1], [19], [25], [17]. We conclude by remarking that, in contrast to the linear case of [19], Lins [20] has shown that in some sense the conjecture is best possible: for any simplex together with a convex subset of the boundary, he constructs a semicontraction whose limit set contains this boundary subset.

References [1]

M. Akian, S. Gaubert, B. Lemmens, and R. Nussbaum, Iteration of order preserving subhomogeneous maps on a cone. Math. Proc. Cambridge Philos. Soc. 140 (2006), 157–176.

[2]

J. C. Alvarez Paiva, Hilbert’s fourth problem in two dimensions. In MASS selecta, Amer. Math. Soc., Providence, RI, 2003, 165–183.

[3]

L. Bartholdi andA. Erschler, Groups of given intermediate word growth. Ann. Inst. Fourier, to appear; preprint, arXiv:1110.3650.

[4]

A. F. Beardon, The dynamics of contractions. Ergodic Theory Dynam. Systems 17 (1997), 1257–1266.

[5]

G. Birkhoff, Extensions of Jentzsch’s theorem. Trans. Amer. Math. Soc. 85 (1957), 219–227.

[6]

A. Całka, On conditions under which isometries have bounded orbits. Colloq. Math. 48 (1984), 219–227.

[7]

T. Foertsch and A. Karlsson, Hilbert metrics and Minkowski norms. J. Geom. 83 (2005), 22–31.

[8]

S. Gaubert and G. Vigeral, A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. Math. Proc. Cambridge Philos. Soc. 152 (2012), 341–363

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A. Karlsson, Non-expanding maps and Busemann functions. Ergodic Theory Dynam. Systems 21 (2001), 1447–1457.

[10] A. Karlsson and G.A. Noskov, The Hilbert metric and Gromov hyperbolicity. Enseign. Math. (2) 48 (2002), 73–89. [11] A. Karlsson, On the dynamics of isometries. Geom. Topol. 9 (2005), 2359–2394. [12] A. Karlsson, A. Metz, and G.A. Noskov, Horoballs in simplices and Minkowski spaces. Internat. J. Math. Math. Sci. 2006, Art. ID 23656, 20 pp. [13] K. Kunugui, Applications des espaces à une infinité de dimensions à la théorie des ensembles. Proc. Imp. Acad. 11 (1935), 351–353. [14] C. Kuratowski, Topologie. I. Espaces métrisables, espaces complets. 2d ed., Monografie Matematyczne 20, Warszawa, Wrocław 1948. [15] B. Lemmens and C. Walsh, Isometries of polyhedral Hilbert geometries. J. Topol. Anal. 3 (2011), 213–241. [16] B. Lemmens, Nonexpansive mappings on Hilbert’s metric spaces. Topol. Methods Nonlinear Anal. 38 (2011), 45–58 [17] B. Lemmens and R. Nussbaum, Nonlinear Perron–Frobenius theory. Cambridge Tracts Math. 189, Cambridge University Press, Cambridge 2012. [18] B. Lemmens and R. Nussbaum, Birkhoff’s version of Hilbert’s metric and its applications in analysis. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 275–303. [19] B. Lins and R. Nussbaum, Iterated linear maps on a cone and Denjoy–Wolff theorems. Linear Algebra Appl. 416 (2006), 615–626. [20] B. Lins, A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains. Math. Proc. Cambridge Philos. Soc. 143 (2007), 157–164. [21] B. Lins, and R. Nussbaum, Denjoy–Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators. J. Funct. Anal. 254 (2008), 2365–2386. [22] C. Liverani, Decay of correlations. Ann. of Math. 142 (1995), 239–301. [23] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 75 (1988), no. 391. [24] R. D. Nussbaum and S.M. Verduyn Lunel, Generalizations of the Perron–Frobenius theorem for nonlinear maps. Mem. Amer. Math. Soc. 138 (1999), no. 659. [25] R. D. Nussbaum, Fixed point theorems and Denjoy–Wolff theorems for Hilbert’s projective metric in infinite dimensions. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 199–249. [26] J.-F. Quint, Convexes divisibles (d’après Yves Benoist). In Séminaire Bourbaki, Volume 2008/2009. Exposés 997—1011, Astérisque 332 (2010), Exp. No. 999, vii, 45–73. [27] H. Samelson, On the Perron–Frobenius theorem. Michigan Math. J. 4 (1957), 57–59. [28] C. Walsh, The horofunction boundary of the Hilbert geometry. Adv. Geom. 8 (2008), 503–529. [29] C. Walsh, The horofunction boundary and isometry group of Hilbert geometry, In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 127–146.

Chapter 10

Birkhoff’s version of Hilbert’s metric and its applications in analysis Bas Lemmens and Roger Nussbaum

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . Birkhoff’s version of Hilbert’s metric . . . . . . . 2.1 Birkhoff’s contraction ratio . . . . . . . . . 2.2 An application to Perron–Frobenius operators 3 Special cones . . . . . . . . . . . . . . . . . . . . 3.1 Simplicial cones . . . . . . . . . . . . . . . 3.2 Polyhedral cones . . . . . . . . . . . . . . . 3.3 Symmetric cones . . . . . . . . . . . . . . . 4 Non-expansive mappings on Hilbert geometries . . 4.1 Periodic orbits . . . . . . . . . . . . . . . . 4.2 Denjoy–Wolff type theorems . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction In the nineteen-fifties Garrett Birkhoff [8] and Hans Samelson [64] independently discovered that one can use Hilbert’s metric and the contraction mapping principle to give an elegant proof of the existence of a positive eigenvector for a variety of linear mappings that leave a cone in a real vector space invariant. Their results can be seen as a direct generalization of Perron’s theorem [59], [60] concerning the existence and uniqueness of a positive eigenvector of square matrices with positive entries. In the past decades the ideas of Birkhoff and Samelson has been further developed by numerous authors. A partial list of contributors include, [12], [13], [19], [20], [23], [33], [35], [36], [37], [38], [42], [52], [62], [68], [73]. It has resulted in a remarkably detailed understanding of the eigenvalues, eigenvectors, and iterative behavior of a variety of linear, and nonlinear, mappings on cones. This body of work belongs to an area in mathematical analysis known as nonlinear Perron–Frobenius theory. A recent introductory account of this field is given in [42].

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Recall that Hilbert’s metric [25] is defined as follows. Let A be a real n-dimensional affine normed space, and denote the norm on the underlying vector space by k k. Consider a bounded, open, convex set A. For x; y 2 , let `xy denote the straight line through x and y in A, and denote the points of intersection of `xy and @ by x 0 and y 0 , where x is between x 0 and y, and y is between x and y 0 , as in Figure 1. ............................ .............. .... .......... ... . . . . . . . . . . ... . . . . . . . ... ..... . . . ... . . . 0 .. ... x .. x y ... .... ... ... . . .... ..... ... .. ....... .. ......... . . .......... ... ............ ....................................

y0

Figure 1. Hilbert’s cross-ratio metric.

On , Hilbert’s metric is defined by

kx 0 yk ky 0 xk ı.x; y/ D log kx 0 xk ky 0 yk

(1.1)

for all x ¤ y in , and ı.x; x/ D 0 for all x 2 . The metric space .; ı/ is called the Hilbert geometry on . In mathematical analysis one uses an alternative version of Hilbert’s metric, which is defined on a cone C in a real vector space V . The version we will use here was popularized by Bushell in [12], and for simplicity we shall refer to it as Birkhoff’s version of Hilbert’s metric. As we shall see, Birkhoff’s version ties together the partial ordering induced by C and the metric. This idea has proved to be very fruitful. The main objective of this survey is to discuss some of the applications of Birkhoff’s version of Hilbert’s metric in the analysis of nonlinear mappings on cones, and to illustrate some of the advantages of Birkhoff’s version by reproving several known geometric results for Hilbert geometries. Among other results, we shall see how Birkhoff’s version can be used to give a simple proof of the well-known fact that the Hilbert geometry on an open n-simplex is isometric to and n-dimensional normed space, whose unit ball is a polytope with n.n C 1/ facets. As many important applications of Hilbert’s metric in analysis are in infinitedimensional Banach spaces, we shall state some of the basic results in an infinitedimensional setting and occasionally point out the difference between the infinitedimensional and finite-dimensional cases. Acknowledgement. This work is partially supported by EPSRC grant EP/J008508/1 (B.L.) and by NSF grant is NSF DMS 1201328 (R.N.).

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2 Birkhoff’s version of Hilbert’s metric To define Birkhoff’s version of Hilbert’s metric we need to recall some elementary concepts from the theory of partially ordered vector spaces. Let V be a (possibly infinite-dimensional) real vector space. A subset C of V is called a cone if (C1) C is convex, (C2) C C for all 0, and (C3) C \ .C / D f0g. If C V satisfies (C1) and (C2) it is called a wedge. A cone C V induces a partial ordering, C , on V by x C y if y x 2 C: If C is merely a wedge, then C is only a pre-order, as it may fail to be anti-symmetric. In analysis one often considers a closed cone C in a Banach space .V; k k/. The cone C is called normal if there exists a constant > 0 such that kxk kyk whenever 0 C x C y. It is known, see Lemma 1.2.5 in [42], that every closed cone in a finite-dimensional normed space is normal, but a closed cone in an infinitedimensional Banach space may fail to be normal. Given a Banach space .V; k k/, we denote the dual space of continuous linear functionals on V by V . If C is a closed cone in a Banach space .V; k k/, the dual wedge is given by, C D f 2 V W .x/ 0 for all x 2 V g: In general C is only a wedge. However, if C is a total cone, i.e., the closure of C C D fx y W x; y 2 C g is V , then C is a cone. In particular, if C is a closed cone in a finite-dimensional vector space V and C has nonempty interior, then C is a closed cone in V with nonempty interior. For 2 C we let † D fx 2 C W .x/ D 1g. It is not hard to show, see Lemma 1.2.4 in [42], that if C is a closed cone with nonempty interior in a finitedimensional normed space, and in the interior of C , then † is a nonempty, compact, convex subset of V . However, in infinite dimensions it may happen that there does not exist 2 C such that † is bounded. A simple example is the Hilbert space, `2 , with the standard positive cone C D f.x1 ; x2 ; : : :/ 2 `2 W xi 0 for all i g. As mentioned earlier, Birkhoff’s version of Hilbert’s metric ties together the partial ordering C and the distance. To define it, let C be a cone in a vector space V . For x 2 V and y 2 C , we say that y dominates x if there exist ˛; ˇ 2 R such that ˛y C x C ˇy. In that case, we write M.x=y/ D inffˇ 2 R W x C ˇyg and m.x=y/ D supf˛ 2 R W ˛y C xg: Obviously, if x; y 2 V are such that y D 0 and y dominates x, then x D 0, as C is a cone. On the other hand, if y 2 C n f0g and y dominates x, then M.x=y/ m.x=y/.

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Using the domination relation one obtains an equivalence relation on C by x C y if y dominates x and x dominates y. The equivalence classes are called the parts of C . Obviously f0g is a part of C . Moreover, if C is a closed cone with nonempty interior, C B , in a Banach space, then C B is a part of C . The parts of a finite-dimensional cone are related to the faces of C . Indeed, if C is a finite-dimensional closed cone, then it can be shown that the parts correspond to the relative interiors of the faces of C , see [42], Lemma 1.2.2. Recall that a face of a convex set S V is a subset F of S with the property that if x; y 2 S and x C .1 /y 2 F for some 0 < < 1, then x; y 2 F . The relative interior of a face F is the interior of F in its affine span. It is easy to verify that if x; y 2 C n f0g, then x C y if, and only if, there exist 0 < ˛ ˇ such that ˛y C x C ˇy. Furthermore, m.x=y/ D supf˛ > 0 W y C ˛ 1 xg D M.y=x/1 :

(2.1)

Birkhoff’s version of Hilbert’s metric on C can now be defined as follows: M.x=y/ d.x; y/ D log m.x=y/ for all x C y with y ¤ 0, d.0; 0/ D 0, and d.x; y/ D 1 otherwise. If C is a closed cone in a Banach space, then d is a genuine metric on the set of rays in each part of the cone as the following lemma shows. Lemma 2.1. If C is a cone in V , then for each x C y C z with y ¤ 0, (i) d.x; y/ 0, (ii) d.x; y/ D d.y; x/, (iii) d.x; z/ d.x; y/ C d.y; z/, (iv) d.x; y/ D d.x; y/ for all ; > 0. Moreover, if C is a closed cone in a Banach space .V; k k/, then d.x; y/ D 0 if, and only if, x D y for some 0. Proof. To prove the first assertion we note that for each 0 < ˛ < m.x=y/ and 0 < M.x=y/ < ˇ we have ˛y C x C ˇy: It follows that y C .ˇ=˛/y, and hence ˇ=˛ 1. Thus M.x=y/=m.x=y/ 1 and hence d is nonnegative. Furthermore, note that by (2.1), d.x; y/ D log M.x=y/M.y=x/ D d.y; x/; which shows that d is symmetric. To show that d satisfies the triangle inequality, we note that for each 0 < ˛ < m.x=y/ and 0 < < m.y=z/ we have ˛y C x and z C y, and hence ˛z C x. This implies that m.x=z/ m.x=y/m.y=z/. In the same way it can be shown that M.x=z/ M.x=y/M.y=z/. Thus, M.x=y/M.y=z/ M.x=z/ ; m.x=z/ m.x=y/m.y=z/

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which proves (iii). For ; > 0, it is easy to verify that M.x=y/ D

M.x=y/

m.x=y/ D

and

m.x=y/;

which gives the fourth assertion. Finally, assume that C is a closed cone in .V; kk/ and x C y with y ¤ 0. As C is closed, m.x=y/y C x C M.x=y/y. If d.x; y/ D 0, then M.x=y/m.x=y/1 D 1, so we get that y C m.x=y/1 x C y from which we deduce that x D m.x=y/y. On the other hand, if x D y for some > 0, then d.x; y/ D 0 by the previous assertion. To understand the relation with Hilbert’s cross-ratio metric, ı, given in (1.1), we consider a cone C in an .n C 1/-dimensional real normed space .V; k k/ and we assume that the interior of C is nonempty. Let H V be an n-dimensional affine hyperplane such that C WD H \ C B is a (relatively) open, bounded, convex set in H . Theorem 2.2. The restriction of d to C coincides with ı. Proof. Consider x ¤ y in C . Let ˛ D m.x=y/ D M.y=x/1 and ˇ D M.x=y/. As C is closed, ˛y C x and x C ˇy. Write u D x ˛y 2 @C and w D y x=ˇ 2 @C . Let `xy denote the straight line through x and y and denote the points of intersection with @C by x 0 and y 0 , as in Figure 2. We know that there exist ; > 1 x0 x

y0

y

u w

Figure 2. Theorem 2.2.

such that

x 0 D y C .x y/

y 0 D x C .y x/:

and

Let be a linear functional on V such that H D fz 2 V W .z/ D 1g. So, u x ˛y y C .x y/ D x 0 D D ; .u/ 1˛ which implies that ˛ D . 1/= . Similarly, x C .y x/ D y 0 D

y x=ˇ w D ; .w/ 1 1=ˇ

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which gives ˇ D =.1 /. Thus, 1 ky x 0 k D D D M.y=x/ kx x 0 k 1 ˛ and

(2.2)

kx y 0 k D D ˇ D M.x=y/; ky y 0 k 1

which shows that d.x; y/ D ı.x; y/ on C . Remark 2.3. From equation (2.2) it follows that log

kx y 0 k D log M.x=y/ ky y 0 k

for all x ¤ y in C . The function FC W C B C B ! R given by FC .x; y/ D log M.x=y/ is called the Funk (weak) metric on C B , see [58], [70]. Remark 2.4. Birkhoff’s version allows one to give a natural definition of Hilbert’s metric on any open, convex, possibly unbounded, subset of an infinite-dimensional Banach space. Indeed, let .V; k k/ be an open, convex set and assume that contains no straight lines, i.e., for each v 2 there exists no u 2 V n f0g such that v C t u 2 for all t 2 R. Let W D R V with norm k.s; v/kW D jsj C kvk for all .s; v/ 2 W . Consider the following set in W : C D f.s; sv/ 2 W W s > 0 and v 2 g: It is easy to verify that C is open, and C C for all > 0. Furthermore its closure, C , is a cone in W . Indeed, suppose that w D .s; u/ 2 C and w 2 C . As s 0, we know that s D 0. Now let wk D .sk ; sk vk / 2 C be such that wk ! w and vk 2 for all k. Then for each t > 0 fixed and k sufficiently large, 0 sk t 1. Thus, for each v 2 and k large, we have .1 sk t /v C sk t vk 2 : Note that k.1 sk t /v C sk t vk .v C t u/k sk t kvk C t ksk vk uk ! 0 as k ! 1, x But for > 0 small we have v C u 2 , as is open. As t was so that v C t u 2 . arbitrary and is convex, we deduce from Proposition 8.5 of [66] that v C t u 2 for all t 0. In the same way we can use the assumption that w 2 C to show that v C t u 2 for all t 0. This, however, contradicts the hypothesis that contains no lines. The restriction of d to f.1; v/ 2 W W v 2 g is a genuine metric, which provides a natural definition of Hilbert’s metric on . Remark 2.5. From an analyst’s point of view Hilbert’s metric on cones has one disadvantage – namely, it is only a metric on the pairs of rays in the cone rather than on the pairs of points in the cone. In applications in analysis one therefore often also

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considers the following variant, which was introduced by Thompson [68]. Given a closed cone C in .V; k k/, Thompson’s metric is defined by ˚ dT .x; y/ D max log M.x=y/; log M.y=x/ for x C y and y ¤ 0, dT .0; 0/ D 0, and dT .x; y/ D 1 otherwise. The reader can verify that dT is a genuine metric on each part of C . It is known, see [68], that if C is a closed normal cone in a Banach space .V; k k/ and P C is a part of C , then .P; dT / is a complete metric space whose topology coincides with the norm topology of the underlying space. Furthermore, if q W C n f0g ! .0; 1/ is a continuous homogeneous (degree 1) function and †q D fx 2 C W q.x/ D 1g, then the metric space .†q \ P; d / is also a complete metric space and its topology coincides with the norm topology. Particularly, interesting examples of homogeneous functions q include q 2 C n f0g and q W x 7! kxk.

2.1 Birkhoff’s contraction ratio The usefulness of Hilbert’s metric lies in the fact that linear, but also certain nonlinear, mappings between cones are non-expansive with respect to this metric. Recall that a mapping f from a metric space .X; dX / into a metric space .Y; dY / is non-expansive if dY .f .x/; f .y// dX .x; y/ for all x; y 2 X: It is said to be a Lipschitz contraction with constant 0 c < 1, if dY .f .x/; f .y// cdX .x; y/ for all x; y 2 X: Note that if C is a cone in V , K is a cone in W , and L W V ! W is a linear mapping, then L.C / K is equivalent to Lx K Ly whenever x C y. A mapping f W C ! K is called order-preserving if x C y implies f .x/ K f .y/. It is said to be homogeneous of degree r if f .x/ D r f .x/ for all x 2 C and > 0. The following result is elementary, but very useful. Proposition 2.6. Let C V and K W be cones. If f W C ! W is orderpreserving and homogeneous of degree r > 0, then M.f .x/=f .y// M.x=y/r and m.x=y/r m.f .x/=f .y// for all x; y 2 C with x C y. Proof. As the proofs of the two inequalities are very similar, we shall only show the first one. If x; y 2 C and x C y, then for each ˇ > 0 with x C ˇy we have that f .x/ K ˇ r f .y/. So, M.f .x/=f .y// ˇ r for all ˇ > M.x=y/, which gives M.f .x/=f .y// M.x=y/r . As an immediate consequence we obtain the following result.

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Corollary 2.7. Let C V and K W be cones. If f W C ! K is order-preserving and homogeneous (of degree 1), then f is non-expansive with respect to d on each part of C . Remark 2.8. It is interesting to note that every order-preserving homogeneous mapping f is also non-expansive with respect to Thompson’s metric. In fact, it can be shown (see [42], Lemma 2.1.7) that an order-preserving mapping f W C ! K is non-expansive under Thompson’s metric if, and only if, f is subhomogeneous, i.e., f .x/ K f .x/ for all x 2 C and 0 1. We should also remark that it is also easy to show that every order-reversing, homogeneous degree 1 mapping is non-expansive under d , see [42], Corollary 2.1.5. From Corollary 2.7 we see that every linear mapping L W V ! W with L.C / K is non-expansive with respect to d on each part of C . Furthermore, if L is invertible and L.C / D K, then L must be an isometry. In fact, in that case we have M.Lx=Ly/ D M.x=y/

and

m.Lx=Ly/ D m.x=y/

for all x C y in C . The non-expansiveness of linear mappings, L W C ! C , on cones provides a way to analyze the eigenvalue problem, Lx D x, on C using contraction mapping arguments. Given cones C V , K W and a linear mapping L W V ! W with L.C / K, the Birkhoff contraction ratio of L is defined by .L/ D inffc 0 W d.Lx; Ly/ cd.x; y/ for all x C y in C g: Moreover, the projective diameter of L is given by .L/ D supfd.Lx; Ly/ W x; y 2 C with Lx K Lyg: Theorem 2.9 (Birkhoff). Let C be a cone in a vector space V and K be a cone in a vector space W . If L W V ! W is a linear mapping with L.C / K, then

1 .L/ ; .L/ D tanh 4 where tanh.1/ D 1. So, if .L/ < 1, then L is a Lipschitz contraction on each part of C , with contraction constant tanh..L/=4/ < 1. In numerous cases one can prove that .L/ < 1. Indeed, Birkhoff showed this for matrices with positive entries and for certain integral operators with positive kernels. In case the linear map L is given by a positive m n matrix A D .aij /, so aij > 0 for all i and j , there exists the following well-known explicit formula, see Appendix A in [42]: api aqj < 1; .A/ D max d.Aei ; Aej / D log max i;j i;j;p;q apj aqi where e1 ; : : : ; en denote the standard basis vectors in Rn .

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Remark 2.10. In [27], [28] Hopf proved a result closely related to Birkhoff’s Theorem 2.9. Hopf was apparently unaware of Birkhoff’s work and did not use Hilbert’s metric. The results of Birkhoff and Hopf have been extended and their connections have been unraveled in numerous papers including, [2], [12], [13], [19], [20]. A detailed exposition of these works can be found in [42], Appendix A. Birkhoff’s Theorem 2.9 can be used to prove the existence and uniqueness of positive eigenvectors of continuous linear mappings that leave a closed, normal cone in a Banach space invariant. In fact, one has the following slightly more general set-up. A mapping L from a cone C V into a cone K W is said to be cone linear, if L.˛x C ˇy/ D ˛Lx C ˇLy for all ˛; ˇ 0 and x; y 2 C . There exist examples, see [10], of cone linear mappings L from a closed cone C in a Banach space V into a closed cone K in a Banach space W such that L is continuous on C , but its linear extension to cl.C C / is not continuous, even if cl.C C / D V . For cone linear mappings there exists the following result, see Theorem A.7.1 in [42]. Theorem 2.11. Let C be a closed normal cone in a Banach space V , and let L W C ! C be a cone linear mapping, which is continuous at 0. If there exists an integer p 1 such that .Lp / < 1 and LpC1 .C / ¤ f0g, then L has a unique eigenvector v 2 C , with kvk D 1, such that Lv D rC .L/v, where rC .f / D lim kLk k1=k C k!1

is the cone spectral radius of L and kLk kC D supfkLk xk W x 2 C with kxk D 1g: Moreover, if we let c D tanh..Lp /=4/ < 1, then d.Lkp x; v/ c k d.x; v/ for all x 2 C n f0g:

2.2 An application to Perron–Frobenius operators In this section we briefly discuss an application of Birkhoff’s contraction ratio, or, more precisely, Theorem 2.11, to so-called Perron–Frobenius operators. These operators play a central role in the study of Hausdorff dimensions of invariant sets given by graph directed iterated function systems, see [55], [57]. For simplicity we restrict attention here to operators arising from iterated function systems rather than the more general case of graph directed iterated function systems. However, the same ideas can be applied in the graph directed case. Let .S; / be a bounded, complete metric space with positive diameter. Let Cb .S / denote the Banach space of bounded, continuous, real-valued functions, f W S ! R, with kf k D sups2S jf .s/j. Write D diam.S /, so 0 < < 1. For 0 < 1 and M > 0 define

K.M; / D ff 2 Cb .S/ W f .s/ f .t /e M.s;t / for all s; t 2 Sg:

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It can be shown, see [57], Lemma 3.2, that K.M; / is a closed normal cone in Cb .S /. By using the assumption that > 0, it is easy to verify that if f 2 K.M; /, then f .s/ 0 for all s 2 S, and f .s/ D 0 for all s 2 S whenever f .t / D 0 for some t 2 S . Furthermore for f 2 K.M; / we have sup f .s/ e M inf f .s/ : s2S

s2S

Now fix M0 > 0 and 0 < 1 and let bi 2 K.M0 ; / for i 2 N. Assume that

i W S ! S are Lipschitz mappings for i 2 N, with ²

³

. i .s/; i .t // Lip. i / D sup W s ¤ t in S c < 1: .s; t /

(2.3)

Assume, in addition, that there exists s 2 S such that X bi .s / < 1: i

Now we define a Perron–Frobenius operator , L W Cb .S / ! Cb .S /, by X .Lf /.t/ D bi .t/f . i .t // for t 2 S:

(2.4)

i

Obviously, kLf k D sup j t2S

X

bi .t/f . i .t//j

i

X

bi .s /e M0 kf k < 1

i

for all f 2 Cb .S/, and hence L is a continuous linear mapping from Cb .S / to Cb .S /. We shall now see how we can use Birkhoff’s Theorem 2.9 to prove that L has a unique eigenvector, v 2 K.M2 ; /, with kvk D 1, for all M2 > M0 =.1 c /. Here c is given in (2.3). Furthermore, for each g 2 K.M2 ; / n f0g,

Lk g v D 0: lim k k!1 kL gk

We will need the following lemma. Lemma 2.12. Let d2 denote the Hilbert metric on K.M2 ; /. If 0 < M1 < M2 , then K.M1 ; / n f0g is a part of K.M2 ; / and supfd2 .f; g/ W f; g 2 K.M1 ; / n f0gg < 1: Proof. Let 2 denote the partial ordering induced by K.M2 ; / on Cb .S /. We need to show that there exists 0 < ˛ ˇ such that ˛f 2 g 2 ˇf

for all f; g 2 K.M1 ; /:

(2.5)

As d2 .f; g/ D d2 .f; g/ for all ; > 0, we may as well assume that kf k D kgk D 1. Furthermore, by interchanging the roles of f and g in (2.5), we see that it suffices

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to prove that there exists ˛ > 0 such that ˛f 2 g for all f; g 2 K.M1 ; / with kf k D kgk D 1. Recall that ˛f 2 g is equivalent to

g.s/ ˛f .s/ .g.t/ ˛f .t//e M2 .s;t /

for all s; t 2 S:

(2.6)

Since f; g 2 K.M1 ; /, we know that

g.s/ g.t /e M1 .s;t/

and

f .s/ f .t /e M1 .s;t /

and

e M1 f .t / 1;

(2.7)

for all s; t 2 S. This implies that

e M1 g.t / 1

for all t 2 S, from which we deduce that

e M1

g.t/ e M1 f .t /

for all t 2 S:

For convenience we shall assume that

˛ < e M1 ; which ensures that g.t/ ˛f .t/ > 0 for all t 2 S . This allows us to rewrite (2.6) as g.s/ ˛f .s/ e M2 .s;t / g.t / ˛f .t/

for all s; t 2 S:

(2.8)

Now using (2.7) we see that for s; t 2 S,

g.t /e M1 .s;t / ˛f .t /e M1 .s;t / g.s/ ˛f .s/ g.t/ ˛f .t/ g.t / ˛f .t /

.g.t/=f .t//e M1 .s;t / ˛e M1 .s;t / : .g.t /=f .t // ˛

This implies for s; t 2 S that

˛.e M1 .s;t / e M1 .s;t / / g.s/ ˛f .s/ e M1 .s;t/ C g.t/ ˛f .t/ e M1 .s;t / ˛

(2.9)

Write r D M2 .s; t / , so r 0 and M1 .s; t / D r, where D M1 =M2 < 1. From (2.9) we see that (2.8) is satisfied if 0 < ˛ < e M1 , can be chosen so that

˛ e M1 ˛

e r e r e r e r

1:

(2.10)

Consider the function W Œ0; 1/ ! R given by

.r/ D

e r e r for r > 0 e r e r

and

.0/ D

2 : 1

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Observe that is continuous on Œ0; 1/ and limr!1 .r/ D 0. A simple calculation gives .1 /e .C1/r 2 C . C 1/e .1/r

0 .r/ D : .e r e r /2 By a power series expansion, for r > 0 we have .1 /e .C1/r 2 C . C 1/e .1/r 1 1 X X . C 1/j 1 r j .1 /j 1 r j 2 C < 0: D .1 / jŠ jŠ j D2

j D2

Thus, achieves its maximum on Œ0; 1/ at 0. Let D .M2 M1 /=.M2 C M1 / and assume that ˛ satisfies 0 < ˛ e M1 . By the previous remarks equation (2.10) will be satisfied if 2 1: 1 1 However, a calculation shows that the left-hand-side of the above expression actually equals 1, so that we can take ˛ D e M1 , and the proof is complete. Note that we have actually shown that for each f; g 2 K.M1 ; / with kf k D kgk D 1, we can choose

M2 M1 M1 e ˛D M2 C M1 This implies that d2 .f; g/ 2 log

and

M2 C M1 M1 ˇD e : M2 M1

M2 C M1 C 2e M1 : M2 M 1

for all f; g 2 K.M1 ; /. We shall also need the following result. Lemma 2.13. Let L W Cb .S/ ! Cb .S/ be given by (2.4), and assume that at least one bi is not identically 0. If M2 > M0 =.1 c / and M1 D M0 C c M2 , then L.K.M2 ; / n f0g/ K.M1 ; / n f0g: Proof. If f 2 K.M2 ; / n f0g, then f B i 2 K.c M2 ; /, because

f . i .s// f . i .t//e M2 .i .s/;i .t // f . i .t //e c

M .s;t / 2

for all s; t 2 S. As bi 2 K.M0 ; /, we have

bi .s/ bi .t/e M0 .s;t /

for all s; t 2 S:

Combining these two inequalities gives bi .s/f . i .s// bi .t/f . i .t //e .M0 Cc

M /.s;t / 2

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for all s; t 2 S , which shows that bi ./f . i .// 2 K.M1 ; / for all i 2 N. Since K.M1 ; / is a closed cone, we find that L.f / 2 K.M1 ; /. It remains to show that L.f / ¤ 0 for f 2 K.M2 ; / n f0g. Note that as f ¤ 0 and f 2 K.M2 ; /, we know that f .s/ > 0 for all s 2 S. As each bi 2 K.M0 ; /, bi .s/ 0 for all s 2 S. Furthermore, there exists j such that bj ¤ 0, and hence bi .t / > 0 for all t 2 S . Thus, L.f /.t/ bj .t /f . j .t // > 0 for all t 2 S , which completes the proof. We can now use Birkhoff’s Theorem 2.9 to prove the following result. Theorem 2.14. Let L W Cb .S/ ! Cb .S/ be given by (2.4), and assume that at least one bi is not identically 0. If M2 > M0 =.1 c /, then there exists a unique v 2 K.M2 ; / with kvk D 1, such that Lv D kLvkv. Furthermore, for each g 2 K.M2 ; / n f0g we have k L g v D 0: (2.11) lim k k!1 kL gk Proof. Let 2 denote the partial ordering from K.M2 ; / and denote the restriction of L to K.M2 ; / by L0 . Note that it follows from Lemmas 2.12 and 2.13 that the projective diameter .L0 / < 1. Now let g 2 K.M2 ; / n f0g. It follows from Lemma 2.13 that L0 g 2 K.M1 ; / and L0 g ¤ 0. As K.M1 ; / K.M2 ; / the same applies to L0 g. So, L20 g 2 K.M1 ; / n f0g, and hence L20 .K.M2 ; // ¤ f0g. It now follows from Theorem 2.11 that there exists a unique v 2 K.M2 ; /, with kvk D 1, such that Lv D kLvkv, and for each g 2 K.M2 ; / n f0g, Equation (2.11) holds. Remark 2.15. As Theorem 2.14 holds for all M2 > M0 =.1 c /, the unique eigenvector v in Theorem 2.14 belongs to K.M0 =.1 c /; /. It can be shown that the elements of K.M; / are Hölder continuous functions with Hölder exponent , and hence v is Hölder with Hölder exponent .

3 Special cones Particularly important examples of finite-dimensional cones arising in analysis are the standard positive cone, RnC D fx 2 Rn W xi 0 for i D 1; : : : ; ng and the Lorentz cone, ƒnC1 D f.s; x/ 2 R Rn W s 2 x12 C C xn2 and s 0g: Other interesting examples are cones of linear operators, such as the cone …n .R/ consisting of positive semi-definite n n matrices in the real vector space Symn of all

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n n symmetric matrices. The cones mentioned above are all examples of so-called symmetric cones. Recall that a cone C , with nonempty interior, in a finite-dimensional inner product space .V; h j i/ is called self-dual if C D C . Furthermore C is called homogeneous if Aut.C / D fA 2 GL.V / W A.C / D C g acts transitively on C B . The interior of a self-dual homogeneous cone is called a symmetric cone. It is well known that the symmetric cones in finite dimensions are in one-toone correspondence with the interiors of the cones of squares of Euclidean Jordan algebras. This fundamental result was proved by Koecher [32] and Vinberg [69]. A detailed exposition of the theory of symmetric cones can be found in [21]. Recall that a Euclidean Jordan algebra is a finite-dimensional real inner-product space .V; h j i/ equipped with a bilinear product .x; y/ 7! xy from V V into V such that for each x; y 2 V , (J1) xy D yx, (J2) x.x 2 y/ D x 2 .xy/, (J3) for each x 2 V , the linear mapping L.x/ W V ! V defined by L.x/y D xy satisfies hL.x/y j zi D hy j L.x/zi for all y; z 2 V: Note that a Euclidean Jordan algebra is commutative, but in general not associative. For example, Symn can be equipped with the usual inner product hA j Bi D tr.BA/ and (Jordan) product A B B D .AB C BA/=2. In this case the interior of the cone of squares in Symn is precisely …n .R/B , defined as …n .R/B D fA 2 Symn W A is positive definiteg D fA2 W A 2 Symn gB : When studying a Hilbert geometry .; ı/, it is often useful to view as the (relative) interior of the intersection of a cone in a vector space and a hyperplane H , and use Birkhoff’s version of Hilbert’s metric. In many interesting cases it gives an alternative formula to compute the distance and provides additional tools to analyze the Hilbert geometry. To illustrate this we need to recall some basic concepts. Let C be a closed cone with nonempty interior in a finite-dimensional vector space V . For u 2 C B define †u D f 2 C W .u/ D 1g, which is a compact convex set in V . Let Eu denote the closure of the set of extreme points of †u . Recall that 2 †u is an extreme point if there exist no ; 2 †u such that D C .1 / for some 0 < < 1. A basic result in convex analysis, due to Minkowski, says that each compact convex set in a finite-dimensional vector space is the closed convex hull of its extreme points, see [63], Corollary 13.5. This result, and its infinite-dimensional extension, is usually called the Krein–Milman theorem. It is easy to show that if C is a closed cone with nonempty interior and u 2 C B , then for x; y 2 V we have x C y if, and only if, .x/ .y/ for all 2 Eu . Indeed, it follows from the Krein–Milman theorem that .x/ .y/ for all 2 C is equivalent to .x/ .y/ for all 2 Eu . Now if x —C y, then y x 62 C . So, by the Hahn–Banach separation theorem, there exist ˛ 2 R and 2 V such that

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.y x/ < ˛ and .z/ > ˛ for all z 2 C . As z 2 C for all 0 and z 2 C , we see that .z/ 0 for all z 2 C , and hence 2 C . Obviously, .0/ D 0, and hence ˛ < 0. Thus, .y/ < .x/. On the other hand, x C y implies .x/ .y/ for all 2 C . So, we have the following result. Lemma 3.1. If C is a closed cone with nonempty interior in a finite-dimensional vector space, and u 2 C B , then for x 2 V and y 2 C B we have M.x=y/ D max

2Eu

.x/ .y/

and m.x=y/ D min

2Eu

.x/ : .y/

(3.1)

Remark 3.2. The identities in (3.1) also hold for closed cones in infinite-dimensional topological vector spaces. Indeed, let C be a closed cone, with nonempty interior in a topological vector space V . Given u 2 C B , one can define the order unit norm, k ku , by kxku D inff˛ > 0 W ˛u C x C ˛ug for all x 2 V . With respect to this norm each linear functional, W V ! R, with .V / Œ0; 1/, is continuous, because j.x/j .u/ for all x 2 V with kxku 1. Now let C D f W V ! R j .V / Œ0; 1/g be the dual of C in .V; k ku / and let †u D f 2 C W .u/ D 1g. Note that †u is a closed subset of the unit ball, B , of .V; k ku / . So, †u is a weak- compact set, as B is weak- compact by the Banach–Alaoglu theorem. If we now let Eu be the weak- closure of the set of extreme points of †u , then Eu is weak- compact and the equalities in (3.1) hold.

3.1 Simplicial cones A cone C in an n-dimensional vector space V is called a simplicial cone if there exist v1 ; : : : ; vn 2 V , linearly independent, such that ˚ Pn C D iD1 i vi W i 0 for all i : A basic example is the standard positive cone RnC . We can apply Lemma 3.1 to derive an explicit formula for Hilbert’s metric on .RnC /B , and use this formula to give a simple proof of the well-known fact that the Hilbert geometry on the open standard Pn x D 1g, is isometric to .n 1/-dimensional simplex, Bn1 D fx 2 .RnC /B W i iD1 an .n 1/-dimensional normed space, see [17], [52], [61]. Indeed, Lemma 3.1, or a direct simple argument, gives M.x=y/ D max xi =yi

and

i

m.x=y/ D min xj =yj j

.RnC /B .

for all x:y 2 Now consider the mapping Log W .RnC /B ! Rn , given by Log.x/ D .log x1 ; : : : ; log xn / for x 2 .RnC /B , and equip Rn with the variation norm, kwkvar D max wi min wj : i

j

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Note that log M.x=y/ D log maxi xi =yi D maxi log xi log yi . Likewise, we see that log m.x=y/ D minj log xj log yj . Thus, the mapping, x 7! Log.x/, is an isometry from ..RnC /B ; d / onto .Rn ; k kvar /. If we let H D fx 2 .RnC /B W xn D 1g, then .H; d / is a genuine metric space, which is isometric to .Bn1 ; d / by Lemma 2.1 (iv). Clearly Log.H / D fx 2 Rn W xn D 0g, which we can identify with Rn1 by projecting out the last coordinate. It follows that .Bn1 ; d / is isometric to the .n 1/-dimensional normed space .Rn1 ; k kH /, where kxkH D maxfx1 ; : : : ; xn1 ; 0g minfx1 ; : : : ; xn1 ; 0g: If we now use Theorem 2.2, we arrive at the following result. Theorem 3.3. The Hilbert geometry .Bn ; ı/ is isometric to .Rn ; k kH /. The unit ball of kkH in Rn is a polytope with n.nC1/ facets. In fact, it is a hexagon, when n D 2, which was already known to Phadke [61], and a rhombic-dodecahedron, when n D 3. If D convfv1 ; : : : ; vnC1 g is a simplex with nonempty interior in an n-dimensional vector space V , then C D f.v; / 2 V R W v 2 and 0g is a simplicial cone 0 D .vnC1 ; 1/ is a basis for V 0 and in V 0 DPV R. Note that v10 D .v1 ; 1/; : : : ; vnC1 0 C D f i i vi W i 0 for all ig. Moreover, the linear mapping A W V 0 ! RnC1 given by X X i vi0 D i ei ; A i

i

where 1 ; : : : ; nC1 2 R and e1 ; : : : ; enC1 are the standard basis vectors in RnC1 , B is invertible and maps C onto RnC1 C . So, A is an isometry from .C ; d / onto nC1 B ..RC / ; d /. By combining this observation with Theorem 3.3 it is easy to show that the Hilbert geometry on B is isometric to .Rn ; k kH /. In [22] Foertsch and Karlsson showed that the only Hilbert geometry isometric to a normed space is the one on an open simplex. See [11] for an early version. Thus, the following result holds. Theorem 3.4. An n-dimensional Hilbert geometry is isometric to a normed space if and only if its domain is an open n-dimensional simplex. In that case it is isometric to .Rn ; k kH /.

3.2 Polyhedral cones In this subsection we shall see how we can use Birkhoff’s version of Hilbert’s metric on polyhedral cones to show that the Hilbert geometry on the interior of a polytope with m facets can be isometrically embedded into the normed space .Rm.m1/=2 ; kk1 /, where kzk1 D maxi jzi j is the supremum norm. It turns out the polytopal Hilbert geometries are the only ones that can be isometrically embedded into a finite-dimensional normed space, see [6], [15], [16], [21]. To prove these results we need to recall some basic concepts concerning polyhedral cones.

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Recall that a closed cone C in Rn is called a polyhedral cone if it is the intersection of finitely many closed half-spaces, i.e., if there exist finitely many linear functionals 1 ; : : : ; k such that C D fx 2 Rn W i .x/ 0 for all i g. A subset F of a polyhedral cone C is called an (exposed) face if there exists a linear functional 2 C such that F D fx 2 C W .x/ D 0g. A face F of C is called a facet if dim F D dim C 1. Furthermore, it is a well-known fact from polyhedral geometry, that if C is a polyhedral cone in Rn with m facets and C B is nonempty, then there exist m linear functional 1 ; : : : ; m such that C D fx 2 Rn W

i .x/

0 for all i g:

and each i defines a facet of C . The facet defining functionals, 1 ; : : : ; m , of a polyhedral cone C correspond to the extreme rays (1-dimensional faces) of the dual cone C of C . If we take u 2 C B , then we can normalize each facet defining functional i so that i .u/ D 1. In that case we have Eu D f 1 ; : : : ; m g. Now using Lemma 3.1 we can show the following result regarding isometric embeddings from [53]. Theorem 3.5. If P is an n-dimensional polytope in Rn with m facets, then the Hilbert geometry on P B can be isometrically embedded into .Rm.m1/=2 ; k k1 /. Proof. Let C D f.x; / 2 Rn R W x 2 P and 0g. Then C is a polyhedral cone in RnC1 D Rn R with nonempty interior and m facets. Denote the facet defining functionals of C by 1 ; : : : ; m . Furthermore, let †B D f.x; s/ 2 C B W s D 1g. So, by Theorem 2.2 the Hilbert geometry .P B ; ı/ is isometric to .†B ; d /. Now define ‰ W †B ! Rm.m1/=2 by ‰ij .x/ D log

i .x/ j .x/

for all 1 i < j m and x 2 †B :

It follows from Lemma 3.1 that M.x=y/ D maxi i .x/= i .y/ for all x; y 2 †B . So, d.x; y/ D log M.x=y/M.y=x/ i .x/ j .y/ D log max i;j i .y/ j .x/ i .x/ i .y/ log D max log i;j j .x/ j .y/ ˇ ˇ ˇ i .x/ i .y/ ˇˇ ˇ log D max ˇ log ˇ; 1i<j m j .x/ j .y/ which shows that d.x; y/ D k‰.x/ ‰.y/k1 for all x; y 2 †B . Thus, ‰ is an isometry of .†B ; d / into .Rm.m1/=2 ; k k1 /. The following result was essentially proved by Foertsch and Karlsson [21]. Theorem 3.6. A Hilbert geometry embeds isometrically into a finite-dimensional normed space if and only if its domain is the interior of a polytope.

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Proof. By the previous theorem it remains to show that a non-polytopal Hilbert geometry cannot be isometrically embedded into any finite-dimensional normed space. For the sake of contradiction, assume that .; ı/ is a non-polytopal Hilbert geometry and that h is an isometry from .; ı/ into a finite-dimensional normed space .V; k k/. The following result by Karlsson and Noskov [31] concerning the behavior of the Gromov product, 2.x j y/p D ı.x; p/ C ı.y; p/ ı.x; y/; will be useful. Suppose that p 2 and .xk / and .yk / are sequences in such that xk ! x 2 @, yk ! y 2 @, and the straight line segment Œx; y 6 @, then there exists a constant R > 0 such that lim sup .xk j yk /p R: k!1

This result will be used to prove the following claim, from which the contradiction will be derived. Claim. If there exist 1 ; : : : ; m 2 @ such that the straight line segment Œi ; j 6 @ for all i ¤ j , then there exist v1 ; : : : ; vm 2 V with kvi k D 1 for all i and kvi vj k 2 for all i ¤ j . To prove the claim let p 2 be fixed. Obviously the mapping x 7! h.x/ h.p/ is also an isometric embedding of .; ı/ into .V; k k/. So, we may as well assume that h.p/ D 0. Now for i D 1; : : : ; m and 0 t < 1 let zi .t / D .1 t /p C t i . Note that the mapping t 7! ı.p; zi .t// is continuous and lim t !1 ı.p; zi .t // D 1. So, for each i and n 2 N there exists 0 < ti;n < 1 such that ı.p; zi .ti;n // D n. For simplicity we write zin D zi .ti;n /. From the result by Karlsson and Noskov it follows that there exists a constant M > 0 such that ı.zin ; zjn / 2n M for all i ¤ j and n N0 , where N0 is a sufficiently large integer. Define uni D n1 h.zin /. Then for each i ¤ j and n N0 we have M 1 1 kuni ujn k D ı.zin ; zjn / .2n M / D 2 : n n n But also kuni k D n1 kh.zin / h.p/k D n1 ı.p; zin / D 1. As the unit sphere in V is n compact, we can find a subsequence .ui k / converging to some vi 2 V for each i . Clearly the limits v1 ; : : : ; vm satisfy kvi k D 1 for all i , and kvi vj k 2 for all i ¤ j , which completes the proof of the claim. To obtain a contradiction we will now show that there exist infinitely many points 1 ; 2 ; : : : 2 @ such that Œi ; j 6 @ for all i ¤ j , which, by the claim, violates the compactness of the unit sphere in V . Without loss of generality we may assume that is contained in a finite-dimensional x of is not a polytope, the polar of , x vector space W and 0 2 . As the closure, , x x D f 2 W W .x/ 1 for all x 2 g;

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x has infinitely many extreme points. As the exposed is also not a polytope. Thus, points are dense in the extreme points, see [67], there exist infinitely many exposed x . Now using the fact that x D x we can find 1 ; 2 ; : : : 2 points 1 ; 2 ; : : : 2 @ x n fi g. Clearly, @ such that i .i / D 1 for all i , and .i / < 1 for all 2 if i ¤ j and 0 < t < 1, then .1 t /i C t j 2 , as ..1 t /i C t j / D x and 0 < t < 1. Thus, the segment .1 t /.i / C t .j / < 1 for all 2 Œi ; j 6 @ for all i ¤ j , which completes the proof. Theorem 3.6 has been strengthened by Colbois andVerovic [15] to quasi-isometries, and by Bernig [6] and Colbois, Vernicos and Verovic [16] to bi-Lipschitz mappings.

3.3 Symmetric cones Let C B be a symmetric cone in .V; h j i/, and let e 2 C B denote the unit in the associated Euclidean Jordan algebra on V . An element c 2 V is called an idempotent if c 2 D c. It is said to be a primitive idempotent if c cannot be written as the sum of two non-zero idempotents. The set of all primitive idempotents in V is denoted by J.V /. A set fc1 ; : : : ; ck g is called a complete system of orthogonal idempotents if (1) ci2 D ci for all i, (2) ci cj D 0 for all i ¤ j , (3) c1 C C ck D e. The spectral theorem ([21], Theorem III.1.1) says that for each x 2 V there exist unique real numbers 1 ; : : : ; k , all distinct, and a complete system of orthogonal idempotents c1 ; : : : ; ck such that x D 1 c1 C C k ck : The numbers i are called the eigenvalues of x. The spectrum of x is denoted by .x/ D f W eigenvalue of xg; and we write C .x/ D maxf W 2 .x/g and

.x/ D minf W 2 .x/g:

It can be shown, see Theorem III.2.1 of [21], that x 2 C B if and only if .x/ .0; 1/. So, one can use the spectral decomposition, x D 1 c1 C C k ck , of x 2 C B , to define the unique square root of x by p p x 1=2 D 1 c1 C C k ck : Similarly, the functions x 7! log x and x 7! x t for t 2 R can be defined on C B . For x 2 V , the linear mapping P .x/ D 2L.x/2 L.x 2 /

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is called the quadratic representation of x. (Recall that L.x/ W V ! V is the linear map given by L.x/y D xy.) Note that P .x 1=2 /x D e for all x 2 C B . It can also be shown that P .x 1 / D P .x/1 for all x 2 C B . In the example of the Euclidean Jordan algebra on Symn the reader can verify that P .A/B D ABA. It is known, see [21], Proposition III.2.2, that if x 2 C B , then P .x/ 2 Aut.C /, and hence M.P .x/w=P .x/y/ D M.w=y/ and m.P .x/w=P .x/y/ D m.w=y/ for all w 2 V and x; y 2 C B . For w 2 V and x 2 C B we write C .w; x/ D C .P .x 1=2 /w/

and

.w; x/ D .P .x 1=2 /w/:

The following formula for Hilbert’s metric on symmetric cones was derived by Koufany [34]. Theorem 3.7. If V is a Euclidean Jordan algebra with symmetric cone C B , then for w 2 V and x 2 C B , M.w=x/ D C .w; x/ and m.w=x/ D .w; x/: In particular, we have for x; y 2 C B ,

C .x; y/ : d.x; y/ D log .x; y/ Proof. Let z D P .x 1=2 /w 2 V and let z D 1 c1 C Ck ck be the spectral decomposition of z. Note that w C ˇx is equivalent to z D P .x 1=2 /w ˇP .x 1=2 /x D ˇe, since P .x 1=2 / 2 Aut.C /. As e D c1 C C ck , this inequality holds if and only if 0 C .ˇ 1 /c1 C C.ˇ k /ck , which is equivalent to ˇ C .z/ D C .w; x/. As M.w=x/ D inffˇ 2 R W w C ˇxg, we deduce that M.w=x/ D C .w; x/. In the same way it can be shown that m.w=x/ D .w; x/. In the example of the Euclidean Jordan algebra on Symn we find for A; B 2 …n .R/ that C .A; B/ D C .P .B 1=2 /A/ D maxf W 2 .B 1=2 AB 1=2 /g D maxf W 2 .B 1 A/g and .A; B/ D minf W 2 .B 1 A/g: Remark 3.8. If we combine Lemma 3.1 and Theorem 3.7 we find for x 2 V and y in a symmetric cone C B that C .x; y/ D max c2Ee

hx j ci hy j ci

and

.x; y/ D min

c2Ee

hx j ci ; hy j ci

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where Ee is the set of extreme points of †e D fx 2 C W hx j ei D 1g. It is known, see [21], Proposition IV.3.2, that Ee D fx 2 †e W x is a primitive idempotentg:

So, C .x; y/ D

max

c2†e \J.V /

hx j ci hy j ci

and

.x; y/ D

min

c2†e \J.V /

hx j ci : hy j ci

These equalities are closely related to the min-max characterization of the eigenvalues of the elements of a Euclidean Jordan algebra by Hirzebruch [26].

4 Non-expansive mappings on Hilbert geometries Many interesting examples of non-expansive mappings on Hilbert geometries arise as normalizations of order-preserving, homogeneous mappings on cones. For example, Bellman operators in Markov decision processes and Shapley operators in stochastic games are order-preserving and homogeneous mappings on RnC after a change of variables, see [5], [7], [51]. These mappings, f W RnC ! RnC , are of the form: n Y

fi .x/ D inf sup ri .˛; ˇ/ ˛2Ai ˇ 2B i

RnC .

P

j D1

p .˛;ˇ /

xj j

;

Here ri .˛; ˇ/ 0, j pj .˛; ˇ/ D 1 and 0 pj .˛; ˇ/ for 1 i n and x 2 1 for all ˛, ˇ, i , and j . The iterates of these operators are used to compute the value of Markov decision processes and stochastic games. Other interesting examples of nonlinear order-preserving mappings on cones are so-called decimation-reproduction operators in the analysis of fractal diffusions [46], [50], and DAD-operators in matrix scaling problems, see [49], [56]. In many applications it is important to understand the iterative behavior of such mappings, f W C B ! C B and of the normalized mappings, g W †B ! †B , given by g.x/ D

f .x/ .f .x//

for x 2 †B D fx 2 C B W .x/ D 1g;

where 2 .C /B . The fact that these mappings are non-expansive with respect to Hilbert’s metric is a very useful tool to analyze their dynamics. In the analysis of the dynamics of a non-expansive mapping f on a Hilbert geometry .; ı/ it is important to distinguish two cases: (1) f has a fixed point in , and (2) f does not have a fixed point in . In the first case, the limit points of each orbit of f lie inside , whereas in the second case all the limit points of each orbit of f lie inside @ by a result of Całka [14]. In the next subsection we will consider the first case.

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4.1 Periodic orbits Before we get started we recall some basic notions from the theory of dynamical systems. A point w 2 is called a periodic point of f W ! if the exists an integer p 1 such that f p .w/ D w. The smallest such p 1 is called the period of w. In particular, w 2 is a fixed point if f .w/ D w. The orbit of x 2 is given by O.x/ D ff k .x/ W k D 0; 1; 2; : : :g. We say that the orbit of x 2 converges to a periodic orbit if there exists a periodic point w of f with period p such that limk!1 f kp .x/ D w. For x 2 we define the !-limit set by x W f ki .x/ ! y for some subsequence ki ! 1g: !.xI f / D fy 2 Note that we allow the limit point y to be in @ even though f need not be defined there. The following result will play an important role. Theorem 4.1. If f W X ! X is a non-expansive mapping on a closed subset X of .Rn ; kk1 / and f has a fixed point in X , then every orbit of f converges to a periodic orbit whose period does not exceed maxk 2k kn . A proof and a discussion of the history of this result can be found in Chapter 4 of [42]. The upper bound given in Theorem 4.1 is currently the strongest known and was obtained by Lemmens and Scheutzow in [40]. It was conjectured by Nussbaum in [53] that the optimal upper bound is 2n , but at present this has been confirmed for n D 1; 2 and 3 only, see [48]. Combining Theorem 4.1 with the isometric embedding result in Theorem 3.5 we obtain the following corollary for non-expansive mappings on polytopal Hilbert geometries. Corollary 4.2. If .; ı/ is a polytopal Hilbert geometry with m facets and f W D ! D is a non-expansive mapping on a closed subset D of .; ı/ with a fixed point, then each orbit of f converges to a periodic orbit whose period does not exceed maxk 2k Nk , where N D m.m 1/=2. It is an interesting open problem to find the optimal upper bound for the possible periods of periodic points of non-expansive mappings on .; ı/ in case is the interior of an n-dimensional simplex. For the 2-simplex, it was shown in [39] that 6 is the optimal upper bound. It is believed that there exists a constant c > 2 such that the periods do not exceed c n if is an n-simplex, but this appears to be hard to prove. Remark 4.3. Corollary 4.2 has the following interesting geometric consequence: It is impossible to isometrically embed a Euclidean plane into any polytopal Hilbert geometry, as it is impossible to isometrically embed a rotation under irrational angle in such Hilbert geometries. Thus, the Euclidean rank of a poytopal Hilbert geometry is 1. This observation complements results by Bletz-Siebert and Foertsch [9], who conjectured that the Euclidean rank of any Hilbert geometry is 1.

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The following, more detailed, results exist for the possible periods of periodic points for order-preserving homogeneous mappings on polyhedral cones, see [1], [40], [41] and [42], Chapter 8. Theorem 4.4. If f W C ! C is a continuous, order-preserving, homogeneous mapping on a polyhedral cone C with nonempty interior and m facets, then the following assertions hold: 1. Every norm bounded orbit of f converges to a periodic orbit whose period does not exceed mŠ ; m mC1 b 3 cŠb 3 cŠb mC2 cŠ 3 where brc denotes the greatest integer q r. 2. In case C is an n-dimensional simplicial cone the set of possible periods of periodic points of f is precisely the set of integers k p for which there exist integers , and 1 q2 kn for some q1 and q2 such that p D q1 q2 , 1 q1 bk=2c 0 k n. 3. If f .v/ D v for some > 0 and v 2 C B , then each orbit of the normalized mapping g W †B ! †B given by g.x/ D

f .x/ .f .x//

for x 2 †B

converges to a periodic orbit whose period does not exceed the upper bound is sharp in case C is a simplicial cone.

m bm=2c

. Moreover,

In particular, we see that on the cone R3C , the set of possible periods of periodic points of order-preserving homogeneous mappings f W R3C ! R3C is f1; 2; 3; 4; 6g. So, it is impossible to have a period-5 point in that case. An example of a mapping on R3C with a period-6 orbit is the mapping f W R3C ! R3C given by 1 0 1 0 x1 .3x1 ^ x2 / _ .3x2 ^ x3 / f @ x2 A D @ .3x1 ^ x3 / _ .3x3 ^ x2 / A for x 2 R3C x3 .3x2 ^ x1 / _ .3x3 ^ x1 / which has x D .1; 2; 0/ as a period-6 point. Here a ^ b D minfa; bg and a _ b D maxfa; bg for a; b 2 R. In view of Corollary 4.2 it is interesting to ask the following question. For which Hilbert geometries .; ı/ do we have that the orbits of each non-expansive mapping f W ! with a fixed point in converge to periodic orbits? Obviously the answer is negative if is the interior of an ellipsoid. However, the following was shown in [39]. Theorem 4.5. If .; ı/ is a strictly convex Hilbert geometry and there exists no 2dimensional affine plane H such that H \ is the interior of an ellipsoid, then every

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orbit of a non-expansive mapping f W ! , with a fixed point in , converges to a periodic orbit. In fact, there exists an integer q 1 such that limk!1 f kq .x/ exists for all x 2 .

4.2 Denjoy–Wolff type theorems In this subsection we briefly discuss the behaviour of fixed point free non-expansive mappings on Hilbert geometries. A more detailed overview of this topic is given by A. Karlsson in Chapter 9 of this volume [30]. Recall that if f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/, then the attractor of f , [ Af D !.xI f /; x2

is contained in @ by Całka’s result [14]. In that case it is interesting to understand the structure of Af in @. This problem was considered by Beardon in [3], [4]. He showed that there is a striking resemblance between the dynamics of fixed point free mappings on Hilbert geometries and the dynamics of fixed point free analytic selfmappings of the open unit disc in the complex plane, which is characterized by the classical Denjoy–Wolff theorem [18], [71], [72]. Theorem 4.6 (Denjoy–Wolff). If f W D ! D is a fixed point free analytic mapping on the open unit disc D in C, then there exists a unique 2 @D such that limk!1 f k .x/ D for all x 2 D, and the convergence is uniform on compact subsets of D. Analytic self-mappings of the open unit disc are non-expansive under the Poincaré metric by the Schwarz–Pick lemma. In [4] Beardon noted that the Denjoy–Wolff theorem should be viewed as a result in geometry, as it essentially only depends on the hyperbolic properties of the Poincaré metric on D. In fact, he showed that the Denjoy– Wolff theorem can be generalized to fixed point free non-expansive mappings on metric spaces that possess sufficient “hyperbolic” properties. As a particular consequence of his results he obtained in [4] the following Denjoy–Wolff theorem result for strictly convex Hilbert geometries. Theorem 4.7. If f W ! is a fixed point free non-expansive mapping on a strictly convex Hilbert geometry .; ı/, then there exists a unique 2 @ such that limk!1 f k .x/ D for all x 2 , and the convergence is uniform on compact subsets of . Beardon’s arguments were sharpened by Karlsson in [29]. It is known that Beardon’s result does not hold for general Hilbert geometries. In fact, for the Hilbert geometry on the open 2-simplex, 2 , Lins [45] showed that if S is a convex subset of

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@2 , then there exists a fixed point free non-expansive mapping f W B2 ! B2 such that [ !.xI f /: SD x2

It was conjectured, however, by Karlsson and Nussbaum that the following is true, see [56]. Conjecture (Karlsson–Nusbaum). If .; ı/ is a Hilbert geometry and f W ! is a fixed point free non-expansive mapping, then there exists a convex set ƒ in @ such that !.x; f / ƒ for all x 2 It turns out that to prove the conjecture it suffices to show that there exists x 2 such that the convex hull of !.xI f / is contained in @, see [56]. At present there exist only partial results for Conjecture 4.2. To begin there exists the following result by Lins [45]. Theorem 4.8. If .; ı/ is a polytopal Hilbert geometry and f W ! is a fixed point free non-expansive mapping, then there exists ƒ @ such that !.xI f / ƒ for all x 2 . To prove this theorem Lins used the fact that a polytopal Hilbert geometry can be isometrically embedded into a finite-dimensional normed space. This property allowed him to show that for each x 2 there exists a horofunction h W ! R such that limk!1 h.f k .x// D 1. For general Hilbert geometries such a horofunction does not always exist, see Remark 3.2 of [44]. So, there seems to be no apparent way to generalize Lins’ arguments to the general case. The following partial result for Conjecture 4.2 is due Karlsson [29]. Theorem 4.9. If f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/ and there exists z 2 such that ı.f k .z/; z/ > 0; k k!1 then there exists a convex ƒ @ such that !.xI f / ƒ for all x 2 . lim

In [31] Karlsson and Noskov showed the following result, which says that the attractor Af of a fixed point free non-expansive mapping f on a Hilbert geometry .; ı/ is a star-shaped subset of @. Theorem 4.10. If f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/, then there exists 2 @ such that for y 2 Af the straight line segment Œy; is contained in @. The following counterpart to Theorem 4.9 was proved by Nussbaum [56].

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Theorem 4.11. If f W ! is a fixed point free non-expansive mapping on a Hilbert geometry .; ı/ and there exists z 2 such that lim inf ı.f kC1 .z/; f k .z// D 0; k!1

then there exists a convex ƒ @ such that !.xI f / ƒ for all x 2 . Other Denjoy–Wolff type theorems for finite and infinite-dimensional Hilbert geometries can be found in [24], [42], [46], [56]. Despite numerous efforts the KarlssonNussbaum conjecture remains one of the most outstanding problems in the field.

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[10] F. F. Bonsall, Linear operators in complete positive cones. Proc. London Math. Soc. 8 (1958), no. 3, 53–75. [11] H. Busemann and B. B. Phadke, A general version of Beltrami’s theorem in the large. Pacific J. Math. 115 (1984), 299–315. [12] P. J. Bushell, Hilbert’s projective metric and positive contraction mappings in a Banach space. Arch. Ration. Mech. Anal. 52 (1973), 330–338. [13] P. J. Bushell, On the projective contraction ratio for positive linear mappings. J. London Math. Soc. 6 (1973), 256–258. [14] A. Całka, On a condition under which isometries have bounded orbits. Colloq. Math. 48 (1984), 219–227. [15] B. Colbois and P. Verovic, Hilbert domains that admit a quasi-isometric embedding into Euclidean space. Adv. Geom. 11 (2011), no. 3, 465–470.

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Part III

Developments and applications

Chapter 11

Convex real projective structures and Hilbert metrics Inkang Kim and Athanase Papadopoulos

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to affine spheres . . . . . . . . . . . . . . . . . . . . . . . . . . Parametrizations of real projective structures and the deformation spaces 4.1 Goldman’s parametrization . . . . . . . . . . . . . . . . . . . . . 4.2 Hitchin’s parametrization . . . . . . . . . . . . . . . . . . . . . . 4.3 Length spectra as parameters . . . . . . . . . . . . . . . . . . . . . 5 Strictly convex manifolds and topological entropy . . . . . . . . . . . . 6 Geodesic currents on strictly convex surfaces . . . . . . . . . . . . . . . 7 Compactification of the deformation space of convex real projective structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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307 309 315 319 320 325 326 327 329

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1 Introduction In what follows, RP n is the n-dimensional real projective space, that is, the set of lines through the origin in RnC1 , and An the n-dimensional affine space, considered as the complement of a hyperplane in RP n . A subset of An is convex if its intersection with each affine line is connected. Let be an open convex subset of An . Then is equipped with a canonical metric, called the Hilbert metric, whose definition we recall in Section 2 below. This metric is invariant by the group of projective transformations of RP n which preserve . Several interesting phenomena were discovered and several good questions arose recently concerning that metric. It is non-Riemannian except in the case where is an ellipsoid, but it shares several properties with the hyperbolic metric (that is, a Riemannian metric of constant negative curvature), especially if is strictly convex. There are also many differences. In the next sections, we will highlight some of these analogies and differences. We start by recalling very classical questions concerning convex sets equipped with their Hilbert geometry. From the observation that the boundary of the unit disk

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is a smooth circle, it was natural to ask whether there exist convex domains in the projective plane with less regular boundary which admit compact quotients by discrete subgroups of the automorphism group of the domain. This question is natural because the unit disk admits such quotients. This existence question was already addressed by Ehresmann in the 1930s [18], and it was answered affirmatively thirty years later by Kac and Vinberg [25]. In modern language, a convex domain in the projective plane which admits a cocompact action is termed divisible. An ellipse in RP 2 or an ellipsoid in RP 3 are well-known examples of convex domains. Their compact quotients are the 2- and 3-dimensional compact hyperbolic manifolds. The theory of hyperbolic manifolds (their construction, classification, their deformation and moduli spaces and their rigidity properties) was born in the works of Klein and Poincaré in the 1880s. This theory rapidly evolved into one of the most beautiful geometric theories that were developed during the twentieth century. Important questions concerning more general convex projective manifolds (quotients of divisible convex domains) were addressed by Benzécri in his thesis [6] (1960). This thesis contains some foundational work on the subject. To find conditions under which a convex domain is divisible is a difficult matter, as is the general question of existence of lattices in semisimple Lie groups. In most cases, the boundary of such a domain is nowhere analytic. Benzécri proved that if the boundary is C 2 , then the original convex set is an ellipsoid, which makes it identified with the hyperbolic space, more precisely, with the Cayley–Klein–Beltrami model of that space. It also follows from the work of Benzécri and from later works that the boundary @ of a divisible convex domain is either a conic (in which case the projective structure arises from a hyperbolic structure) or this boundary is nowhere C 1C for some > 0. Explicit examples of quotients of projective manifolds obtained by reflections along convex polyhedra in the sphere were obtained by Vinberg in the early 1970s. These constructions may be considered as projective analogues of Poincaré’s examples of hyperbolic manifolds that use reflections along convex polyhedra in hyperbolic space. A convex real projective manifold is the quotient of a convex set RP n by a discrete group of projective transformations, and the structure is termed strictly convex if is strictly convex. The moduli space of convex projective structures generalizes the Teichmüller spaces of hyperbolic surfaces. There are many interesting recent developments on convex projective manifolds and their deformations. Hitchin [23] discovered a component in the space of representations of fundamental groups of surfaces in SL.n; R/, for every n 2, which contains Teichmüller space, and whose elements have several properties in common with Teichmüller space. He called these components Teichmüller components, and today such a component is called a Hitchin component. In the case n D 3, such components consist of real projective structures on surfaces. Johnson and Millson [20] showed that there are non-trivial continuous deformations of higher-dimensional hyperbolic structures through strictly convex projective structures. Conversely, one might expect that if a compact manifold admits a strictly convex projective structure, then it is a deformation of a hyperbolic structure. But Benoist [3] constructed in dimension four an example of a manifold

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which admits a strictly convex real projective structure but no hyperbolic structure. Later, Kapovich [26] generalized the method to show that such examples exist in any dimension 4. In this chapter, we study some relations between hyperbolic geometry and differential projective geometry. Some of the natural questions that appear in this setting are the following. • To what extent does Hilbert geometry generalize hyperbolic geometry? • What are the relations and the common properties between spaces of deformations of convex projective structures and Teichmüller spaces? • Are there compactifications of deformation spaces of convex projective manifolds that are analogous to compactifications of Teichmüller spaces? • How does the hyperbolic behavior of geodesic flows of Hilbert manifolds generalize the hyperbolic behavior (and in particular the Anosov theory) of negatively curved Riemannian manifolds? The exposition is by no means complete and thorough. In some cases, we just record the results together with some references known to us, hoping to arouse the reader’s interest in these questions. The hyperbolic-geometry aspects of Hilbert geometry, from the dynamical point of view, are more thoroughly discussed in Chapter 7 of this volume [17]. We would like to thank Sarah Bray, Bill Goldman, Ludovic Marquis and Marc Troyanov for valuable comments and corrections on a preliminary version of this chapter.

2 Preliminaries We recall some basic notions of projective geometry on manifolds. The automorphism group of RP n , denoted by PGL.n; R/ – the group of projective transformations – is the quotient of the linear group GL.n C 1; R/ by the action of the nonzero scalar transformations. It is sometimes convenient to work on the sphere Sn , which is a double-sheeted cover of RP n . The sphere is equipped with the induced projective structure whose automorphism group (also called the projective automorphism group) is the group SL˙ .n C 1; R/ of real .n C 1/ .n C 1/ matrices of determinant ˙1. This is also the group of volume-preserving affine transformations of RnC1 . The projective lines (or, more simply, the lines) in RP n are the projections of the great circles of the sphere Sn by the canonical map Sn ! RP n . This set of lines is preserved by the group of projective transformations. The projective lines play simultaneously the role of lines of a geometry defined in the axiomatic sense, and the role of geodesics for the Hilbert metric, as we shall recall below.

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Let be a subset of RP n . We say that is convex if the intersection of with any line of RP n is connected. We say that is a properly convex (or a proper convex) subset of RP n if it is convex and contained in the complement of a hyperplane. An affine patch in RP n is the complement of a hyperplane. An affine patch can be seen as the affine n-dimensional space An . Let be a properly convex subset of RP n . The convexity of as a subset of projective space (that is, the intersection of with each projective line is connected) is equivalent to the convexity of as a subset of affine space (that is, the intersection of with each affine line is connected). Let be a properly convex subset of RP n contained in an affine patch A. We equip A with a Euclidean norm j j. We shall also denote by j j the associated Euclidean distance. The space A plays now the role of a Euclidean space Rn in which sits. We present a brief summary of some notions associated to the Hilbert geometry of that we shall use in this chapter. For x ¤ y 2 , let p, q be the intersection points of the line xy with @ such that p, x, y, q are in this order. The Hilbert distance between x and y is defined by d .x; y/ D

jp yjjq xj 1 log : 2 jp xjjq yj

The value of d .x; y/ does not depend on the choice of the Euclidean metric j j on An . For x D y, we set d .x; y/ D 0. This metric coincides with the familiar hyperbolic metric is the case where @ is an ellipse (in dimension 2) or ellipsoid (in dimension 3). This metric of the ellipse or ellipsoid is the so-called “projective model”, or Cayley–Klein–Beltrami model of hyperbolic geometry. The Hilbert metric is Finsler, and it is not Riemannian unless @ is an ellipsoid. The Finsler norm is given, for x 2 and a vector v in the tangent space of at x, by kvkx D

1 1 1 C jvj 2 jx p j jx p C j

(2.1)

where p ˙ are the intersection points with @ of the oriented line in defined by the vector v based at x and where j j is our chosen norm on the affine patch. This norm is reversible, that is, it satisfies kvkx D k vkx . The Finsler metric associated to this Finsler norm is the metric on defined by taking the distance between two arbitrary points to be the infimum of the lengths of C 1 paths joining them, where the length of a path is defined by integrating the norms of tangent vectors using Formula (2.1). It is an easy exercise to show that this Finsler metric on is the Hilbert metric. The (projective) automorphism group of is the group of projective transformations of RnC1 that preserve . In other words, we have Aut./ D ff 2 PGL.n; R/ j f ./ D /g: The Hilbert metric of is invariant by the group Aut./. A recent result by Walsh described in Chapter 5 of this volume ([50]) says that the isometry group Aut.; d /

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of the Hilbert metric d coincides with the group Aut./ except in a few special cases where Aut.; d / is an order-two extension of Aut./. We now associate a Borel measure on using the Finsler structure. The construction is analogous to the one used in Riemannian geometry. We follow the presentation of Marquis in [40]. Let Vol be a Lebesgue measure on A normalized by Vol.fv 2 A W jvj < 1g/ D 1. We define a measure on by setting, for each Borel subset A A, Z d Vol.x/ ; .A/ D A Vol.Bx .1// where Bx .1/ D fv 2 Tx W kvkx < 1g, the norm k k being the one given by (2.1). This measure turns out to be the Hausdorff measure induced by the Hilbert metric [9]. In particular, it is independent of the choice of the Euclidean norm of A. It is called the Busemann volume, and also the Hilbert volume. It is invariant by the action of Aut./. From the definition, we have the following, for any two convex domains 1 2 of Rn : 1 2 (1) kvk x kvkx for every tangent vector v based at a point x in 1 ;

(2) d2 .x; y/ d1 .x; y/ for every x and y in 1 ; (3) Bx1 .1/ Bx2 .1/ for every x in 1 ; (4) 2 .A/ 1 .A/ for any Borel set A in 1 . For more details on the notion of measure associated to a general Finsler metric, we refer the reader to [1] and [9]. We now consider projective structures on manifolds. Definition 2.1. A real projective structure on an n-dimensional differentiable manifold M is a maximal atlas with values in the n-dimensional projective space RP n whose transition functions are restrictions of projective automorphisms of RP n . Equipped with such a structure, the manifold M becomes a real projective manifold. We shall sometimes delete the adjective “real” since all the projective structures we consider in this chapter are real. An isomorphism between two n-dimensional projective manifolds is a homeomorphism between the underlying manifolds which, in each projective chart, is locally the restriction of an element of the projective transformation group of RP n . To each real projective structure is associated, by the general theory of G-structures of Ehresmann [18], a developing map and a holonomy homomorphism. The developing map arises from the attempt to define a global chart for the structure. It is constructed by starting with a coordinate chart around a given point and trying to extend it by analytic continuation. There is an obstruction for doing so if the manifold is not simply connected, and in fact, the only obstruction is the fundamental group of M . (When we come back to the point we started with, by analytically continuing

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along a nontrivial path, we generally end up with a different germ of a chart into RP n than the one we started with.) Thus, instead of obtaining a map from our manifold M to RP n , we end up with a map z ! RP n ; dev W M z is the universal covering of M . The holonomy homomorphism is an injective where M homomorphism hol W 1 .M / ! PSL.n C 1; R/ satisfying dev B D hol. / B dev: This developing map is completely determined by its restriction to an arbitrary open subset of M . When we change the initial coordinate chart in the above construction, the resulting map differs from the previous one by post-composition by an element of PGL.n; R/. The result is that although the developing map and the holonomy homomorphism depend on some choices (namely, the choice of the initial chart), there are nice transformation formulae relating the various maps and homomorphisms obtained. In particular, the holonomy homomorphism is well defined up to a conjugation by an element of PGL.n; R/. In the case where the developing map is a homeomorphism onto its image, we can z /=hol.1 .M //. We refer the reader to the paper by Ehresmann [18] write M D dev.M for the general theory of developing maps and holonomy representations associated to geometric structures. Benzécri [6], Kuiper [32] and subsequently Koszul [30] considered thoroughly the case of real projective structures. Thurston, starting in the 1970s, included this theory as an important part of the general theory of geometrization of low-dimensional manifolds, see [48]. Goldman [21], motivated by ideas of Thurston, developed the theory of moduli spaces of projective structures on surfaces. In their paper [47], Sullivan and Thurston give an example of a projective structure on the torus whose developing map is not a covering of projective space. Talking about the sources of differential projective geometry, one has also mention the work of Chern on the Gauss–Bonnet formula and characteristic classes that motivated several later works [13]. In what follows, we shall use the more restrictive notion of convex real projective structure. This is the case where the developing map sends homeomorphically the z of M onto a convex subset of some Rn sitting in RP n as the comuniversal cover M plement of an affine hyperplane. It turns out that such a structure is more manageable than a general real projective structure, and in particular, one can introduce the Hilbert metric into the playground. There is a lot of classical and more recent activity on convex projective structures and their deformation spaces. In what follows, we shall use the terminology convex to mean properly convex. There is a characterization of such a structure, which we can also take as a definition:

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Definition 2.2. A convex real projective manifold M is an object of the form = where is a convex domain in RP n containing no projective line and a discrete group of the projective automorphism group Aut./ of . We call M strictly convex if is strictly convex i.e., if is convex and @ does not contain any line segment. The relation between Definitions 2.1 and 2.2 is made through the natural identifications between and the universal cover of M and between and the fundamental group of M . In dimension two, we have more knowledge about convex projective structures, and in particular there is the following in Goldman [21] (Proposition 3.1): Proposition 2.3. Let M be a projective structure on a surface. Then, the following three properties are equivalent: (1) M is projectively equivalent to a quotient = where is a convex open subset of RP 2 and a discrete group of projective transformations of RP 2 which acts freely and properly discontinuously; z ! RP 2 is a diffeomorphism onto a convex subset (2) the developing map dev W M 2 of RP ; (3) every path in M is homotopic relative endpoints to a geodesic path (that is, a path which in coordinate charts is contained in a line of RP 2 ). We shall be interested in quotients of convex sets, and we make right now the following definition: Definition 2.4. An open (properly) convex set is said to be divisible if there exists a discrete subgroup Aut./ such that = is compact. There is a natural identification between two convex real projective manifolds 1 = 1 and 2 =2 , defined by the condition that there exists a projective transformation g of RP n such that g.1 / D 2 and g1 g 1 D 2 . This equivalence relation is used in the definition of the moduli space and the deformation space of convex real projective structures. We are mainly interested in the deformation space of convex projective structures on surfaces, which, like the Teichmüller space of hyperbolic (or of conformal) structures, is a space of equivalence classes of marked convex projective structures. We recall the concept of marking, which originates in Teichmüller theory. We start with a fixed topological surface S0 . Definition 2.5. A marked convex projective structure on S0 is a pair .f; S / where S is a surface homeomorphic to S0 equipped with a convex projective structure, and f W S0 ! S is a homeomorphism. A marked convex projective structure on S0 induces a convex projective structure on the base surface S0 itself by pull-back. Conversely, a convex projective structure on

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S0 can be considered as a marked convex projective structure, by taking the marking to be the identity homeomorphism of S0 . Definition 2.6. The deformation space of convex projective structures on S0 is the set of equivalence classes of pairs .f; S/, where S is a convex projective surface homeomorphic to S0 and where two pairs .f; S / and .f 0 ; S 0 / are considered to be equivalent if there exists a projective homeomorphism f 00 W S ! S 0 that is homotopic to f 0 B f 1 . Equivalently, we can define the deformation space of convex projective structures on a surface (or, more generally, on a manifold) as the space of homotopy classes of convex projective structures on that surface (or manifold). This space is equipped with a natural topology arising from the C 1 topology on developing maps. This topology is Hausdorff (see [21], p. 793). We shall say more about the deformation space of real projective structures on surfaces in §4. The basic elements of the theory of convex real projective structures on closed surfaces and their deformations are due to Kuiper [32], Kac–Vinberg [25] and Benzécri [6], and a complete theory has been developed (including the case of surfaces with boundary) by Goldman [21]. We shall elaborate on Goldman’s parametrization of the deformation space in §4. For any convex real projective manifold M D = , the Hilbert metric on descends to a metric on M called the Hilbert metric of M . For a strictly convex real projective structure M D = , as in the hyperbolic case, there exists a unique Hilbert geodesic in each homotopy class of a loop, unless this loop represents a parabolic element of the fundamental group, in which case the curve is homotopic to a puncture (or cusp) of S . Lemma 2.7. For any convex real projective surface S of finite type and of negative Euler characteristic (the surface may have geodesic boundary components and cusps), the area of S with respect to the Hausdorff measure induced by the Hilbert metric is uniformly bounded below independently of the topology of S . The proof uses a pair of pants decomposition of S. We recall that a (topological) pair of pants in S is an embedded surface which is homeomorphic to a sphere with three holes, where a hole is either a puncture or a boundary component, such that the following two conditions hold: • Each boundary component of P is a simple closed curve in S which is not homotopic to a point or to puncture of S. • There is no embedded annulus in S whose two boundary components are the union of boundary component of P and a boundary component of S. Note that the holes of P might be boundary components of S, and they can also be punctures, and the latter are considered as boundary curves of length zero.

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A topological pair of pants decomposition P of S is a union of disjoint simple closed curves in that surface such that the closure of each connected component of the complement of P in S is a topological pair of pants in S . Any surface of negative Euler characteristic admits a topological pair of pants decomposition. It is easy to see, using an Euler characteristic argument, that for a closed surface Sg of genus g 2, there are 2g 2 pairs of pants in a pair of pants decomposition. Given a surface equipped with a metric, a geodesic pair of pants decomposition is a topological pair of pants decomposition in which each curve which is not homotopic to a puncture is a closed geodesic. On any surface with finitely generated fundamental group equipped with a hyperbolic metric or with a Hilbert metric, any topological pair of pants decomposition is homotopic to a geodesic pair of pants decomposition. Furthermore, for hyperbolic metrics and for strictly convex structures, the closed geodesics in each free homotopy classes are unique, so every topological pair of pants decomposition is homotopic to a unique geodesic pair of pants decomposition. We now sketch the proof of Lemma 2.7. Proof. Take a geodesic pair of pants P in S. One can decompose P into two ideal triangles. Therefore, it suffices to give a bound for the area of an ideal triangle. Let T be a lift of an ideal triangle in with three ideal vertices p1 , p2 , p3 . Choose three projective lines P1 , P2 , P3 containing p1 , p2 , p3 disjoint from . They form a triangle 4 containing . Now by a comparison argument, it suffices to lower bound the area of T in 4. This is an exercise in projective geometry; see [14]. There are several papers where the reader can find concise introductions to the basics of the modern theory of projective geometry. We refer to the survey by Benoist [4] and to the sections on preliminaries in the thesis of Marquis [40]. Let us note that in studying divisible convex sets of finite co-volume, we do not lose a lot if we restrict ourselves to strictly convex sets. The following theorem is due to Marquis [41]. Theorem 2.8. Let be a proper open convex subset of RP 2 . If is not a triangle and admits a finite-volume quotient surface, then is strictly convex.

3 Relation to affine spheres An affine sphere is a smooth hypersurface in RnC1 characterized by the condition that its affine normal lines meet at a common point. The family of affine spheres is invariant under the group of affine transformations of RnC1 . In fact, it is the simplest interesting such family. Thus, it is not surprising that affine spheres are useful in the study of affine, and also of projective structures. (Recall that affine geometry is projective geometry where a hyperplane at infinity in RP n has been fixed.) Affine spheres also appear

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naturally in the solutions of certain PDEs, namely, the Monge–Ampère equations, and also in the study of hyperbolic surfaces. In fact, it is known that on a compact hyperbolic surface equipped with a cubic differential, there is a unique associated equiaffine metric called the Blaschke metric. (An object is called equiaffine if it is invariant by the group of volume-preserving affine transformations.) This, together with the Cheng–Yau classification of complete hyperbolic affine spheres, gives a new parametrization of the space of real projective structures on the surface and of the Hitchin component of the representations of the fundamental group of the surface into SL.3; R/, see [23], [33], [36]. In this section, we review some intricate relations between affine spheres and Hilbert metrics. We shall also refer to the relation between cubic differentials and affine spheres in §4.2 below. Let M RnC1 be a transversely oriented smooth hypersurface with a trivial bundle E D M RnC1 . Choose a transverse vector field over M so that E D TX M ˚ L where TM is the tangent bundle to M and L a trivial line bundle over M spanned by . If r is the standard affine flat connection on RnC1 , its restriction on E gives the following equations (Gauss and Weingarten): rX Y D DX Y C h.X; Y /; rX D S.X / C .X / for any tangent vector fields X and Y to M . The equations are obtained by splitting at each point x 2 M the tangent space to RnC1 at that point to TMx ˚ L. Here, D is a torsion-free connection on TM , h a symmetric 2-form on TM , S a shape operator and a 1-form. If M is locally strictly convex, i.e., if it can be written locally as the graph of a function with positive definite Hessian, then there exists a unique transverse vector field such that (1) D 0, (2) h is positive definite, and (3) jdet.Y1 ; ; Yn ; /j D 1 for any h-orthonormal frame Yi of TM . (see Proposition 2.1 of [5]). The vector field is called the affine normal, D the Blaschke connection, and h the affine metric on M . These are equiaffine notions. Definition 3.1. A hypersurface M in RnC1 is said to be an affine sphere with affine curvature 1 if the shape operator satisfies S D Id. In more geometric terms, an affine sphere in RnC1 is a smooth hypersurface characterized by the fact that its affine normal lines meet at a common point, called the center of the affine sphere (which could be at infinity, and in that case the affine sphere is said to be improper). The affine normal field is an affine invariant of the surface and the condition of being an affine sphere is therefore affinely invariant. This makes the family of affine spheres invariant under the group of affine transformations of RnC1 . Examples of affine spheres include ellipsoids and quadric hypersurfaces. There are

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two kinds of proper locally convex affine spheres, the hyperbolic, all of whose normals point away from the center, and the elliptic, all of whose normals point towards the center. An affine sphere is not necessarily affinely equivalent to a Euclidean sphere and, in fact, there are infinitely many non-equivalent affine spheres. See [38], [43] and [39] for surveys on affine spheres. Another family of objects which is well-known to be invariant under the group of affine transformations is the family of straight lines. Affine spheres were introduced at the beginning of the twentieth century by Ti¸ ¸ teica, and they were studied later on by Blaschke, Calabi, Cheng–Yau and others; see the survey by Loftin [38] for some historical background. The definition of the affine metric using the invariance of the affine normal was derived by Blaschke [7]. There are relations, discovered by Blaschke and by Calabi, between the theory of affine spheres and the real Monge–Ampère equations, and there is also a relation between affine spheres and the theory of convex real projective structures. The work of Cheng–Yau on affine spheres, combined with work of Wang [51] on PDEs in the setting of affine differential geometry, was also used by Labourie and Loftin to parametrize equivalence classes of representations in SL.3; R/ by cubic differentials on a Riemann surface (see §4.2 below). The work of Cheng–Yau on the Monge–Ampère equations associates to each properly convex subset of RnC1 an affine sphere in this space. Our aim in the rest of this section is to give an idea of how Hilbert geometry fits into this picture, in particular through the following two results: • For any proper convex set , the construction of affine spheres by Cheng and Yau leads naturally to a Riemannian metric on which is bi-Lipshitz equivalent to the Hilbert metric at the level of norms, cf. Proposition 3.3 below. • In the case of a strictly convex real projective surface, there is a comparison between the Hilbert volume on that surface with the affine volume (see Corollary 3.4). In both cases, the constants that appear in the comparison (that of the norms and that of the volumes) are universal. We briefly recall the construction by Cheng and Yau of an affine sphere in RnC1 associated to properly convex subset of this space. A cone C RnC1 is a subset which is invariant by the action of the positive reals by homotheties. A convex cone C RnC1 is a cone which is (the closure of) the inverse image of a convex set in RP n by the canonical projection RnC1 n f0g ! RP n . To any bounded open convex subset Rn , there is an associated convex cone C./ D ft.1; x/jx 2 ; t > 0g RnC1 which is a connected component of the inverse image of a convex set by the natural map RnC1 n f0g ! RP n with fiber RC . A group of projective transformations which acts properly discontinuously on also acts on the cone C ./. This is a consequence of the fact that any representation of a discrete group into PSL.n C 1; R/ lifts to a representation into the group SL˙ .n C 1; R/ of invertible .n C 1/ .n C 1/ matrices whose determinant is ˙1.

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Cheng and Yau [12] proved the following, which solved a conjecture of Calabi [11]: Theorem 3.2. If C RnC1 is an open convex cone containing no lines, then there exists a unique embedded hyperbolic affine sphere H whose center is the origin, which has affine curvature 1, and which is asymptotic to the boundary of C . The fact that C contains no lines is equivalent to the fact that the convex set is proper. The affine invariants of H are invariants of . The projection map induces a homeomorphism between H and . To the bounded open convex set Rn is associated the affine sphere defined as ²

HD

³

1 .1; x/ j x 2 u.x/

where u is the unique convex solution of the real Monge–Ampère equation detD 2 u D .1=u/nC2 satisfying [email protected] D 0: The projection map RnC1 n f0g ! RP n induces a diffeomorphism between H and , and the affine metric h on H induces a Riemannian metric on , still denoted by h and called the affine metric. This metric gives rise to a measure h on . Now we have two measures on , one is h , coming from the affine metric and the other is , coming from the Hilbert metric. Benoist and Hulin proved in [5] that the affine metric is bi-Lipschitz equivalent to the Hilbert metric: Proposition 3.3 (Benoist–Hulin). There exists a constant c > 0 such that for any properly convex set , x 2 and X 2 Tx , 1 kX kF kXkh ckX kF ; c where the subscript F denotes the Finsler norm of the Hilbert metric and h the affine metric. Benoist and Hulin deduce this result from the cocompactness of the action of SL .n C 1; R/ on the set of pairs .x; / and the continuous dependence on .x; / of both affine and Hilbert metrics. The result for Hilbert metrics is contained in Benzécri’s thesis, [6], Chapter V, where the author introduces several spaces he calls body spaces and form spaces (“espaces de corps” and “espaces de formes”). One of these spaces is, for n 1, the space of pairs .x; /, where RP n is a properly convex open subset and x is a point in . It is equipped with the Hausdorff topology. More precisely, the topology on the set of pairs .x; / is the product topology, where ˙

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on the first factor the topology is induced by the canonical metric on projective space (which is the quotient metric of the canonical metric on the sphere) and on the second factor the topology is induced by the Hausdorff distance on the compact subsets of projective space. (To use precisely the last notion, one needs to replace a subset by its closure.) Benzécri considers then the natural quotient of this space by the action of the group SL˙ .n C 1; R/ and he proves that this quotient is compact and metrizable (see [6], Théorème 2, p. 309). This implies the following: Corollary 3.4 ([5] Proposition 2.6). There exists a universal constant C (that depends only on the dimension) such that for any convex real projective manifold M , we have, for any Borel subset B of M , 1 VolH .B/ VolA .B/ C VolH .B/; C where H denotes the Hilbert volume and A the affine volume. In particular we have, in the case where M is a finite volume quotient, 1 VolH .M / VolA .M / C VolH .M /: C Thus, for a convex real projective manifold, having finite affine volume and having finite Hilbert volume are equivalent properties. One should mention that in the special case of surfaces and under the additional assumption that the boundary of the convex set is smooth, Loewner and Nirenberg solved the Monge–Ampère equation and they also constructed a Riemannian metric on the convex set which is invariant by projective transformations. By the uniqueness of the solution to the Monge–Ampère equation, this metric is the same as the one constructed by Cheng and Yau using affine spheres. The relation between the structure equations of affine spheres and RP 2 structures is explained in detail in the survey [39] by Loftin and McIntosh. The asymptotic affine sphere and an invariant Riemannian metric associated to a Hilbert geometry are also mentioned in Chapter 8 of this volume [42] (§4.1 and 4.2). We shall say more on affine spheres in §4.2 below.

4 Parametrizations of real projective structures and the deformation spaces The classification of convex real projective structures on surfaces of nonnegative Euler characteristic is due to Kuiper, cf. [31] and [32]. In this section, we describe parameter spaces for these structures. We start, in the first subsection, with Goldman’s parametrization, which was inspired by Thurston’s exposition of the Fenchel–Nielsen parametrization of Teichmüller space associated to a pants decomposition of a hyper-

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bolic surface. The parametrization consists of the length and the twist parameters of the pants curves. The Teichmüller space of a surface sits naturally as a subset of the deformation space of real projective structures, and Goldman’s parameters are a generalization of the Fenchel–Nielsen parameters. We then describe Hitchin’s parametrization, which also generalizes the parameters for Teichmüller space, when this space is considered as a connected component of the character variety of the fundamental group of the surface in SL.2; R/. Hitchin’s parameter space is a component of the character variety of representations of the fundamental group of the surface in SL.n; R/. Hitchin’s work gave rise to several other works by various authors, and it is also related to several questions addressed in this chapter. We shall mention some of these relationships below. In the final subsection, we mention a set of parameters due to the first author of this chapter that use the length spectra of the Hilbert metrics. This is also a generalization of a classical parametrization of hyperbolic structures by geodesic length spectra. It is an interesting question to study more carefully the structure of the parameter spaces and their nature (analytic, algebraic, etc.)

4.1 Goldman’s parametrization Our goal in this subsection is to give a brief description of Goldman’s parameters for the deformation space of convex real projective structures on surfaces of negative Euler characteristic. Recall that by using the Cayley–Klein–Beltrami model of hyperbolic geometry, a hyperbolic structure (in the sense of a Riemannian metric of constant curvature equal to 1) on a closed surface S of negative Euler characteristic is a special case of a convex real projective structure. This is a structure of the type = where Rn RP n is the interior of an ellipse and a subgroup of projective transformations of RP n that preserve . For any topological surface S of negative Euler characteristic, a classical and useful set of parameters for the space of isotopy classes of hyperbolic structures on S (that is, for the Teichmüller space of S ) is provided by the Fenchel–Nielsen coordinates associated to pair of pants decompositions. These parameters consist of the set of lengths of the pants curves of this decomposition, rendered geodesic for the hyperbolic structure, together with the twist parameters along these curves that measure the way in which the pairs of pants are glued together. (One needs a convention to measure the twists, whereas the length parameters are intrinsic.) In the case of a closed surface of genus g 2, an Euler characteristic argument shows that the number of curves in any pants decomposition of S is 3g 3. Thus, the length and twist parameters of S make a total of 6g 6 parameters, which is indeed the dimension of the Teichmüller space of S . These parameters are the so-called Fenchel–Nielsen parameters. For convex real projective structures on surfaces, there are analogues of the Fenchel– Nielsen parameters associated to a pair of pants decomposition, and we now describe

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them briefly. This parametrization of the space of equivalence classes (the so-called deformation space) of convex real projective structures on a closed surface of genus g 2 was obtained by Goldman [21], who showed that this deformation space, D.S /, is homeomorphic to an open cell of dimension 16.g 1/. In this projective setting, the pants curves are made geodesic with respect to the Hilbert metric (in the coordinate charts, the curves are made affine straight lines). The length of a simple closed geodesic in a given homotopy class is uniquely defined, and this set of lengths is part of the parameters associated to a pair of pants decomposition. However, in the setting of projective structures, the other parameters of the Fenchel–Nielsen coordinates are more complicated to describe that in the case of Teichmüller space; they are described in terms of 3 3 matrices that represent the curves on the surface. In what follows, we shall give an idea of these parameters. We recall that the fundamental group of a surface equipped with a convex real projective structure acts freely and properly discontinuously on the convex set which is the image of the associated developing map. Thus, instead of talking about parameters for the equivalence classes of convex real projective structures on a given closed surface S, one can talk about parameters for the equivalence classes of properly convex open subsets of RP 2 equipped with a properly discontinuous free action of a group isomorphic to the fundamental group 1 .S /. In this way, the deformation space can be viewed as an open subset of the character variety Hom.1 .S/ ! PSL.3; R//=PSL.3; R/ of representations of the fundamental group 1 .S / into the Lie group PSL.3; R/ and where the quotient is by the action of PSL.3; R/ on the space of representations by conjugation. The Teichmüller space T .S / becomes a subspace of D.S /. In the general case of a compact surface S of finite type with n boundary components with negative Euler characteristic .S/, denoting by D.S / the deformation space of convex real projective structures on S , Goldman proves the following (see [21], Theorem 1): Theorem 4.1. The space D.S/ is an open cell of dimension 8 .S /, and the map which associated to each convex projective surface S the germ of its structure near @S is a fibration of D.S/ over an open 2n-cell with fiber an open cell of dimension 8 .S / 2n. We shall explain below the behavior of the projective structures near the boundary. In the rest of this subsection, we give a brief description of Goldman’s parameters. The parameters are based on pairs of pants decompositions of the surface. In what follows, we shall assume that the pairs of pants decompositions that we consider are all geodesic. The parametrization of convex real projective structures on Sg is then done by first describing the structures on individual pairs of pants and then gluing, as we usually do in Teichmüller theory for the parametrization of hyperbolic structures. It turns out however that in the case of general real projective structures, the parameters associated to the pair of pants and to their gluing are more complex than in the case of hyperbolic structures.

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In this way, one is led to the question of understanding convex projective structures on pairs of pants, and this naturally requires the study of convex structures on surfaces with boundary. Recall that a convex projective structure on a surface S is a maximal atlas with values in RP 2 whose transition functions are restrictions of projective automorphisms of RP n . We need to consider convex projective structure on surfaces with boundary. This is also defined by an atlas with values in RP 2 , and we furthermore require that for each open set U in S which is the domain of a chart satisfying U \ @S 6D ; and for each arc in U which is in @S , the image of this arc by the chart map is contained in a projective line of RP 2 . The boundary components therefore become closed geodesics, and we require that each such closed geodesic has a geodesically convex collar neighborhood whose holonomy has distinct positive eigenvalues. Such structures are sometimes called structures “with standard convex projective collar neighborhood”. The image by the holonomy map of a loop representing a boundary component, which is well defined up to conjugacy, is a matrix in SL.3; R/ which has three distinct and positive real eigenvalues whose product is equal to 1. Such a matrix is termed by Goldman positive hyperbolic. The matrix in SL.3; R/ associated to such a boundary component is well defined up to conjugacy and it plays the role of the Fenchel–Nielsen length parameter in the hyperbolic setting. Thus, we have a Fenchel–Nielsen type length parameter associated to a closed geodesic which is two-dimensional in the case of convex projective structures. One can glue projective structures on surfaces with boundary with standard convex projective collar neighborhoods by identifying the collar neighborhoods of boundary components using projective isomorphisms. Goldman proves in [21] that if we glue in this way a finite number of convex projective structures on surfaces with boundary which all have negative Euler characteristic, then the resulting projective structure is convex. On each pair of pants, a convex real projective structure is determined by 3 2 C 2 parameters. Here, 3 is the number of boundary curves of the pair of pants, and near each boundary curve, the projective structure is determined by the two real parameters which we mentioned above. Also, whereas a hyperbolic structure on a pair of pants is completely determined by the “length parameters” associated to the boundary components (which are the lengths of these boundary components, including boundary component of length zero, which correspond to cusps), a general convex real projective structure on a pair of pants is not determined by the sole length parameters associated to the boundary, but there are two more extra real parameters involved, called “twist parameters”. To be more precise, recall first that the projective transformations of RP 2 can be represented by 3 3 matrices. These matrices act on the vector space R3 , and their action on RP 2 is the quotient action. The whole discussion can be reduced then to considerations on matrices and their actions on R3 , or, more precisely, on the set of lines through the origin of that vector space (which is the projective space RP 2 ). Note also that the group acting by matrices on R3 is in fact the group SL˙ .3; R/ and not PSL.3; R/, but we shall not worry here about the difference between the two groups.

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Thus, the holonomy of each closed geodesic on the surface S is positive hyperbolic. In particular, the set of parameters for SL.3; R/-conjugacy classes of such a matrix is an open 2-cell. Goldman, in [21], gives three equivalent sets of coordinates for the space of conjugacy classes of such matrices, and he makes a complete study of the dynamics of the action of a transformation of RP 2 representing such an element. We are now interested in the automorphisms of RP 2 that preserve . For a hyperbolic element of Aut./, the attractive and repulsive fixed points belong to the boundary of the convex set . A lift to of the geodesic in the quotient surface S is preserved by the action of the corresponding affine transformation, and that action is indeed hyperbolic in the sense that it is a translation along that geodesic, with an attractive and a repelling fixed point as endpoints. A positive hyperbolic isometry representing a boundary component of a pair of pants in the decomposition has three distinct eigenvalues 1 > 2 > 3 and it is conjugate to a diagonal matrix with these eigenvalues, 1 , 2 , 3 , in that order, on the diagonal. The action on RP 2 of such a matrix has three fixed points. Using the coordinates of R3 , these points correspond to the line passing through .1; 0; 0/, which is an attracting fixed point, the line passing through .0; 1; 0/, which is a saddle point, and the line passing through .0; 0; 1/, which is repelling. Now the three lines in RP 2 passing through the pairs of such points divide the space into four triangles which are invariant by the action of the given diagonal matrix. Goldman in [21] describes in detail the action of this matrix on these triangles. In the above matrix representation, the first parameter for the projective structure on the pair of pants which is associated to a closed curve which is the boundary of a pair of pants is log 13 , which is the Hilbert metric length of the closed geodesic. The other parameter is 3 log 2 . (We are following Goldman’s exposition, and the factor 2 is a convenient normalization.) These are the six parameters associated to the boundary curves of a pair on pants. The extra two parameters are called by Goldman “interior parameters”. In conclusion, we have the following proposition: Proposition 4.2 (Goldman [21]). The deformation space of RP 2 structures on a pair of pants is an open cell of dimension 8. When one glues two convex projective structures on pairs of pants along simple closed curves, there are two new parameters involved, associated to a simple closed curve C obtained after gluing. One of these parameters is called the twisting parameter, and the other one is called the vertical twist parameter. The two parameters combined are the analogues of the single Fenchel–Nielsen twist parameter of the case of hyperbolic surfaces. In fact, in the case where the projective structure is a hyperbolic structure, the twisting parameter is the usual twist parameter. More precisely, given a diagonal hyperbolic isometry .1 ; 2 ; 3 / representing C , the matrices 2 1 3 2 t 3 e 3 t 0 0 e 0 0 2 6 7 t D 4 0 1 0 5 ; Ot D 4 0 e 3t 0 5 1 0 0 e t 0 0 e 3 t

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commute with the diagonal matrix .1 ; 2 ; 3 / where 1 , 2 , 3 are the three eigenvalues which we described above and the parameters associated to these two matrices are used to glue two structures along C . Here, t is the twisting parameter and O t is the vertical parameter. Suppose that two pairs of pants P1 and P2 are glued along C . Once a projective structure on P1 is given, the two parameters corresponding to the length and 2 for C are already determined for P2 , and the twisting and the vertical parameters are needed to glue along C . In any case, the total parameters are 8.2g 2/, since there are 2g 2 pairs of pants and 8 parameters on each pair of pants. The vertical parameter gives rise to a so-called bulging deformation of the projective structure. This deformation is associated to a measured geodesic lamination on a hyperbolic surface, and it deforms the underlying RP 2 structure of that surface. The bulging deformation is an extension of the earthquake deformation (which itself is the extension of the Fenchel–Nielsen deformation from closed geodesics to measured geodesic laminations) and of the bending deformation in PSL.2; C/ (which is a complexification of the Fenchel–Nielsen deformation) to the context of convex real projective structures. The bulging deformation was introduced by Goldman in [21], who also wrote a recent paper on that subject [22]. Let us recall this deformation. For a closed hyperbolic surface S and a closed geodesic on S, after conjugation, we may assume that 2

./ D t0

e t0 4 D 0 0

3 0 0 1 0 5; t0 0 e

t0 > 0;

as an element of SL.3; R/. Now let t be the one-parameter subgroup generated by . /. If is an ellipse so that S D = .1 .S //, the eigenspaces Rv1 , Rv2 , Rv3 corresponding to e t , 1, e t respectively, define three points C , 0 , in RP 2 where ˙ are the attracting and the repelling fixed points of . / on @, and 0 is outside of . Consider a triangle 4 passing through these three points with left and right vertices corresponding to C , , and top vertex to 0 . Then the dynamics of . / on 4 is from the right vertex to the left and top vertices, and from the top vertex to the left vertex. Inside 4, the orbits of t are arcs of conics tangent to 4, one of which is the segment C of @ from to C . See Figure 1.2 in [21]. The time-t earthquake map of S along is given by the partial right Dehn twist along , which amounts to moving the right hand side of the lifts of by the amount t in . This can be realized by conjugating the action of the right hand side of S n D S1 [ S2 (if is separating) by t , .1 .S1 // h. /i t .1 .S2 // t1 , and correspondingly, using an HNN extension for the non-separating case. Obviously this earthquake deformation does not change the domain . To deform the domain, we perform the bulging deformation which we already mentioned. We want to replace C by another conic which is an orbit of t tangent to 4. This can be

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realized by conjugating the right hand side of S n (if is separating) by 2 1 3 e 3 t 0 0 2 6 7 Ot D 4 0 0 5; e 3t 1 0 0 e 3 t and correspondingly using an HNN extension for the non-separating case. What O t does to the domain is stretching it in the 0 direction, which entails to move the boundary arc C to one of the conics traced by t outside if t > 0, to one inside if t < 0. When t ! ˙1, the domain degenerates to the one containing edges of 4. By Theorem 2.8, the degenerate structures have infinite Hilbert area. Geometrically, the degenerate structure has an infinitely long cylinder attached along . The following is an open question: Question. Is the topological entropy function decreasing to some non-zero number in the above bulging deformation? And what is that number?

4.2 Hitchin’s parametrization Hitchin parametrized a specific component, called the Hitchin component, of the character variety of representations of 1 .S/ in SL.n C 1; R/ [23]. This is the component that contains the representations of hyperbolic structures, and it has some properties which are analogous to those of the Teichmüller space, which is a component of the character variety of representations in SL.2; R/. In the case n D 2 and if a complex structure on S is fixed, Hitchin identified the Hitchin component with the vector space H 0 .S; K 2 ˚ K 3 / of holomorphic quadratic and cubic differentials over the Riemann surface S. Here K is the canonical line bundle of S. In his work, Hitchin used the techniques of Higgs bundles, which are holomorphic vector bundles equipped with so-called “Higgs fields” that appeared in earlier works of Hitchin and of Simpson in their study of Teichmüller space. The Hitchin component for n D 2 (that is, representations in SL.3; R/)) was described again by Labourie [33] and Loftin [36] independently using affine spheres, a notion which we considered in Section 3. Both Labourie and Loftin showed that the space of equivalence classes of convex real projective structures on a closed oriented surface of genus 2 is parametrized by the space of conformal structures equipped with holomorphic cubic differentials. Using the theorem of Riemann–Roch gives another proof of the result of Goldman that this space of equivalence classes is an open cell of dimension 16.g 1/ (cf. Theorem 4.3). The parametrization is by a fiber bundle whose base (Teichmüller space) has dimension 3g 3 and each fiber has dimension 5g 5. This parametrization has also the advantage of equipping the deformation space of convex real projective structure with a natural complex structure. Labourie made in [33] the relation between these cubic differentials and those which appear in Hitchin’s parametrization.

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The proof of the correspondence established by Labourie and Loftin is based on the works of Cheng–Yau [12] that we mentioned above and the work of Wang [51]. It starts with the fact that S can be written as the quotient = where is a bounded convex domain in R2 and a subgroup of SL.3; R/ acting properly discontinuously on , and it makes use of the fact that can be canonically identified with the affine sphere H asymptotic to the open cone C R3 that sits over it, cf. §3. The group can be lifted to a group acting linearly on RnC1 and preserving the affine sphere. The natural projection C ! induces a diffeomorphism between H and . The affine metric associated to the affine sphere H induces a Riemann surface structure on = , and the cubic differential on this Riemann surface is essentially obtained by taking the difference between the Levi-Civita connection of the affine metric on H and the Blaschke connection of H . This follows from the work of Wang [51] who related convex projective structures on a surface to holomorphic data. Wang worked in the setting of affine differential geometry. He gave a condition in terms of the affine metric for a two-dimensional surface to be an affine sphere that involves conformal geometry. See also Labourie [33] and Loftin [37]. The work of Labourie and Loftin has been extended to the case of noncompact surfaces of finite Hilbert volume by Benoist and Hulin [5]. For the relation between cubic differentials and the differential geometry of surfaces, we refer the reader to [39].

4.3 Length spectra as parameters Let M be a manifold equipped with a metric g and let be the set of free homotopy classes of simple closed curves on S . The marked length spectrum of .M; g/ is the function on which associates to each element the infimum of the lengths of closed curves in that free homotopy class. In the above definition, the adjective marked refers to the fact that we are not only considering the set of lengths associated to the elements of , but we keep track of each element of with its associated length parameter. In what follows, all length spectra that we consider are marked, and we shall sometimes denote marked the length spectrum by the term length spectrum. The first author of this chapter showed in [27] that a normalized version of the length spectrum for the Hilbert metric is a set of parameters for convex projective structures on surfaces, up to dual structures. Here a dual structure is the real projective x f0gg. structure induced from the dual cone D fw W hw; vi > 0 for all v 2 (Note that length spectra cannot distinguish between dual structures.) This result is an analogue of the fact that the projectivization of length spectra of hyperbolic structures on a surface of finite type can be used as parameters for the Teichmüller space of that surface. Thurston used these parameters on Teichmüller space in his construction of his boundary whose elements are projective equivalence classes of measured foliations (or, equivalently, of measured laminations), see [49] and [19]. The analogous result for convex projective structures on the surface was also used in order to define a boundary for the deformation space of these convex structures. In fact, Kim’s parameters are

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expressed in terms of logarithms of eigenvalues of hyperbolic 3 3 matrices; see [27] for the details. We shall get back to this matter in §6.

5 Strictly convex manifolds and topological entropy We already mentioned that strictly convex real projective structures have several properties which are similar to properties of negatively curved Riemannian manifolds. Let us give a few examples. Benoist [2] proved the following fundamental theorem about strictly convex projective manifolds, relating the regularity of the boundary of the universal covering to the large-scale geometry of a group that divides it. Theorem 5.1. Let be a divisible convex set, divided by a group . Then the following are equivalent: (1) .; d / is Gromov-hyperbolic; (2) is strictly convex; (3) @ is C 1 ; (4) The geodesic flow on = is Anosov. Let us make a few comments on these statements. Although the Hilbert metric is generally non-Riemannian, there is a natural notion of geodesic flow associated to the projective manifold = , which is the quotient by of the geodesic flow on . The geodesic flow of is defined on the unit tangent bundle T1 ./ ' .T n f0g/=RC . This is the space of pairs .x; / where x is a point in and is a vector of length one based at x (remember that for strictly convex manifolds, the projective lines are exactly the Euclidean lines) with respect to the Hilbert metric. The geodesic flow t W T1 ./ ! T1 ./ is then obtained by moving the pair .x; / unit length in the direction specified by . The quotient flow, denoted by the same name, t W T1 .=/ ! T1 .= /, is the geodesic flow associated to = . We recall that a C 1 flow f t W W ! W on a Riemannian manifold W is said to be Anosov if there is a splitting of the tangent space at each point v into three subspaces Tv W D E u .v/ ˚ E s .v/ ˚ R .X / where E u , E s have positive dimension, E u expanding under the flow, E s contracting and X the generator of the flow. A typical Anosov flow is the geodesic flow on the unit tangent bundle of a manifold of negative curvature. Anosov flows have interesting dynamical properties, for instance, the union of periodic orbits are dense. The definition of an Anosov flow can be transcribed in a straightforward way to the setting of the Finsler manifolds and =.

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We also recall that Gromov hyperbolicity is a property of the large-scale geometry of a metric space. In the case of a geodesic metric space (that is, a metric space in which distances between points are realized by length of curves joining them), the property says that triangles are uniformly ı-thin, that is, that there exists a ı > 0 such that for any triangle, any side is contained in the ı-neighborhood of the two others. See [24] and [15]. It follows from Theorem 5.1 that convex real projective structures on a given closed manifold M are either all strictly convex or they are all non-strictly convex. The topological entropy of a flow t W W ! W on a compact manifold W with distance function d is defined as follows. For all t 0, let d t be the function on W W defined by d t .x; y/ D max d. s .x/; s .y// 0st

for any two points x and y in W . It is not difficult to see that d t satisfies the properties of a distance function. For any > 0 and t 2 R, we consider coverings of W by open sets of diameter less than with respect to the distance d t and we let N. ; t; / be the minimal cardinality of such a covering. The topological entropy of is then defined as htop . / D lim lim sup 1t log N. ; t; / : !0

t!1

For a strictly convex compact manifold M , one can consider the associated geodesic flow on the unit tangent bundle T1 M of M . The strict convexity ensures that in the local projective charts, any geodesic is contained in a projective line, and therefore there are unique geodesics between any two given points. Benoist started the study of this flow in [2] and he showed that it is Anosov and topologically mixing, generalizing properties which were known to hold for the geodesic flow associated to a hyperbolic manifold. We recall that a flow f t on a space M is said to be topologically mixing if for any two open sets A and B in M , there exists a real number t0 such that for every t > t0 , we have f t .A/ \ B 6D ;. An Anosov flow is not necessarily topologically mixing and vice versa. There are several interesting global questions concerning the long-term behaviour of orbits and more generally the dynamics and the ergodic theory of Anosov flows. Some of these questions are considered by Crampon [16] who continued the study initiated by Benoist. Crampon showed the following result on entropy: Theorem 5.2. Let t be the geodesic flow of the Hilbert metric on a strictly convex projective compact manifold M of dimension n. Its topological entropy satisfies htop . t / .n 1/; with equality if and only if M is a hyperbolic manifold with constant curvature 1.

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Some of the results of Crampon are developed in detail in Chapter 7 of this volume [17]. The regularity of the boundary of the convex set plays an important role in the study of the geodesic flow.

6 Geodesic currents on strictly convex surfaces We recalled in §4.3 that it was proved in [27] that if two strictly convex compact real projective manifolds have the same length spectrum with respect to the Hilbert metric, then they are projectively equivalent, up to dual structures. Thus, the length spectrum can be used as a parameter space for real projective structures up to dual structures. The length spectrum, and more generally, the set of geodesics in the manifold, or the space of geodesics in the universal cover equipped with the action of the fundamental group, can be studied from various points of view: dynamical, measure theoretical, etc. We shall consider more particularly the case where the manifold is two-dimensional. We start by recalling the notion of geodesic current. Let S D = be a closed surface of genus at least two equipped with a strictly convex real projective structure. A geodesic current on S is a D 1 .S /-invariant Borel measure on the set [email protected] @ n Diag/=Z2 ; where Diag is the diagonal set and where Z2 acts by interchanging the coordinates. Equivalently, a geodesic current is an invariant transverse measure for the geodesic flow on the unit tangent bundle of S. The unit tangent bundle T1 S of S can be seen as the quotient of the set of bi-infinite geodesics by the action of . The term “geodesic” denotes here the lines in the sense of projective geometry, but we also recall that in the case where is strictly convex (and in fact, it suffices that @ does not contain two affinely independent non-empty open segments), the set of projective geodesics coincides with the set of geodesics (in the sense of distance-minimizing curves) for the associated Hilbert metric. The equivalence between the above two definitions of a geodesic current is based on the fact that from any two distinct points in @ there is a unique projective geodesic in having these points as endpoints. The invariant transverse measures for the geodesic flow of S are also the -invariant invariant transverse measures of the geodesic flow on . There are methods for obtaining invariant measures for geodesic flows and we shall mention the Bowen–Margulis measure below. In the setting of Riemannian manifolds of negative curvature, there are classical methods for constructing transverse measures for geodesic flows, and the theory of such transverse measures is well developed. The methods have been adapted by several authors to the case of Hilbert geometry. For more information on this subject, we refer the reader to the chapter by Crampon in this volume [17].

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Let W 1 .S/ ! SL.3; R/ be a holonomy representation associated to a real projective structure on S. We now explain a method of constructing geodesic currents which originates in the work of Ledrappier [34]. We start with the fact that for a strictly convex projective structure, the boundary @ is C 1 , and we recall the definition of a Busemann cocycle B in this context, cf. [27]. For a fixed base point o 2 and for 2 @, choose a unit speed geodesic ray r.t / with r.0/ D o and r.1/ D , and let B .o; y/ D lim .d .y; r.t // t / : t!1

The map t 7! d .y; r.t// t is non-increasing and bounded from below, therefore the above limit exists. This function is called the Busemann function associated to the geodesic ray r. Since the geodesic ray between a point in and a point in @ is unique, one can think of the Busemann function as a function on @ that depends on the choice of a basepoint in . The function changes by an additive constant when we change the basepoint. When is the unit disc (i.e. the hyperbolic space), the value of B .o; y/ is the signed distance between the two horospheres based at and passing by o and by y. One also talks about the Busemann cocycle B .o; r 1 o/ associated to a point 2 @; the reason for this terminology is the cocycle property expressed in Equation (6.1) below. We refer the reader to [44] for a systematic presentation of several properties of Busemann functions. The group , being isomorphic to the fundamental group of a closed surface of genus 2, is hyperbolic in the sense of Gromov (meaning that its Cayley graph with respect to some – or, equivalently to any – finite generating system, equipped with the word metric, is a hyperbolic geodesic metric space). Such a group has a well-defined Gromov boundary. Such a boundary carries a natural Hölder structure, see [24] and [15]. We shall also use the canonical identification between the two boundary spaces @ and @. Since the surface S is closed, each element 2 1 .S / ' is hyperbolic, that is, it acts on without fixed point leaving invariant a unique geodesic of that space and acting as a translation along that geodesic, with one attractive and one repelling fixed point in @, the two endpoints of the invariant geodesic. The translation length of , denoted by `./, is defined as `./ D inf d .x; .x//: x2

Checking the translation length of a group element is generally a pleasant exercise. The attracting fixed point at infinity of is denoted by C . For each 2 @, consider a Busemann cocycle .; / 7! B .o; 1 o/;

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where o 2 is a fixed base point. Then B .o; 11 01 o/ D B1 .1 o; 01 o/ D B1 .o; 01 o/ C B1 .1 o; o/ D B1 .o; 01 o/ C B .o; 11 o/: Hence, if we set c.; / D B .o; 1 o/, we have c.0 1 ; / D c.0 ; 1 / C c.1 ; /:

(6.1)

This map c W @ ! R is a Hölder cocycle, that is, besides the cocycle property given by the preceding equation, c.; / is a Hölder map for every 2 . The period of c at is defined to be `c ./ D c.; C /: The reason for this terminology is that two Hölder cocycles are cohomologous if and only if they have the same periods (see Theorem 1.a of [34]). In our case, `c ./ D B C .o; 1 o/ D `. / D `. 1 / D `c . 1 /: Hence the set of periods of c is just the length spectrum of the real projective structure with respect to the Hilbert metric. Such a cocycle c (that is, a cocycle satisfying the property `c ./ D `c . 1 /), is said to be even. The exponential growth rate of a Hölder cocycle c is defined as hc D lim sup s!1

log #fŒ 2 Œ W `c . / sg s

where Œ denotes the conjugacy class of . In [34], it is shown that if 0 < hc < 1 then there exists an associated Patterson–Sullivan measure on @, i.e., a probability measure on @ such that d 1 ./ D e hc c. ;/ : d In the case at hand, hc is just the growth rate of lengths of closed geodesics in the Hilbert metric. We recalled the definition of the topological entropy of a flow in §5. Now we need the notion of volume entropy of a metric. This is a measure of the asymptotic growth rate of volumes of metric balls. We recall that the Hilbert metric on the convex set equips this set with a notion of volume, viz., the Hausdorff measure associated to the metric. It is called the Hilbert volume (see §2). This notion of volume descends to the quotients of by properly discontinuous actions of groups of projective transformations. We choose a point x in . The volume entropy, hvol of the metric is then defined as 1 hvol D lim log vol.B.x; r// r!1 r

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if this limit exists, where B.x; r/ denotes the open ball of center x and radius r. The geodesic flow on the unit tangent bundle of a strictly convex real projective surface is C 1 and Anosov. Therefore a classical theorem of Bowen [10] applies to conclude that the topological entropy hvol is equal to the growth rate of lengths of closed geodesics. More precisely, if N.T / denotes the number of closed geodesics in = and htop the topological entropy of the flow, we have N.T / '

e htop T : htop T

It is shown in [16] that 0 < hvol hhyp D 1 where hhyp is the topological entropy of a hyperbolic structure. Now we summarize some general facts about the theory of Hölder cocycles and Patterson–Sullivan measures associated to real projective structures. Let be a properly discontinuous action of a group of projective transformations on a convex set . A family of finite Borel measures fx gx2 defined on @ is said to be an ˛-conformal density (or a conformal density of dimension ˛) for if any two metrics in this family are equivalent (that is, if they have the same measure-zero sets) and if they satisfy the following properties: (1) (2)

dy ./ D e ˛B .x;y/ where B .x; / dx that B .x; x/ D 0 for all x; y 2 ; dx ./ dx

D

dx . 1 / d 1 x

is the Busemann function based at such

for all x; y 2 and 2 ;

(3) x D x for all x 2 and 2 . d

Condition (1) expresses the fact that the Radon–Nikodym cocycles dyx ./ of the family of measures fx gx2 are expressed in terms of the Busemann cocycles of the convex set . We refer the reader to the paper [45] of Patterson and [46] of Sullivan for the original ideas behind the introduction of conformal densities. Given an ˛-conformal density fx gx2 on @, the measure d U.; / D d Ux .; / D e 2ı. /. ;/x dx ./dx ./ is a -invariant measure on @ @ which is independent of x 2 , where .; /x denotes the quantity .; /x D B .x; z/ B .x; z/ for any z in and where ı./ is equal to the volume entropy of the associated Hilbert metric. (The quantity .; /x is also related to the Gromov product of and with basepoint x.) Finally d Udt is a geodesic flow invariant measure on T1 M , where M D = , and it is called a Bowen–Margulis measure. In the classical theory of dynamical systems, the Bowen–

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Margulis measure is an invariant measure for a hyperbolic system (in particular, for an Anosov flow). In the case of a mixing Anosov flow (like the geodesic flow of a compact hyperbolic manifold), the Bowen–Margulis measure maximizes entropy. In this sense, it provides a very good description of the complexity of the dynamical system. There is a construction of the Bowen–Margulis measure using the Patterson– Sullivan techniques and we shall talk about this below. Since we are interested in the group itself rather than , a Patterson–Sullivan measure is d o 1 .x/ D e hc c. ;x/ do for some Hölder cocycle c. Henceforth we will omit the base point o. Two measures and 0 are equivalent if and only if two associated cocycles c and c 0 have the same periods [34]. The Patterson–Sullivan geodesic current associated to the Hölder cocycle c is d mc .x; y/ D e 2hc .x;y/o d.x/d.y/: In [34], it is shown that there is a 1-1 correspondence between Hölder cocycles and Patterson–Sullivan geodesic currents. From now on, we denote by G the set of Patterson–Sullivan geodesic currents defined by Hölder cocycles. The space of geodesic currents, as a space of measures, is equipped with a natural weak topology of convergence on continuous functions. Note that if is a dual structure of , since has the same marked length spectrum as , the cocycle defined by is the same as the one defined by . By associating a Hölder cocycle c D B. / .o; / and the Patterson–Sullivan geodesic current to the projective structure , we obtain a map from the moduli space D of strictly convex structures on S P W D ! G: This map is 1-1 on Teichmüller space and 2-1 elsewhere [27]. The preimage of a point consists in two dual projective structures. The interested reader will find more details in [29].

7 Compactification of the deformation space of convex real projective structures Extensive work has been done on the compactification of Teichmüller space and of Riemann’s moduli space, and more recently on that of the character varieties of representations of fundamental groups of surfaces into various Lie groups. Several possible compactifications and boundary constructions of Teichmüller space have been obtained. Some of them use hyperbolic geometry (Thurston’s compactification, the horofunction boundary construction, etc.), other compactifications use complex structures (Teichmüller’s compactification, the Bers compactification, etc.), and others use

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algebraic geometry (e.g. the Morgan–Shalen compactification). There are also combinatorial compactifications. Each compactification captures some essential properties of the space. It is then natural to try to construct compactifications of moduli spaces of projective structures. A compactification of the deformation space of strictly convex real projective structures on a manifold has been constructed by the first author in [28]. It is related to Hilbert geometry in the sense that it uses the length spectrum of this metric, in the same way Thurston’s geometric compactification Tx of Teichmüller space T uses the hyperbolic length spectrum. The compactification of the deformation space D of strictly convex real projective structures is however more mysterious than Thurston’s compactification of Teichmüller space. Loftin developed in [37] another compactification of D which is based on holomorphic cubic differentials on degenerate (noded) surfaces. Since Teichmüller space T is a subspace of D, one can naturally expect that the compactification of D obtained by using the length spectrum should include the Thurston boundary consisting of projective measured laminations or, equivalently, of actions of the fundamental group of the surface on R-trees. The compactification in [28] is done in terms of the geometry of X D SL.3; R/=SO.3/ by regarding a real projective structure as a holonomy representation W 1 .S / ! SL.3; R/. A boundary point of the compactification is then either a reducible representation or a limit representation which acts on the asymptotic cone of X. Specifically, if i W 1 .S / ! SO.2; 1/ SL.3; R/ is a sequence of hyperbolic structures which converges to a projective lamination in Thurston’s compactification, the limit action of i converges to the affine building R T , where T is the real tree dual to the measured lamination . In this way, the space of projective measured laminations appears naturally as a subset of the boundary of the deformation space of strictly convex real projective structures. On the other hand, the geodesic currents introduced in Section 6 are natural generalizations of measured laminations. Hence we may also use the space G of geodesic currents to compactify D. On G , one can define an intersection form i W G G ! RC ; so that i.˛; ˇ/ is the total mass of the product measure ˛ ˇ on the set of pairs of transversal geodesics on S . This is a natural generalization of the intersection form i.1 ; 2 / when 1 and 2 are measured geodesic laminations. The geodesic current m associated to the Hilbert metric m of a strictly convex real projective structure is properly normalized as: i.m ; m / D

Area.m/; 2

like in the hyperbolic case [8]. This compactification by geodesic currents satisfies the following (see [29]):

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Theorem 7.1. The two compactifications of the space of strictly convex real projective structures on a closed surface S, the one defined via the marked length spectrum and the other via geodesic currents, are naturally homeomorphic. Sketch of proof. Look at the following diagram: L

/ P .R / DE OC EE EE EE E ` P P EEE E" PG,

where is the set of conjugacy classes of elements in 1 .S / and L is the marked length spectrum map relatively to the Hilbert metric. The map ` is defined using the periods of the corresponding Hölder cocycles. The diagram commutes, and the map ` is injective, hence the compactifications of the images of L and P P are homeomorphic. Corollary 7.2. Thurston’s compactification Tx by projective measured laminations is contained in P P .D/. Proof. When restricted to T , if i mi ! ˛ in G and the projective structures mi diverge, then by the properness of the map T ! G , i ! 0 and i.˛; ˛/ D lim i.i mi ; i mi / D lim 2i i.mi ; mi / D lim 2i 2j .S /j D 0: i !1 i!1 i!1 2 Hence ˛ must be a measured lamination since measured laminations are characterized by self-intersection number being zero. In this way, we see again Thurston’s compactification sitting inside the compactification of real projective structures. We believe that whenever the Hilbert metric area of a diverging sequence is bounded above, the sequence will converge to a projective measured lamination. Acknowledgement. The first author gratefully acknowledges the partial support of NRF grant (2010-0024171). The second author is supported by the French ANR project FINSLER.

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[10] R. Bowen, Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30. [11] E. Calabi, Complete affine hyperspheres I. In Symposia Mathematica, Vol. X, Academic Press, London 1972, 19–38. [12] S. Cheng and S. Yau, On the regularity of the Monge–Ampère equation [email protected] [email protected] @sxu / D F .x; u/. Comm. Pure Appl. Math. 30 (1977), 41–68. [13] S.-S. Chern, On the curvatura integra in a Riemannian manifold. Ann. of Math. (2) 46 (1945) 674–684. [14] B. Colbois and C. Vernicos, L’aire des triangles idéaux en géométrie de Hilbert. Enseign. Math. (2) 50 (2004), no. 3, 203–237. [15] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes: Les groupes hyperboliques de Gromov. Lecture Notes in Math. 1441, Springer-Verlag, Berlin 1990. [16] M. Crampon, Entropies of strictly convex projective manifolds. J. Mod. Dyn. 3 (2009), no. 4, 511-547. [17] M. Crampon, The geodesic flow of Finsler and Hilbert geometries. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 161–206. [18] Ch. Ehresmann, Sur les espaces localement homogènes, Enseign. Math. 35 (1936), 317–333. [19] A. Fathi, F. Laudenbach, and V. Poénaru (eds.), Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979). [20] D. Johnson and J. Millson, Deformation spaces associated to compact hyperbolic manifolds. In Discrete groups in geometry and analysis (New Haven, Conn., 1984), Progr. Math. 67, Birkhäuser, Boston 1987, 48–106.

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[21] W. Goldman, Convex real projective structures on compact surfaces. J. Differential Geom. 31 (1990), 791–845. [22] W. Goldman, Bulging deformations of RP 2 -manifolds. Preprint, arXiv:1302.0777. [23] N. Hitchin, Lie groups and Teichmüller space. Topology 31 (1992), 449–473. [24] M. Gromov, Hyperbolic groups. In Essays in group theory, Publ. Math. Sci. Res. Inst. 8, Springer-Verlag, New York 1987, 75–263. [25] V. Kac and E. Vinberg, Quasi-homogeneous cones. Mat. Zametki 1 (1967), 347–354; English transl. Math. Notes 1 (1968), 231–235. [26] M. Kapovich, Convex projective structures on Gromov-Thurston manifolds. Geom. Top. (2007), 1777–1830. [27] I. Kim, Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds. J. Differential Geom. 58 (2001), 189–218; erratum ibid. 86 (2010), 189. [28] I. Kim, Compactification of strictly convex real projective structures, Geom. Dedicata. 113 (2005), 185–195. [29] I. Kim, Deformation of strictly convex real projective structures via geodesic currents, Patterson–Sullivan measures, and cubic differentials. In preparation. [30] J.-L. Koszul, Variétés localement plates et convexité. Osaka J. Math. 2 (1965), 285–290. [31] N. H. Kuiper, Sur les surfaces localement affines. In Géométrie différentielle, Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, CNRS, Paris 1953, 79–87. [32] N. H. Kuiper, On convex locally-projective spaces. In Convegno Internazionale di Geometria Differenziale (Italia, 20–26 Sett. 1953), Edizioni Cremonese, Roma 1954, 200–213. [33] F. Labourie, Flat projective structures on surfaces and cubic holomorphic differentials. Pure Appl. Math. Q. 3 (2007), 1057–1099. [34] F. Ledrappier, Structure au bord des variétés à courbure négative. In Semin. Theor. Spectr. Geom. 13, Année 1994–1995, Université de Grenoble I, Institut Fourier, Saint-Martind’Hères 1995, 97–122. [35] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations. In Contributions to analysis, a collection of papers dedicated to Lipman Bers, Academic Press, New York 1974, 245–272. [36] J. Loftin, Affine spheres and convex RP 2 manifolds, Amer. J. Math. 123 (2) (2001), 255–274. [37] J. Loftin, The compactification of the moduli space of convex RP 2 -surfaces I. J. Differential Geom. 68 (2004), 223–276. [38] J. Loftin, Survey on affine spheres. In Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM) 13, Internat. Press, Somerville, MA, 2010, 161–191. [39] J. Loftin and I. McIntosh, Cubic differentials in the differential geometry of surfaces. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume V, European Mathematical Society, to appear. [40] L. Marquis, Les pavages en géométrie projective de dimension 2 et 3, Thèse, Université de Paris-Sud, Orsay 2009.

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[41] L. Marquis, Surface projective convexe de volume fini. Ann. Inst. Fourier 62 (2012), no. 1, 325–392. [42] L. Marquis, Around groups in Hilbert geometry. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 207–261. [43] K. Nomizu and T. Sasaki, Affine differential geometry. Cambridge Tracts Math. 111, Cambridge University Press, Cambridge 1994. [44] A. Papadopoulos, Metric spaces, convexity and nonpositive curvature. 2nd edition, IRMA Lect. Math. Theor. Phys. 6, European Mathematical Society, Zürich 2014. [45] S. J. Patterson, Lectures on measures on limit sets of Kleinian groups. In Analytical and geometric aspects of hyperbolic space, Symp. Warwick and Durham, 1984, London Math. Soc. Lecture Note Ser. 111, Cambridge University Press, Cambridge 1987, 281–323. [46] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. [47] D. Sullivan and W. P. Thurston, Manifolds with canonical coordinate charts: some examples. Enseign. Math. (2) 29 (1983), no. 1–2, 15–25. [48] W. P. Thurston, The geometry and topology of three-manifolds. Mimeographed notes, Princeton University, 1976. [49] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417–431. [50] C. Walsh, Gauge-reversing maps on cones, and Hilbert and Thompson isometries. Preprint, arXiv:1312.7871. [51] C. P. Wang, Some examples of complete hyperbolic affine 2-spheres in R3 . In Global differential geometry and global analysis (Berlin 1990), Lecture Notes in Math. 1481, Springer-Verlag, Berlin 1991, 271–280.

Chapter 12

Weil–Petersson Funk metric on Teichmüller space Hideki Miyachi, Ken’ichi Ohshika and Sumio Yamada

Contents 1 2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Funk metric in Rd and its representations . . . . . . . . . . Weil–Petersson geometry . . . . . . . . . . . . . . . . . . . . . . 3.1 Teichmüller space . . . . . . . . . . . . . . . . . . . . . . 3.2 Weil–Petersson metric and geodesic completions of .T ; d / . 3.3 Tx as a convex subset in D.Tx ; / . . . . . . . . . . . . . . . 3.4 The Weil–Petersson Funk metric F2 . . . . . . . . . . . . 4 Properties of the Funk metric F2 . . . . . . . . . . . . . . . . . . 4.1 Density of strata in the Weil–Petersson visual sphere . . . . 4.2 Non-degeneracy of the weak metric F2 . . . . . . . . . . . 4.3 Weil–Petersson Funk topology and Weil–Petersson topology 4.4 Mapping class group invariance and the translation distances 5 Three Funk-type metrics on Teichmüller spaces . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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339 340 341 341 342 343 344 345 345 347 347 348 348 350

1 Introduction Given a closed surface †g of genus g > 1, the Deligne–Mumford–Mahler compactified moduli space Mxg of complex structures on † has been actively studied. It is a projective variety, that is, an algebraic subset of the complex projective space. There is a natural metric called Weil–Petersson metric on both moduli and Teichmüller spaces. The Weil–Petersson completion Tx of the Teichmüller space T is the space whose quotient space by the action of the mapping class group of † is the compactification of the moduli space. In 1937, Teichmüller showed that the space T is diffeomorphic to an open ball of dimension 6g 6, the number of parameters first claimed by Riemann. By going from moduli space to Teichmüller space, we lose the structure of projective variety, and we gain instead the structure of a “convex” set. Indeed the Teichmüller space Tg is a very intriguing and important convex set naturally arising from the theory of moduli of Riemann surfaces.

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In this chapter, we will make the Weil–Petersson convex structure precise, and will introduce the Funk-type distance function, called the Weil–Petersson Funk metric [30]. In particular, this approach provided an answer to a problem raised by A. Papadopoulos [13] “Realize Teichmüller space as a bounded convex set somewhere and study the Hilbert metric on it”, even though the resulting convex set is not bounded, where the Hilbert metric in question is the arithmetic symmetrization of the Weil– Petersson Funk metric. Additionally, a set of new results is presented concerning the properties of the Weil–Petersson Funk metric, juxtaposed with another set of results recently obtained by K. Fujiwara [9]. Those properties are interesting as they suggest further interactions between the Weil–Petersson geometry and the curve complex geometry. In the last section, we will give a short survey of other Funk-type metrics defined on Teichmüller spaces, further emphasizing the convex-geometric approach to Teichmüller theory. We hope that this will demonstrate the ubiquity of the Funk-type metrics associated with convex bodies naturally arising in various contexts. The authors thank Scott Wolpert for his interest and suggestions on this work. We also thank Athanase Papadopoulos and Marc Troyanov for inviting us to contribute to this volume as well as for their valuable comments.

2 The Funk metric in Rd and its representations Let be an open bounded convex subset in a Euclidean space .Rd ; d / where d is the standard Euclidean metric. We set the presentation in [14] as our reference for the Funk and Hilbert metrics of , and we also refer to the first part of the paper [29] and Chapter 2 of the present volume. There are three different descriptions of the Funk metric. The first one is the original definition: d.x; b.x; y// ; F1 .x; y/ D log d.y; b.x; y// where for x 6D y in , the point b.x; y/ is the intersection of the boundary @ with the Euclidean ray fx Ct xy W t > 0g from x though y and where xy is the unit tangent vector in Rd pointing from x to y. When x D y, we set F .x; y/ D 0. The second description is the variational interpretation of the above value using the geometry of supporting hyperplanes: F2 .x; y/ D sup log 2P

d.x; / ; d.y; /

where P is the set of all supporting hyperplanes of . This is given in [30]. In the literature, a variational characterization of Hilbert metrics appears in the work of Birkhoff and Nussbaum ([12]), where the supporting hyperplanes are treated as elements of the dual space, but the Funk metric was not mentioned there.

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Finally, the Finsler structure p;x ./ is given by the following function (the Minkowski functional) on vectors in each tangent space to at x: p;x ./ D sup

2P

h .x/; i ; d.x; /

where is the unit vector in Tx perpendicular to, and directed toward . This is a weak norm on each tangent space which is defined in such a way that the Funk distance is described as the infimum of length of curves: Z b p;.t / .P .t // dt; F3 .x; y/ D inf

a

the infimum being taken over all the piecewise C 1 curves satisfying .a/ D x and .b/ D y. For any convex domain Rd , the three quantities F1 .x; y/, F2 .x; y/, F3 .x; y/ are all equal to each other, and we set F .x; y/ WD F1 .x; y/ D F2 .x; y/ D F3 .x; y/ for every x and y in .

3 Weil–Petersson geometry 3.1 Teichmüller space Let † be a compact smooth surface with negative Euler characteristic. The Teichmüller space T D T .†/ of † is the quotient space of the space M1 of finite-area metrics of constant curvature 1 on † by the group Diff 0 † of diffeomorphisms of † homotopic to the identity. Each metric determines a conformal structure on † and the Teichmüller space of † is canonically identified with the space of marked conformal structures on the interior of †. The Weil–Petersson metric on the Teichmüller space T is the L2 metric on the surface † for deformation tensors of the hyperbolic metric G, Z hh1 ; h2 iWP D hh1 .x/; h2 .x/iG.x/ dG .x/: †

Here, the deformation tensors h1 , h2 are traceless and divergence-free with respect to G, which insures that the deformation of G preserves the constant curvature condition as well as the L2 -perpendicularity to the diffeomorphism fibers. We denote by dWP .p; q/ the Weil–Petersson distance between the points p and q in T induced from the Riemannian structure of the Weil–Petersson metric. It is known ([20], [7]) that .T ; dWP / is an incomplete metric space. We denote by .Tx ; dWP / the metric completion of .T ; dWP /.

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In [27], [28], the third author has studied the geometry of the Weil–Petersson completed Teichmüller space Tx of the surface †. It was shown there that the metric completion Tx is a CAT(0) space [2], that is, a non-positively curved metric space in the sense of Alexandrov. The completion Tx has a stratification by its boundary sets. We will discuss the stratification in the next section. The above definition of the Weil–Petersson metric came later; it was introduced by Fischer and Tromba in the 1970s. Historically the tangent space at p 2 T , consisting of deformation tensors of the hyperbolic metric p, is canonically identified with the space of harmonic Beltrami differentials ²

HB.p/ D

³

'N j ' is a holomorphic quadratic differential for p ; p

where p is the conformal factor of the hyperbolic metric representing p with respect to an isothermal coordinate system. The Weil–Petersson metric is a Riemannian metric on T defined by the scalar product Z hv1 ; v2 ip D v1 .z/v2 .z/ x .z/ dxdy; †

for v1 ; v2 2 HB.p/ and z D x Ciy. The Weil–Petersson metric is Kähler with respect p to the canonical complex structure on T , which corresponds to multiplication by 1 on holomorphic quadratic differentials.

3.2 Weil–Petersson metric and geodesic completions of .T ; d/ In the Weil–Petersson geometric setting, given a closed surface † of genus g > 1, the Weil–Petersson metric completion ([27], [28]) Tx is identified with a stratified space known as the augmented Teichmüller space [1], [11], S Tx D 2C .†/ T ; where C .†/ is the curve complex of †, which can be identified as the index set of nodal surfaces † specifying the location of the nodes. The stratum T is the Teichmüller space of the nodal surface † . Let be the set of homotopy classes of non-contractible, non-peripheral simple closed curves on †, i.e. the vertex set of C.S /, namely D f 2 C .†/ j jj D 1g: Here j j stands for the number of nodes in † . Hence, each element in represents a single node. In [27], [28], the third author showed that each frontier stratum Tx is embedded totally geodesically into Tx , which is a natural extension of a preceding result [21] by S. Wolpert which states that T is Weil–Petersson geodesically convex. In [29], a new space was introduced which can be viewed as a Weil–Petersson geodesic completion, called the Teichmüller–Coxeter complex D.Tx ; /. The space is a development of the original space Tx by a Coxeter

Chapter 12. Weil–Petersson Funk metric on Teichmüller space

343

group W generated by reflections across the frontier strata fTx g, a quotient space of W Tx by an equivalence relation. Here stands for a simple complex of groups [2]. It was shown by Wolpert [23] that two intersecting strata of the same dimension meet at a right angle (in the sense of the Alexandrov angle between Weil–Petersson geodesics), making the development D.Tx ; / a so-called cubical complex [2], [8]. This feature then is used to show that the development is also a CAT(0) space. The Coxeter complex setting allows to view Teichmüller space as a convex set in an ambient space D.Tx ; /, bounded by a set of complex-codimension one “supporting hyperplanes” fD.Tx ; / j 2 g of the frontier stratum fTx g with each representing a single node. Every boundary point is contained in at least one of the set of supporting hyperplanes fD.Tx ; / j 2 g. In this picture, each D.Tx ; / is a totally geodesic set, metrically and geodesically complete, and when D.Tx1 ; / and D.Tx2 ; / intersect along D.Tx1 [2 ; /, they meet at a right angle. One can view the translates of fD.Tx ; / j 2 g by the action of the Coxeter group W as forming a right-angled grid structure in D.Tx ; /, whose lattice points are the orbits by the Coxeter group W of the set fTx j j j D 3g 3g with indexing the maximal set of nodes on the surface.

3.3 Tx as a convex subset in D.Tx ; / In this setting, for each in , one can consider a half space, namely the set H in D.Tx ; /, containing Tx and bounded by D.Tx ; /. We note the fact obtained by Wolpert [23] that the Weil–Petersson metric completion Tx is the closure of the convex hull of the vertex set fTx j j j D 3g 3g, which suggests an interpretation of the Teichmüller space as a simplex. We can summarize the above discussion as S T Tx D 2 H with @Tx D.Tx ; /; where every boundary point b 2 @Tx belongs to D.Tx ; / for some in . Each half space H is bounded by the “supporting hyperplane” D.Tx ; /. 3.3.1 Geometry of the nearest point projection. We have the following general facts about the nearest point projection to a complete convex subset of a CAT.0/ space. Proposition ([2], II.2.4). Suppose that X is a CAT.0/ space, and that C is a complete convex subset of X. Then there exists a well-defined nearest point projection W X ! C satisfying the following. (1) For each x 2 X, the projection .x/ of x satisfies d.x; .x// D d.x; C /. (2) If x 0 lies on the geodesic segment Œx; .x/, then .x 0 / D .x/. (3) For x 62 C and y 2 C and y ¤ .x/, then †.x/ .x; y/ =2, where † denotes the Alexandrov angle.

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(4) The projection map x 7! .x/ is a retraction of X to C . We are interested in the case where X D Tx and C D Tx , as each Tx lies in Tx as a complete convex set. We use the symbol … to denote the nearest point projection W Tx ! Tx . Actually there is another nearest point projection, which we also denote by … from D.Tx ; / to D.Tx ; /. The former projection … W Tx ! Tx is the restriction of … W D.Tx ; / ! D.Tx ; /. It follows from the proposition that for each point x 2 Tx D.Tx ; /, there exists a nearest point projection … .x/ 2 D.Tx ; /, and that the Weil–Petersson geodesic x… .x/ meets D.Tx ; / perpendicularly, its length uniquely realizing the distance inf y2Tx d.x; y/ D d.x; … .x//. Also note that when x 2 T Tx , the foot … .x/ lies in T Tx , due to the fact that the frontier sets intersect perpendicularly in terms of the Weil–Petersson Alexandorov angle. In other words, when the projection is restricted to T , as … jT W T ! Tx , we have .… jT /1 ŒT D T for any 2 , and .… /1 ŒTx n T \ T D ;. It follows that for x 2 T and y 2 Tx with y ¤ … .x/, we have †… .x/ .x; y/ D =2, in light of the statement .3/ of the above proposition. It was shown in [9] that the Weil–Petersson geodesic segment x… .x/ is also the F2 -geodesic of infinite F2 -length provided that the stratum Tx is the closest to x among all the strata.

3.4 The Weil–Petersson Funk metric F2 We now transcribe the Euclidean Funk geometry as well as its compatible Finsler structure in the previous section to the Weil–Petersson setting. We exhibited three equivalent ways of writing down the Funk distance, which we called F1 ; F2 and F3 . In the Weil–Petersson setting, these definitions a priori differ from each other, and they are related by inequalities. We define the Weil–Petersson Funk metric F2 on T as F2 .x; y/ D sup log 2

d.x; Tx / ; d.y; Tx /

where d is the Weil–Petersson distance defined on Tx . That this is indeed a metric will be shown below, and in particular it will be convenient for that to use the formula given in Equation 3.1 below. The fact that F2 satisfies the triangle inequality will be proved using that second formula, and the fact that it is non-negative and non-degenerate is the content of Theorem 4.4. In order to make the analogy with the Euclidean setting more obvious, and in order to make clearer the viewpoint that Teichmüller space T is a convex body in the ambient space D.Tx ; / with supporting hypersurfaces fD.Tx ; g2 , we can instead define the metric as d.x; D.Tx ; // F2 .x; y/ D sup log : (3.1) d.y; D.Tx ; // 2

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Chapter 12. Weil–Petersson Funk metric on Teichmüller space

This follows immediately from the content of Subsection 3.3.1. Note that F2 .x; y/ satisfies the triangle inequality, for F2 .x; y/ C F2 .y; z/ D sup log 2

d.x; D.Tx ; // d.y; D.Tx ; // C sup log d.y; D.Tx ; // 2 d.z; D.Tx ; //

sup log 2

D sup log 2

d.y; D.Tx ; // d.x; D.Tx ; // C log d.y; D.Tx ; // d.z; D.Tx ; //

d.x; D.Tx ; // D F2 .x; z/: d.z; D.Tx ; //

As in the Euclidean case, the equality holds when the stratum D.Tx ; / achieving the value of the supremum for x and y coincides with the one for y and z. However, those strata are no longer characterized in terms of Weil–Petersson geodesic rays, and this prevents us from stating that the Weil–Petersson geodesics are Funk distance minimizers. Recall that in the Euclidean setting, the relevant supporting hypersurfaces are the ones containing the boundary points b.x; y/ and b.y; z/ where the rays hit the boundary set @. We also remark that there is no definite statement for the convexity of the Funk distance F2 .x; y/ in the y variable, unlike the Euclidean counterpart.

4 Properties of the Funk metric F2 4.1 Density of strata in the Weil–Petersson visual sphere For x 2 T , we denote by ;x the shortest Weil–Petersson geodesic from x to Tx . Recall that the endpoint of the Weil–Petersson geodesic segment is the nearest point projection … .x/ of x. We call … .x/ 2 Tx the foot of ;x . Theorem 4.1 (Density). For any point x 2 T , the set of Weil–Petersson geodesic rays f ;x g 2 is dense in the unit tangent sphere at x, where each ray, parameterized by arc-length, is uniquely represented by a unit tangent vector. Proof. The following argument is due to Wolpert [24]. Let c W Œ0; 1/ ! Tx be any Weil–Petersson geodesic ray issued from x. We shall show that c can be approximated arbitrarily closely by rays i ;x for a sequence of strata fi g with i 2 . An arc-length-parameterized Weil–Petersson geodesic ray in the Teichmüller space is said to be forward geodesic flow transitive, transitive for short, if its projection to the moduli space Mg lifted to its unit tangent bundle U TMg has dense image. By Burns–Masur–Wilkinson [5], any Weil–Petersson geodesic ray can be approximated by transitive geodesics. Let fci W Ji ! T g be a sequence of arc-length parameterized

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geodesics converging uniformly on every compact set to the geodesic ray c. There is a decreasing sequence of positive numbers i converging to zero such that both the distance between ci .0/ and c.0/ and the distance between the unit tangent vector of ci at ci .0/ and that of c at c.0/ are less than i , the latter distance being measured using the parallel transport along the geodesic connecting ci .0/ and c.0/. Since each ci is transitive, there exist a sequence of diverging positive numbers fti g, y 2 T and 2 , such that the tangent vector of ci at ti is sufficiently close to a translate of the tangent vector v0 of ;y at y by an element gi of the mapping class group Mod.†/ so that the geodesic arc i connecting ci .ti / to gi .… .y// (D …gi ./ .gi y/) forms an angle greater than =2 i with Tgi . / and the tangent vectors at ci .ti / of i and of ci form an angle less than i . Now, consider the geodesic arc connecting x and ci .ti /, which we denote by ˛i . Since the tangent vectors of c at c.0/ D x and of ci at ci .0/ are within distance i , by the CAT.0/ property, there is a sequence ıi converging to 0 such that the angle formed by ˛i and ci at ci .ti / is less than ıi , which implies that the angle formed by ˛i and i at ci .ti / is less than

i C ıi . On the other hand, as ci converges to c uniformly on any compact set, the CAT.0/ geometry, applied to the geodesic triangle 4.x; ci .ti /; c.i // where c.i / is the projection of ci .ti / on c, implies that the angle between the tangent vectors of c and of ˛i at x is less than ıi . We consider the quadrilateral formed by ˛i , i , gi . /;x and the geodesic segment …gi ./ .x/ …gi ./ .gi y/ on Tgi ./ . Then using the CAT.0/ property again, we see that the angle formed by gi ./;x and ˛i at x is less than i C ıi . Therefore, the angle formed by c and gi ./;x at x is less than i C 2ıi . This shows that c is approximated by rays in f ;x g2 arbitrarily closely. Remark 4.2. K. Fujiwara in [9] independently obtained a similar result, where it was shown that the set of Weil–Petersson geodesic rays ;x translated by the pseudoAnosov elements of the mapping class group is dense in the unit tangent sphere. Remark 4.3. Instead of taking the nearest point projections … W T ! Tx with 2 (or equivalently jj D 1), one can consider the projections … W T ! Tx with j j D 3g 3. For such a , Tx Tx , which is equal to T , is a zero-dimensional stratum representing a nodal surface † with maximal number of nodes, so that the nodal surface is a union of thrice-punctured spheres. A result by J. Brock [4] states that in the Weil–Petersson visual sphere VT .x/, of each point x 2 T , the Weil–Petersson geodesic rays ;x are dense. Here, the visual sphere VT .p/ at p is the set of all finite and infinite unit-speed geodesic rays emanating from p, and it is homeomorphic to the unit tangent sphere at p, where the correspondence is by assigning the unit tangent vector at p to each geodesic. As each … W T ! T is a projection to a point, the proof follows much more readily than that of the density theorem we have stated, which uses the heavy machinery of the Weil–Petersson geodesic flow ergodicity.

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Chapter 12. Weil–Petersson Funk metric on Teichmüller space

4.2 Non-degeneracy of the weak metric F2 Having the knowledge of the density of rays ;x with 2 in the visual sphere VT .x/, we can proceed to show: Theorem 4.4. The Funk metric F2 is non-negative and non-degenerate. Proof. Given a pair of distinct points x; y 2 T , we apply Theorem 4.1 to the geodesic arc connecting them. We then see that for any > 0, there is a simple closed curve 2 such that d.y; ;x / < . Take y 0 2 ;x with d.y; y 0 / < . Note that … .y 0 / D … .x/. Since the development D.Tx / is CAT.0/, d.… .x/; … .y// D d.… .y 0 /; … .y// d.y 0 ; y/ : Therefore, d.x; y/ C d.y 0 ; Tx / d.x; y 0 / C d.y 0 ; Tx / d.x; Tx / > : d.y 0 ; Tx / C 2 d.y 0 ; Tx / C 2 d.y; Tx / Therefore, if we take sufficiently small so that 3 < d.x; y/, we have F2 .x; y/ log

d.x; y/ C d.y 0 ; Tx / d.x; Tx / log > 0; d.y; Tx / 2 C d.y 0 ; Tx /

(4.1)

and the statement is proved.

4.3 Weil–Petersson Funk topology and Weil–Petersson topology It is well known that on the Teichmüller space T , the metric topologies (namely the topology whose open sets are realized as unions of the metric balls) with respect to the Teichmüller distance and the Weil–Petersson distance are equivalent. As for the comparison between the Weil–Petersson metric topology and the Weil–Petersson Funk metric topology, we have the following qualitative statement. Proposition 4.5. The identity map W .T ; d / ,! .T ; F2 / is locally Lipschitz. Proof. Let K be a compact set in T . Then, there is an M D M.K/ > 0 such that d.x; Tx / M for all x 2 K and all 2 . Let x; y be points in K. For any 2 , we have d.y; Tx / C d.x; y/ d.x; y/ d.x; y/ d.x; Tx / ; log log 1 C C log M d.y; Tx / d.y; Tx / d.y; Tx / for some constant C D C.K/ > 1. By taking supremum over 2 , we have C d.x; y/. Therefore the identity map is C =M -Lipschitz in K. F2 .x; y/ M

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4.4 Mapping class group invariance and the translation distances By definition, the Weil–Petersson Funk metric, F2 is invariant under the mapping class group action as the mapping class group Map.†/ is a subgroup of the permutation group of : F2 .x; y/ D F2 .gx; gy/ for all g 2 Map.†/. Namely each element g of Map.†/ is an F2 -isometry. Thurston [18] and Bers [3] classified the elements of the mapping class group into three categories; elliptic, reducible, and irreducible/pseudo-Anosov. For a metric space .X; d / and an element g of the isometry group Isom.X /, we define the translation length ı.g/ by ı.g/ D inf d.x; gx/: x2X

Fujiwara showed [9] that for each pseudo-Anosov mapping class g and a point x 2 T , there exists E > 0 such that for all n ¤ 0, log jnj E F2 .x; g n x/ log jnj C E; while F2 .x; g n x/ is bounded if g is a Dehn twist. Furthermore if g is a pseudo-Anosov mapping class, there exists Q > 0 such that for all n ¤ 0, log jnj Q ı.g n / log jnj C Q: For sufficiently large n, ı.g n / > 0 and the infimum is achieved by some x 2 T . If is a Dehn twist, ı.g n / D 0 for each n.

5 Three Funk-type metrics on Teichmüller spaces In this section, we make a comparison among three metric structures defined on Teichmüller spaces. The first is the Teichmüller metric, which is defined as follows. Definition 5.1. Let ŒG1 and ŒG2 be two conformal structures (uniformized by hyperbolic metrics Gi ) on †. The Teichmüller distance between ŒG1 and ŒG2 is given by 1 dT .ŒG1 ; ŒG2 / D inf log K; 2 f where the infimum is taken over the set of all K > 0 such that there exists a K-quasiconformal homeomorphism f W .†; ŒG1 / ! .†; ŒG2 / that is isotopic to the identity map. Here the Teichmüller space is regarded as the space of conformal structures, where the uniformization is not relevant.

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Kerckhoff [10] showed that the Teichmüller distance can be alternatively defined as dT .ŒG1 ; ŒG2 / D

Ext ŒG1 . / 1 ; sup log 2 2 Ext ŒG2 . /

where Ext ŒG ./ is the extremal length of the homotopy class of simple closed curves in †. Recall [15] that the extremal length of is defined as 1=Mod† . / where Mod† . / is the supremum of the moduli of the topological cylinders embedded in † with core curves in the class . Secondly, Thurston’s asymmetric metric [19] is defined as follows. Definition 5.2. Let G1 and G2 be the two hyperbolic metrics on †. A distance function can be defined as ` .G1 / T .G1 ; G2 / D sup log : ` .G2 / 2 It is called Thurston’s asymmetric metric. This quantity was shown by Thurston to be equal to the following number L.G1 ; G2 / D inf Lip./; Id†

where the infimum is taken over all diffeomorphisms in the isotopy class of the identity, and Lip./ is the Lipschitz constant of the map : dG2 ..x/; .y// : dG1 .x; y/ x¤y2†

Lip./ D sup

For this reason, the quantity T .G1 ; G2 / is sometimes called Thurston’s Lipschitz metric. Thurston in his paper ([19], Chapter 4) emphasizes the underlying convex geometry for the metric. In particular, the space of measured geodesic laminations is embedded into the cotangent space TG T for a fixed point G in T as the boundary set of a convex body, where the embedding is given by the differential of the logarithm of the geodesic length function of measured geodesic laminations d log length W PL.†/ ! TG T ; where the map only registers the projective classes of measured geodesic laminations, for one is taking the logarithmic derivative of the length. Now recall the new metric we have introduced above: The Weil–Petersson-Funk metric F2 on T is defined as F2 .x; y/ D sup log 2

for x and y in T .

d.x; Tx / d.y; Tx /

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Now the analogy among the Teichmüller metric, the Thurston metric and the Weil– Petersson-Funk metric is clear: For each of them we have an embedding ‚ W T ! RC where the target space has a weak metric

²

³

x d.x; y/ D sup max log ; 0 y

for x D .x /2 , and each of the three Funk-type metrics is the pulled-back metric ‚ d defined on T T . Note that the weak metric d is the Funk metric for the convex set RC in R . Lastly we remark that the infinite-dimensional orthant RC with its Hilbert distance d.x; y/ is known to be isometric to an infinite-dimensional weak Minkowski space .R ; p/, namely a linear space with a weak Minkowski norm p, where the map is given by fx g 2 7! flog x g2 . (Compare also with Theorem 6.1 of Chapter 4 in this volume for a similar result in Hilbert geometry.) Hence the three different embeddings of the Teichmüller space T into the orthant RC induce isometric embeddings of the Teichmüller space into the infinite-dimensional weak Minkowski space .R ; p/ with respect to the Hilbert metrics obtained by taking the arithmetic symmetrization of the three Funk-type metrics. This observation provides an interesting starting point for a comparative study of the three Funk-type metrics on the Teichmüller space: the Teichmüller metric, the Thurston metric and the Weil–Petersson-Funk metric. Acknowledgement. This work is partially supported by MEXT KAKENHI 21540177 (H.M.), by JSPS KAKENHI 22244005 (K.O.) and by JSPS KAKENHI 24340009 (S.Y.).

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K. Fujiwara, Geometry of the Funk metric on Weil–Petersson spaces. Math. Z. 274 (2013), 647–665.

[10] S. Kerckhoff, The asymptotic geometry of Teichmüller space. Topology 19 (1981), 23–41. [11] H. Masur, The extension of the Weil–Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43 (1976), 267–304. [12] R. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 75 (1988), no. 391. [13] A. Papadopoulos, Problem 13 in Problem Session “Teichmüller Theory”. In Oberwolfach Reports Vol. 7, European Mathematical Society, Zürich 2010, 3085–3157. [14] A. Papadopoulos and M. Troyanov, Weak Finsler structures and the Funk weak metric. Proc. Cambridge Philos. Soc. 147 (2009), 419–437. [15] A. Papadopoulos and G. Théret, On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, European Mathematical Society, Zürich 2007, 111–204. [16] B. B. Phadke, A triangular world with hexagonal circles. Geom. Dedicata 3 (1975), 511–520. [17] H. Royden, Automorphisms and isometries of Teichmüller space. In 1971 Advances in the theory of Riemann surfaces, Ann. Math. Studies 66, Princeton University Press, Princeton, N.J., 1971, 369–383 [18] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417–431. [19] W. P. Thurston, Minimal stretch maps between hyperbolic surfaces. Preprint, arXiv:9801039v1. [20] S. Wolpert, Noncompleteness of the Weil–Petersson metric for Teichmüller space. Pacific J. Math. 61 (1975), 573–577. [21] S. Wolpert, Geodesic length functions and the Nielsen problem. J. Differential Geom. 25 (1987), 275–296. [22] S. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space. In Surveys in differential geometry, Vol. VIII, Surv. Differ. Geom. 8, International Press, Cambridge, MA, 2003. [23] S. Wolpert, Behavior of geodesic-length functions on Teichmüller space. J. Differential Geom. 79 (2008), 277–334. [24] S. Wolpert, Personal communication. [25] S. Wolpert, The Weil–Petersson metric geometry. In Handbook of Teichmüller theory (A. Papadopoulos ed.), Volume II, European Mathematical Society, Zürich 2009, 47–64. [26] S. Wolpert, Families of Riemann surfaces and Weil–Petersson geometry. CBMS Regional Conf. Ser. in Math. 113, Amer. Math. Soc., Providence, RI, 2010. [27] S. Yamada, Weil–Petersson completion of Teichmüller spaces and mapping class group actions. Preprint, arXiv:0112001.

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[28] S. Yamada, On the geometry of Weil–Petersson completion of Teichmüller spaces. Math. Res. Lett. 11 (2004), 327–344. [29] S. Yamada, Weil–Petersson geometry of Teichmüller–Coxeter complex and its finite rank property. Geom. Dedicata 145 (2010), 43–63. [30] S.Yamada, Convex bodies in Euclidean and Weil–Petersson geometries. Proc. Amer. Math. Soc. 142 (2014), 603–616 [31] S. Yamada. Local and global aspects of Weil–Petersson geometry. In Handbook of Teichmüller theory (A. Papadopoulos ed.), Volume IV, European Mathematical Society, Zürich 2012, 43–111.

Chapter 13

Funk and Hilbert geometries in spaces of constant curvature Athanase Papadopoulos and Sumio Yamada

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 On analogies between the Euclidean and the non-Euclidean geometries 3 The Funk metric in hyperbolic and in spherical geometries . . . . . . . 4 Properties of the Funk metric . . . . . . . . . . . . . . . . . . . . . . 5 The Finsler structure of the Funk metric . . . . . . . . . . . . . . . . . 6 An example of a hyperbolic Funk metric . . . . . . . . . . . . . . . . 7 The cross ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Hilbert metric in Hn and in S n . . . . . . . . . . . . . . . . . . . 9 Projective transformations as isometries of Hilbert metrics . . . . . . . 10 Hilbert metrics as models of the hyperbolic plane . . . . . . . . . . . . 11 Examples of Hilbert metrics . . . . . . . . . . . . . . . . . . . . . . . 12 On Hilbert’s Problem IV . . . . . . . . . . . . . . . . . . . . . . . . . 13 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

353 354 358 360 361 363 364 366 367 368 369 370 370 377

1 Introduction It is well known that convex sets and their properties can be studied not only in Euclidean space but also in other metric spaces and in particular in the spaces of constant curvature: the sphere S n (or more conveniently the upper-hemisphere U n S n ) and the hyperbolic space Hn . The Funk and Hilbert metrics of convex subsets of S n and Hn can be defined in a way which is parallel to the way they are defined in the Euclidean space Rn . In particular, in each of these spaces of constant curvature, there is an invariant under the so-called projectivity (or perspectivity) transformation group which generalizes the cross ratio defined on real projective space. We developed this point of view in our papers [21] and [22] and we highlighted there some analogies and some differences between the three constant curvature settings, Rn , S n and Hn . The goal of the present chapter is to give a comprehensive survey of

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these results and to work out a few examples. We also provide a historical perspective for this theory. This study can be included in the setting of a generalized form of Hilbert’s Problem IV.

2 On analogies between the Euclidean and the non-Euclidean geometries The analogy we describe in this chapter between the Euclidean, spherical and hyperbolic worlds is in the lineage of several observations that were made by various people along several decades (even centuries), stressing on formal analogies between results and formulae in the three constant curvature geometries. In this section, we review such analogies. This will motivate our introduction of the non-Euclidean Funk and Hilbert metrics. We start with the observation that it is remarkable that one can pass from certain trigonometric formulae of Euclidean geometry to corresponding formulae in spherical (respectively hyperbolic) geometry by replacing side lengths by the sines (respectively the hyperbolic sines) of these lengths or by similar formal transformations. The simplest and most well-known example of such an occurrence is the following result known as the sine rule.1 Consider, in any one of the three geometries of constant curvature 0, 1 or 1, a triangle ABC with sides a, b, c opposite to the vertices A, B, C . We have, in Euclidean geometry, a b c D D ; sin A sin B sin C in spherical geometry, sin b sin c sin a D D ; sin A sin B sin C and in hyperbolic geometry, sinh b sinh c sinh a D D : sin A sin B sin C From these formulae, one can get other sets of formally analogous formulae in the three geometries, since many trigonometric formulae can be obtained by using the sine rule alone. As an example, the length ` of the side of a regular n-gon inscribed in a circle of radius R is given by the following: in Euclidean geometry, ` D R sin ; 2 n 1 These results and other results stated below are valid for spaces of constant curvature 0, 1 and 1. For spaces of different constant curvature, a normalization factor is needed. For the proof, and for other trigonometric formulae in hyperbolic trigonometry we refer the reader to [1], where the proofs are given in a model-free setting. The advantage of such a setting is that the proofs can be done at the same time in the spherical and in the hyperbolic geometry cases.

Chapter 13. Funk and Hilbert geometries in spaces of constant curvature

in spherical geometry, sin

355

` D .sin R/ sin ; 2 n

and in hyperbolic geometry, ` D .sinh R/ sin : 2 n Another formula with similar patterns in the three geometries is the one of the for circumference of the circle of radius r. In the Euclidean plane it is r, on the sphere it is 2 sin r and in the hyperbolic plane2 it is 2 sinh r. These formulae can be obtained by taking limits in the preceding formulae for n-gons as the number of sides tends to infinity. In the appendix, we review some other results of this kind. Finally, let us mention the famous relation known as Ptolemy’s Theorem, which gives a necessary and sufficient condition for a quadrilateral to be inscribed in a circle. In this context, a circle is a geometric circle, i.e. it is an equidistant locus from a point. This result says that a quadrilateral with side lengths a, b, c, d and diagonal lengths e and f is inscribed in a circle if and only if we have, in Euclidean geometry, sinh

ef D ac C bd; in spherical geometry, sin e sin f D sin a sin c C sin b sin d; and in hyperbolic geometry, sinh e sinh f D sinh a sinh c C sinh b sinh d where a, b, c, d denote the doubles of the side lengths and e, f denote the doubles of the diagonal lengths (see Figure 1). There are interesting generalizations of these relations in the papers [30] and [31] by Valentine and in the paper [7] by Guo and Sönmez. The following is a brief list of observations that were made by various people concerning the analogies between the three geometries. The Alsatian mathematician Johann Heinrich Lambert (1728–1777), in his Theorie der Parallellinien, [25] and [12], written in 1766, made the observation that certain properties of hyperbolic geometry (which, for him, was hypothetical) can be obtained from corresponding properties of spherical geometry by replacing certain distances p by the same distances multiplied by the imaginary number 1. In fact, if we take as a model of spherical geometry a sphere of radius r in the 3-dimensional Euclidean space,3 then, by a classical theorem attributed to Albert Girard (1595–1632), the area 2 This

formula in the case hyperbolic geometry was known to Lobachevsky and it is contained in several of his memoirs; see e.g. [13], p. 34. It is also contained in a letter by Gauss to Schumacher, dated July 31, 1831. 3A sphere of radius r has constant curvature 1=R2 . We recall however that Lambert did not use this language since he did not have the notion of curvature.

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a

b

f

e d

c

Figure 1

of a triangle on that sphere is a multiple of the angular excess; more precisely, it is equal to the quantity (2.1) r 2 .˛ C ˇ C / where ˛, ˇ, are the three angles of the triangle (in radians). p If in Equation 2.1 for the area we replace the radius r by an “imaginary radius” 1r, we obtain r 2 .˛ C ˇ C / D r 2 . ˛ ˇ /; which is the area of triangle of angles ˛, ˇ, in a hyperbolic space of constant curvature. Lambert declares at this occasion that one “is almost led to the conclusion that the third hypothesis4 occurs on a sphere of imaginary radius”. In the 19th century, spherical and hyperbolic geometry formulae were commented by F. A. Taurinus (1794–1874) (a contemporary of Gauss and Lobachevsky), who pointed out in 1825 simple passages between hyperbolic and spherical trigonometry using transformations similar to those we mentioned. In his memoir, which is also called Theorie der Parallellinien [25], he obtained fundamental trigonometric formulae for hyperbolic geometry (which was, from his point of view, like it was for Lambert, purely hypothetical), again by working on a sphere of “imaginary radius”, and he called the geometry that he so obtained the logarithmic-spherical geometry (“logarithmisch-sphärische Geometrie”) 4 The “third hypothesis”, in Lambert’s memoir, is the hypothesis under which the angle sum in some triangle, or, equivalently, in any triangle, is less than two right angles. Lambert made this hypothesis at some point in his memoir Theorie der Parallellinien with the aim of deriving a contradiction. He did not find the contradiction he was seeking for but his development of the subject constitutes a substantial part of the field of hyperbolic geometry. We refer the reader to [12] for a critical edition with a French translation and a mathematical commentary of Lambert’s work.

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Lobachevsky, in his Elements of geometry [14], in his Geometrical Researches on the Theory of Parallels [15], in his Pangeometry [13] and in other works, noted an analogous passage between the formulae of hyperbolic trigonometry and those of spherical trigonometry which consists in replacing, in the trigonometric equations of spherical geometry, p p the edge lengths a, b, c of a triangle by the imaginary quantities p in these equations the terms a 1, b 1, c 1, and at the same time, replacing p p sin ….a/, cos ….a/ and tan ….a/ by 1= cos a, 1 tan a and 1= 1 sin a respectively. (Here, … denotes Lobachevsky’s angle of parallelism function, see [14], [15] and the exposition in [1].) Beltrami, in his Saggio di interpretazione della geometria non-euclidea [2], showed that the trigonometric formulae for the pseudo-sphere (which is a model he had constructed for the hyperbolic plane) can be obtained from those ofpthe usual sphere by considering the pseudo-sphere as a sphere of imaginary radius 1, and he attributed this observation to E. F. A. Minding and to D. Codazzi. Let us quote Beltrami from [2] (Stillwell’s translation [26], p. 18): By an observation of Minding (Vol. XX of Crelle’s Journal), the ordinary formulae for spherical triangles are converted into those for geodesic triangles on a surface p of constant negative curvature by inserting the factor 1 in the ratio of the side to radius and leaving the angles unaltered, which amounts to changing the circular functions involving the radius into hyperbolic functions. For example, the first formula of spherical trigonometry cos becomes cosh

b c b c a D cos cos C sin sin cos A R R R R R

c b c b a D cosh cosh C sinh sinh cos A: R R R R R

Here, in the case of spherical geometry, R denotes the radius of the sphere, which means that we take the sphere of constant curvature R2 , and in the case of hyperbolic geometry, the plane has constant curvature R2 . We note that it is because of such a “duality” between the trigonometric formulae for spherical and for hyperbolic geometry that Beltrami chose, for hyperbolic geometry, the name “pseudo-spherical geometry”. Klein, in his Über die sogenannte Nicht-Euklidische Geometrie [11] made a similar observation (Stillwell’s translation [26], p. 99): The trigonometric formulae that hold for our measure result from the formulae of spherical trigonometry by replacing sides by sides divided by ci .

Story, in his paper On the non-Euclidean trigonometry [27], working in Klein’s setting of projective geometry, showed that some trigonometric formulae of hyperbolic geometry can be deduced from those of spherical geometry by replacing each length by the corresponding length divided by 2ik and each angle by the corresponding angle divided by 2ik 0 , where k and k 0 are two constants which can be imaginary and which are related to Klein’s constants in [11].

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Coxeter discovered a passage between formulae for volumes of hyperbolic polyhedra and formulae for volumes of spherical polyhedra, see [5]. In this work, Coxeter made a relation between computations of Lobachevsky in hyperbolic geometry and formulae discovered by Schläfli on the sphere. Thurston, in his notes on geometry (see [28] and [29]) continues this tradition of using the term “sphere of imaginary radius” to describe hyperbolic space. This refers usually to the so-called hyperboloid model of n-dimensional hyperbolic geometry embedded in .n C 1/-dimensional Euclidean space, a model that bears a close analogy with the n-sphere embedded in the same Euclidean space. Indeed, whereas the sphere in this model is the space of unit norm vectors for the quadratic form q.x/ D x02 C x12 C C xn2 , the hyperbolic space is a connected component of the space of vectors of norm i for the quadratic form q.x/ D x02 C x12 C xn2 . In both models, lines (and more-generally, k-dimensional planes, for 1 k n 1) are the intersections with the model with planes (or .k C 1/-planes) passing through the origin, etc.

3 The Funk metric in hyperbolic and in spherical geometries We start by some remarks on convexity. In hyperbolic n-space Hn , convexity is defined as in Euclidean space: we say that a set Hn is convex if for any two points in , the geodesic segment joining them is contained in . To define convexity for an open subset of the sphere S n , one has to take into account the fact that geodesic segments between two points are not unique. Even if we restrict ourselves to geodesics of shortest length, there may be two distinct such geodesics joining some pairs of points of the sphere, namely, diametrically opposite points. For this reason, we shall always assume that all subsets of S n that we consider are contained in an open hemisphere. Another way of proceeding would be to work in elliptic space (the quotient of the sphere by its order-two canonical involution) rather than on the sphere itself, but we prefer to work on the sphere. In any case, and modulo this complication, a convex set in the sphere (or in projective space) is a subset whose intersection with any line (that is, any great circle) is connected. In the rest of this section, will be an open convex subset of one of the three constant curvature spaces, Rn , S n or Hn . We do not assume that is bounded but we shall always assume that is not equal to the whole space Rn or Hn and, in the case of the sphere, we shall assume that is contained in an open hemisphere. At some point below, we shall assume, for reasons that will become clear, that our convex subset S n has diameter < =2 with respect to the angular metric of the sphere. We shall specify this whenever needed. We now define the Funk metric. Let us first consider the case of the hyperbolic space.

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Given two distinct points x and y in a convex set in Hn or S n , we denote by R.x; y/ D fexpx .txy / j t > 0g the geodesic ray starting at x and passing through y. Here xy is the unit vector at x tangent to the arc-length parameterized geodesic in Hn or S n connecting x to y. The ray R.x; y/ is the image of the exponential map ft 7! expx txy ; t 0g. Note that for the convex sets under consideration, each exponential map restricted to the component of exp1 x ./ containing the origin of the tangent space is a diffeomorphism onto . (This point is only relevant in the spherical case because of the existence of cut loci.) In the three geometries (Euclidean, hyperbolic and spherical), a hyperplane is a complete totally geodesic codimension-one subspace, and an (open) half-space is a connected component of the complement of a hyperplane in one of these spaces. (Note that on the sphere S n , hyperplanes are codimension-one great spheres and that half-spaces are called hemispheres.) n n n T Any convex set in one of the three geometries R , H or S can be represented as .b/2P H.b/ where H.b/ is a half-space bounded by a hyperplane .b/ tangent to @ at a boundary point b. We call these totally geodesic submanifolds .b/ supporting hyperplanes of . We denote by P the set of all supporting hyperplanes of and by P .b/ P the set of supporting hyperplanes of at the boundary point b 2 . Definition 3.1 (The hyperbolic Funk metric). For a pair of points x and y in Hn , the Funk (asymmetric) distance from x to y is defined as 8 d.y; b.x; y// for x ¤ y. This implies that the Funk distance F .x; y/ is always non-negative. Given a point x in the convex set S n and a hyperplane .b/ supporting at a boundary point b 2 @, we have d.x; .b// =2. The Funk metric on a convex subset S n is an asymmetric metric and it is also given by the following variational formula: F .x; y/ D sup log 2P

sin d.x; / sin d.y; /

for x 6D y.

4 Properties of the Funk metric Assume as before that is an open convex subset of any one of the three constant curvature geometries, let d denote the distance function in that geometry and let F .x; y/ denote the Funk metric of . The following propositions either follow easily from the definitions, or they are proved in [22]. Proposition 4.1. The Funk metric is unbounded. Proposition 4.2. We have the following: (1) The d -geodesics of are also Funk geodesics.

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(2) All the Funk geodesics of are d -geodesics if and only if is strictly convex, that is, if and only if the boundary of does not contain any nonempty open d -segment. We recall that a path s W Œa; b ! .X; d / in a metric space .X; d / is said to be geodesic if for any a < t1 < t2 < t3 < b, the equality d.s.t1 /; s.t2 // C d.s.t2 /; s.t3 // D d.s.t1 /; s.t3 // is satisfied. The reader should remember that in writing such an equality, when the metric is not symmetric (as in the case of the Funk metric), the order of the variables in d.; / must be respected. A geodesic ray is a geodesic s W Œ0; 1Œ! .X; d / whose restriction to any segment is a geodesic path. Proposition 4.3. The image of any d -geodesic segment starting at a point x in and ending at a point on @ is the image of an F -geodesic ray. Proposition 4.4. The Funk metric F separates points (that is, we have F .x; y/ > 0 () x 6D y) if and only if is bounded. Proposition 4.5. If is bounded, then, we have, for every x in and for every sequence xn in , F .x; xn / ! 0 () d.x; xn / ! 0 () F .xn ; x/ ! 0: Having a geodesic metric space, it is natural to ask whether the distance function to a point is convex, this being a kind of non-positive curvature property. In [22] we prove the following: Proposition 4.6. In spherical geometry, the Funk metric F .x; y/ defined on a convex set S n is convex in the y-variable. In hyperbolic geometry, the statement does not hold. In Euclidean geometry, the Funk metric F .x; y/ is convex in the y-variable, see [32].

5 The Finsler structure of the Funk metric In each of the three geometries, there is an infinitesimal linear structure for the Funk metric F , which provides a description of this metric as a Finsler metric. Recall that a family of linear functionals p;x defined on the family of tangent spaces Tx of a manifold is called a Finsler structure for a metric d on if the distance d is equal to the infimum of the lengths of all the piecewise C 1 paths W Œ0; 1 ! with .0/ D x; .1/ D y, that is, Z 1 p;.t / .P .t //dt: F .x; y/ D inf

0

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We now describe the Finsler norm of tangent vectors to for the Funk metric. We first recall the Euclidean setting. In this setting, the Funk metric is induced by the tautological Finsler structure in the sense of [19]. The norm function (also called the Minkowski functional) on each tangent space is given by the following: p;x ./ D sup

2P

kk d.x; T .x; ; //

where T .x; ; / is the point where the geodesic ray in Rn from x with initial velocity intersects the hyperplane . In this formula, it is easy to see that the supremum is achieved when the point T .x; ; / coincides with a boundary point b 2 @, namely 2 P .T .x; ; //. Using similarity of Euclidean triangles, this can be written as p;x ./ D sup

2P

h; i d.x; /

where is the unit tangent vector at x with direction opposite to the gradient vector of the functional d.; /. In the hyperbolic setting, the value of the Minkowski functional for the Funk metric F is given by p;x ./ D sup

2P

kk tanh d.x; T .x; ; //

(5.1)

h .x/; iHn tanh d.x; /

(5.2)

or, equivalently, by p;x ./ D sup

2P

where, as in the Euclidean case, T .x; ; / is the point where the geodesic ray in Hn from x with initial tangent vector intersects the hyperplane and .x/ is the unit tangent vector at x whose direction is opposite to the one of the gradient vector of the functional d.; / at x. We note that the gradient vector field’s integral curves are the geodesics which meet the supporting hyperplane perpendicularly. The details are given in [22]. Note that in either representation it is easy to see that the Minkowski functional is convex in 2 Tx Rd , since the functional p;x is convex (in fact it is linear) in for each fixed 2 P , and since by taking the sup over , convexity is preserved. We recall that in the Euclidean setting, the Funk metric for a convex set corresponds z in the sense to a linear structure which we called the tautological Finsler structure of [19]: z D f.x; / 2 Rn j x 2 and x C 2 g: It is called tautological because the fiber over each point x of is the set itself, with the point x as origin. In the hyperbolic setting, to display such a property, we need to incorporate the exponential map expx W Tx Hn ! Hn . Namely, the Finsler structure

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363

corresponds to the following subset of the total space of the tangent bundle T Hn : z D f.x; / 2 T Hn j x 2 and expx 2 g; where the fiber over each point x of is the set exp1 x ./, with the point x identified z is given as the origin of the tangent space Tx Hn . The Finsler norm of elements of by the Minkowski functional p;x . The Minkowski functional formula for the spherical Funk metric is analogous to that of the hyperbolic case; it is simply obtained by replacing the hyperbolic sine and cosine functions by the usual sine and cosine functions respectively. We will omit its explicit description here. As a tautological Finsler structure, it corresponds to the following subset of the total space of the tangent bundle to the sphere: z D f.x; / 2 T S n j x 2 and expx 2 g:

6 An example of a hyperbolic Funk metric We take to be a half-plane in H2 . In the upper half-plane model, and up to an isometry of this hyperbolic plane, we can regard the positive y-axis as the unique supporting hyperplane for (which in this case is a line), which we call . Thus the hyperbolic half-plane is described in the complex coordinate z as fz W Im.z/ > 0; Re.z/ > 0g H2 , see Figure 2. Now we consider the nearest point projection of a point p in the first quadrant onto the line and we calculate the value of sinh d.p; /.

f .p/

p

Figure 2

The nearest point projection of p to is called the foot of p on and denoted by f .p/. The hyperbolic distance between p and f .p/ is the length of the hyperbolic geodesic connecting these two points, which in this model is an arc A of a Euclidean circle centered at the origin, since the circle meets the x-axis and the y-axis perpendicularly.

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Denoting the polar coordinates of p by .r; /, we have f .p/ D .r; =2/. Let

W t 7! .r cos t; r sin t / be the parametrized geodesic segment A. Then the norm of the tangent vector k .t /0 k to the path at t is computed using (1) .t /0 D .r sin t; r cos t /; (2) the hyperbolic metric tensor ds 2 D .dx 2 C dy 2 /=y 2 ; (3) y D r sin t . This gives k .t /0 k D 1= sin t . Thus the hyperbolic length of the arc A is Z =2 Z =2 1 0 dt D log j tan. =2/j: k .t / k dt D L.A/ D sin t Note that as 0 < < =2, L.A/ is indeed positive. Hence we have sinh d.p; / D sinh L.A/ D

1 t .p/2 ; 2t .p/

where tan. .p/=2/ D t .p/. From this, we can compute the Funk distance F .p; q/ between two points p and q in H2 . If the ray R.p; q/ from p through q does not hit the boundary line of , then F .p; q/ D 0. If the ray R.p; q/ from p through q hits at some point b.p; q/, then

t .q/ 1 t .p/2 F .p; q/ D log t .p/ 1 t .q/2

or, equivalently, by using elementary geometry,

1 m.q/2 m.p/ F .p; q/ D log 1 m.p/2 m.q/

where m.p/ is the absolute value of the slope of the Euclidean line segment connecting p and f .p/

7 The cross ratio We define the cross ratio for four collinear ordered (and thus distinct) points A1 , A2 , A3 , A4 , in the Euclidean case, by ŒA2 ; A3 ; A4 ; A1 e WD

A2 A4 A3 A1 ; A3 A4 A2 A1

in the hyperbolic case, as ŒA2 ; A3 ; A4 ; A1 h WD

sinh A2 A4 sinh A3 A1 ; sinh A3 A4 sinh A2 A1

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and in the spherical case, by ŒA2 ; A3 ; A4 ; A1 s WD

sin A2 A4 sin A3 A1 ; sin A3 A4 sin A2 A1

where in each case Ai Aj stands for the distance (in the given space) between the pair of points Ai and Aj , which is equal to the length of the line segment joining them. In the spherical geometry case, we assume that the four points are contained in an open hemisphere. We note that in the modern formulation, the cross ratio is defined using the algebraic length, which makes it a signed quantity. We do not need this fact here. In [22], we showed that these three cross-ratios are the manifestation of the same entity; they are obtained from each other via projection maps between familiar representations of the three geometries in RnC1 . More specifically, consider the sphere S n as the set of unit vectors in RnC1 with respect to the Euclidean norm 2 x12 C C xn2 C xnC1 D1

and the hyperbolic space Hn as the set of “vectors of imaginary norm i ” with xnC1 > 0 in RnC1 with respect to the Minkowski norm 2 x12 C C xn2 xnC1 D 1:

These models of the two constant curvature spaces are called “projective”. The geodesics in the curved spaces are realized as the intersection of the unit sphere with the two-dimensional subspace of RnC1 through the origin of this space. In [22], we proved the following: Theorem 7.1 (Spherical case). Let Ps be the projection map from the origin of RnC1 sending the hyperplane fxnC1 D 1g RnC1 onto the open upper hemisphere U n of S n . Then Ps preserves the values of the cross ratios; namely, for a set of four ordered pairwise distinct points A1 , A2 , A3 , A4 aligned on a great circle in the upper hemisphere, we have ŒPs .A2 /; P2 .A3 /; Ps .A4 /; Ps .A1 /s D ŒA2 ; A3 ; A4 ; A1 e : Theorem 7.2 (Hyperbolic case). Let Ph be the projection map from the origin of RnC1 sending the unit disc of the hyperplane fxnC1 D 1g RnC1 onto the hyperboloid Hn RnC1 . Then the map Ph preserves the cross ratios; namely, for a set of four ordered pairwise distinct points A1 , A2 , A3 , A4 aligned on a geodesic in the upper hyperboloid, we have ŒPh .A2 /; Ph .A3 /; Ph .A4 /; Ph .A1 /h D ŒA2 ; A3 ; A4 ; A1 e : We also mention the following: Proposition 7.3. The Euclidean, spherical and hyperbolic cross ratios are projectivity invariants.

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We elaborate on the meaning of this statement in the case of spherical geometry: consider four ordered distinct line segments l1 , l2 , l3 , l4 intersecting at a point A and let l be a geodesic line intersecting l1 , l2 , l3 , l4 at points A1 , A2 , A3 , A4 respectively (Figure 3). We assume as usual that all these line segments are contained in an open hemisphere, so that they are all distance minimizing paths between their endpoints. Then we have

2 2

2 2

sin A2 AA4 sin A3 AA1 sinh A2 A4 sinh A3 A1 D : sinh A3 A4 sinh A2 A1 sin A3 AA4 sin A2 AA1 Thus, the cross ratio depends only on the angles that the four lines make. The details are given in the appendix.

A4 A3 A2 A1

l1

l2

l3

l4

Figure 3

Proposition 7.3 can also be deduced from the perspectivity invariance of the Euclidean cross ratio and then using the projections Ph and Ps .

8 The Hilbert metric in Hn and in S n We start with the case of the hyperbolic space Hn . Let be again an open convex (as before, possibly unbounded) proper subset of Hn . Definition 8.1 (The hyperbolic Hilbert metric). The Hilbert metric of is defined by 1 F .x; y/ C F .y; x/ H.x; y/ D 2 1 sinh d.x; b.x; y// sinh d.y; b.y; x// D log 2 sinh d.y; b.x; y// sinh d.x; b.y; x// 1 D logŒx; y; b.x; y/; b.y; x/h : 2

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The pair .; H.x; y// satisfies all the axioms of a metric space except, perhaps, for the separation axiom (which says that if x 6D y then H.x; y/ > 0). The separation axiom is satisfied if and only if does not contain a bi-infinite geodesic of Hn . In particular, if is bounded, then H.x; y/ is a genuine metric. The hyperbolic geodesic segment connecting x and y is a Funk geodesic realizing both lengths F .x; y/ and F .y; x/. This implies that the hyperbolic geodesic line segment is also a Hilbert geodesic. Proposition 8.2 (Hilbert geodesics). For any convex subset of Hn , we have the following: (1) the geodesics for the hyperbolic metric are Hilbert geodesics; (2) the hyperbolic geodesics are the unique Hilbert geodesics joining their endpoints if and only if satisfies the following property: there does not exist two hyperbolic geodesic segments in @ of nonempty interior which span a 2-dimensional totally geodesic subspace in . The case of the spherical Hilbert metric, as the reader might have guessed, is treated in analogy with case of the hyperbolic Hilbert metric. We define the spherical Hilbert metric H.x; y/ as an arithmetic symmetrization of the Funk spherical metric, that is, we set 1 H.x; y/ D F .x; y/ C F .y; x/ 2 sin d.x; b.x; y// sin d.y; b.y; x// 1 D log 2 sin d.y; b.x; y// sin d.x; b.y; x// 1 D logŒx; y; b.x; y/; b.y; x/s : 2 Thus, the formula in the spherical case is obtained from that of the hyperbolic case by replacing the sinh function by the sine function. We assume here that the convex set is contained in an open hemisphere. Unlike the case of the Funk spherical metric, we do not need to assume that is contained in a subset of the sphere of diameter < =2, for the value H.x; y/ as defined is always non-negative. We will see below (Theorem 9.1) that all isometries of the Hilbert metric are induced by isometries of the hyperbolic metric.

9 Projective transformations as isometries of Hilbert metrics Proposition 7.3 saying that the generalized cross ratio is a projective invariant implies that a projective self-transformation of Rn , Hn or S n induced by a linear projective transformation of RnC1 , induces an isometry of the Hilbert metric H of a convex set in any one of these geometries; namely, the following two spaces are isometric

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z Hz / .; H / Š .;

z D ./. Clearly every isometry of the space forms Rn , Hn and S n is a where projective transformation. One can ask whether there are isometries of Hilbert metrics that are not induced by isometries of the underlying Riemannian manifolds. In Rn , the set of linear projective transformation is the general linear group GL.n; R/, a strictly larger group than the group of the linear isometries O.n/. From the fact, whose proof is given in the appendix (Proposition 13.5), that in hyperbolic geometry each projective transformation is an isometry of the space, we have the following: Theorem 9.1. Given a convex set in the hyperbolic space, a diffeomorphism is z H z / with z D ./ if and only an isometry of the Hilbert metric: .; H / Š .; if is the restriction of a hyperbolic isometry to . On the other hand, in the spherical case, a projective transformation of S n RnC1 can be regarded as an element of the general linear group GL.n C 1; R/. Therefore we have the following: Theorem 9.2. Given a convex set in S n , each element of GL.n C 1; R/ induces z H z / where z D ./. an isometry of the Hilbert metric: .; H / Š .; We note that by Theorems 7.1 and 7.2, which state that the projection maps Ph and Ps are natural with respect to perspectivities, each bounded convex set Hn corresponds to a convex set Ph1 ./ Rn with the hyperbolic Hilbert metric H isometric to the Euclidean Hilbert metric HP 1 ./ , and that each convex set 0 h

U n S n corresponds to a convex set Ps1 .0 / Rn with H0 isometric to HPs1 .0 / . This observation in effect says that among the three geometries of Rn , Hn and S n , the Hilbert geometries of bounded convex sets on each space are mutually identifiable. In light of this fact, the isometries of a Hilbert metric for a bounded convex set are always realized as that induced from the isometries of Hn via the identifications through the perspectivities Ps and Ph .

10 Hilbert metrics as models of the hyperbolic plane The projection maps Ps and Ph described in §7 between the Euclidean plane Rn D fxnC1 D 1g RnC1 , the upper hemisphere U n S n RnC1 and the hyperboloid Hn RnC1 are natural with respect to the cross ratios defined in each of these spaces n fxnC1 D 1g a disk of radius R > 0 centered (Theorems 7.1 and 7.2). Denote by DR at the origin of the Euclidean plane. By restricting the projection map Ps to the disk

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n DR , we obtain a map from this disk into the sphere S n . For R < 1, by restricting n , we obtain a map from this disk into the hyperboloid/hyperbolic space Hn . Ph to DR These maps are natural with respect the corresponding cross ratios, and therefore they n equipped with are isometries with respect to the Hilbert metrics. Since the disk DR its Hilbert metric is a model of hyperbolic geometry (the Beltrami–Klein model), we obtain in this way new models of hyperbolic space which sit inside hyperbolic space and inside the sphere respectively. We call such models generalized Beltrami–Klein models, since they are defined using the spherical and the hyperbolic cross ratios respectively. Thus we identify, in each of the positively/negatively curved spaces, a nested family of geodesic balls as models of the hyperbolic space. Note that the limit of these models in S n and in Hn as the radius R goes to =2 and 1 respectively, is the upper hemisphere U n and the entire space Hn whose Hilbert metrics, if we define them by the formula we used for proper open convex subsets, are metrics that are identically zero.

11 Examples of Hilbert metrics Let us first recall the Hilbert metric of a Euclidean planar example, described in detail by Phadke in the paper [23] (although the expression “Hilbert metric” is not used there). This is the case where is the quarter-plane fx > 0; y > 0g R2 . Denoting by .x1 ; x2 / and .y1 ; y2 / the coordinates of two points x and y in , we have, for x 6D y, the following formula for the Hilbert distance between them: ²ˇ ³ x1 ˇˇ ˇˇ x2 ˇˇ ˇˇ x1 y2 ˇˇ 1 ˇ H .x; y/ D max ˇlog ˇ; ˇlog ˇ; ˇlog ˇ : 2 y1 y2 y1 x2 Using our non-Euclidean cross ratios and the natural projections among the three geometries, we can write formulae for a quarter-plane in the hyperbolic plane H2 , and the tri-rectangular geodesic triangle of the upper hemisphere U S 2 using the obvious coordinates. Namely for the spherical case, they are the latitude values x1 , x2 of the images of the nearest point projection to a pair of longitude great circles separated by =2 in the upper hemisphere. In the hyperbolic case, the coordinates x1 , x2 are the distances measured from the vertex of the quarter plane to the feet of the nearest point projections to the pair of perpendicular geodesics bounding the quarter plane. In these spaces transported by the projection maps, we have ²ˇ ³ sin x1 ˇˇ ˇˇ 1 sin x2 ˇˇ ˇˇ sin x1 sin y2 ˇˇ ˇ H .x; y/ D max ˇlog ; log ; log ˇ ˇ ˇ ˇ ˇ 2 sin y1 sin y2 sin y1 sin x2 in the spherical setting, and ²ˇ ³ 1 sinh x1 ˇˇ ˇˇ sinh x2 ˇˇ ˇˇ sinh x1 sinh y2 ˇˇ ˇ H .x; y/ D max ˇlog ˇ; ˇlog ˇ; ˇlog ˇ 2 sinh y1 sinh y2 sinh y1 sinh x2 in the hyperbolic setting.

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We finally recall that by a special case of a result due to Nussbaum [4], de la Harpe [8] and Foertsch–Karlsson [6], the Hilbert metric of a convex open subset of R2 is isometric to a Minkowski normed space (that is, a translation-invariant metric on R2 arising from a norm) if and only if is a triangle.5 Since any triangle in the projective plane P 2 is projectively equivalent to the quarter-plane , the above formula in the Euclidean case applies to an arbitrary plane triangle. We conclude that the non-Euclidean analogues that we exhibit are formulae for non-Euclidean Hilbert metrics which are isometric to (linear) Minkowski metrics.

12 On Hilbert’s Problem IV Hilbert’s Fourth Problem, from the collection of mathematical problems he presented in 1900 at the Second International Congress of Mathematicians in Paris, is titled: “Problem of the straight line as the shortest distance between two points”. The problem asks for the characterization and the study of individual metrics (symmetric but also asymmetric) on subsets of Euclidean space for which the Euclidean straight lines are geodesic; see [10] and the survey [16] in this volume. It is natural to ask an analogue of Hilbert’s problem in the non-Euclidan context that we consider here, that is, to study the metrics on subsets of the sphere and of hyperbolic space for which the spherical and the hyperbolic straight lines respectively are geodesics. The projection maps defined in §7 between the Euclidean plane Rn , the upper hemisphere U n S n and the hyperbolic plane Hn send lines to lines, and one can capture the projective geometry in a manner almost identical to the one in the Euclidean situation, and in some sense the study of the Hilbert metrics on convex subsets of the non-Euclidean spaces of constant curvature can be reduced to the Euclidean one. However this does not hold in the case of Funk metrics.

13 Appendix In this appendix, we present the proofs of some results in the projective geometry of Hn and S n , in particular on the projective invariance of cross ratio. We start with a few results in the setting of hyperbolic geometry. The proofs are based on the sine rule and they mimic the proofs of corresponding results in Euclidean geometry. From our point of view, the fact that is most interesting is the formal analogy between the results and the formulas in the three cases: Euclidean, hyperbolic and spherical.

5 This result attributed to Nussbaum, de la Harpe, and Foertsch–Karlsson was known to Busemann 1967. See Chapter 1, p. 30 of this Handbook [17].

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Proposition 13.1. Let ABC be a hyperbolic triangle. We join A by a geodesic to a point D on the line BC . Then we have

1 1

sin DAC sin B sinh DC D : sinh BD sin BAD sin C A

B

C

D

Figure 4

Proof. The proof is independent of whether the point D is between B and C , or C is between D and B, or B is between C and D. Applying the sine rule in the triangle DAC , we have

1

sin CAD sinh DC D : sinh AD sin C Applying the sine rule in the triangle BAD, we have sin B sinh AD : D sinh BD sin BAD From the last two equations, we obtain

1

1 1

sin DAC sin B sinh DC D : sinh BD sin BAD sin C This proves Proposition 13.1. Proposition 13.2. Consider four ordered distinct geodesic lines l1 , l2 , l3 , l4 intersecting at a point A and let l be a geodesic line intersecting l1 , l2 , l3 , l4 at points A1 , A2 , A3 , A4 respectively. Then we have

2 2

2 2

sin A2 AA4 sin A3 AA1 sinh A2 A4 sinh A3 A1 D : sinh A3 A4 sinh A2 A1 sin A3 AA4 sin A2 AA1 Proof. We refer to Figure 5. Using Proposition 13.1, we have

2 2

sin A2 AA4 sin A3 sinh A2 A4 D : sinh A3 A4 sin A3 AA4 sin A2

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A4 A3 A2 A1

l1

l2

l3

l4

Figure 5

Using again Proposition 13.1, we have

2 2

sin A3 AA1 sin A2 sinh A3 A1 D : sinh A2 A1 sin A2 AA1 sin A3 This implies

2 2

2 2

sin A2 AA4 sin A3 AA1 sinh A2 A4 sinh A3 A1 D : sinh A3 A4 sinh A2 A1 sin A3 AA4 sin A2 AA1 This proves Proposition 13.2. Corollary 13.3 (Cross ratio invariance in hyperbolic geometry). Consider four ordered distinct geodesics l1 , l2 , l3 , l4 in the hyperbolic plane that are concurrent at a point A and let l be a geodesic that intersects these four lines at points A1 , A2 , A3 , A4 respectively. Then the cross ratio of the ordered quadruple A1 , A2 , A3 , A4 does not depend on the choice of the line l 0 . Proof. By Proposition 13.2, the cross ratio of the quadruple A1 , A2 , A3 , A4 depends only on the angles that are made by the lines l1 , l2 , l3 , l4 . The result of Corollary 13.3 is a hyperbolic analogue of the result saying that the Euclidean cross ratio is a projective invariant. We now present the theorem of Menelaus in hyperbolic geometry with a proof based on the sine rule.6 This theorem 6 Menelaus (2nd century A.D.) in his Spherics gave the theorem in the Euclidean and in the spherical cases. There are several proofs of that theorem, and the proof that we give is not the original proof given by Menelaus, since he did not have the sine rule at his disposal. In fact, Menelaus did not formulate his theorem in terms of sines, but of chords. The notion of sine was introduced by the Arabs in the ninth century, who also discovered the sine rule, in Euclidean and in spherical geometry. In fact, Menelaus deduces the proof for the spherical case from the proof in the Euclidean case, by reasoning in the sphere embedded in the three-dimensional Euclidean space, see [24]. This theorem of Menelaus is sometimes referred to – especially by historians of mathematics – as the sector figure theorem, because this is the way ancient authors (like Ptolemy) referred to it. Menelaus’

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will be used below in the proof of the triangle inequality for the hyperbolic Funk metric. Proposition 13.4 (Menelaus’ Theorem in the hyperbolic plane). Let ABC be a triangle in the hyperbolic plane and let A0 , B 0 , C 0 be three points on the lines containing the sides BC , AC , AB. Then, the points A0 , B 0 , C 0 are aligned if and only if we have the relation sinh AC 0 sinh BA0 sinh CB 0 D1 sinh AB 0 sinh BC 0 sinh CA0 Proof. The proof is analogous to the proof of the corresponding Euclidean theorem (see [20] in this volume). Here we just prove the “only if” direction in order to show how one can use the hyperbolic sine rule instead of proportionality in the Euclidean case. We consider the case of Figure 6. A

B0

C0

C

A0

B

Figure 6. The theorem of Menelaus says that three points A0 , B 0 , C 0 on the lines containing the sides BC , AC , AB of a triangle are aligned if and only they satisfy the relation in Proposition 13.4.

In the triangle AB 0 C 0 , the sine rule gives

2 D sinh AC : sinh AB 2 C B sin A sin AB 0 C 0 0

0

0

0

In the triangle BC 0 A0 , the sine rule gives

2 D sinh BA : sinh BC 2 sin B AC

sin B C 0 A0 0

0

0

0

In the triangle CA0 B 0 , the sine rule gives

1 D sinh CB : sinh CA 2 sin A BC sin CA0 B 0

0

0

0

0

Spherics has been conceived as the non-Euclidean counterpart of Euclid’s Elements. It survived only in Arabic texts. A German edition exists [18], based on a manuscript of Ibn ‘IrNaq (10th century), and a new critical English edition based on several other Arabic manuscripts is in preparation [24].

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2

2

2

Multiplying the three sides of the last three equations and using the fact that sin AB 0 C 0 D sin A0 B 0 C and sin B C 0 A0 D sin AC 0 B 0 , we obtain the desired equality 1D

sinh AC 0 sinh BA0 sinh CB 0 : sinh AB 0 sinh BC 0 sinh CA0

The spherical analogue of all these results can be done in a similar way; the function sinh should be replaced in this case by the function sin. We now give an alternative proof of the triangle inequality for the hyperbolic Funk metric F .x; y/. This will also provide an application of the non-Euclidean cross ratio and it will further illustrate the formal analogies between the formulae and the reasonings in the three constant curvature geometries. The proof uses a drawing (Figure 7) and Menelaus’ Theorem (cf. Proposition 13.4). We present it here, using the notation of [33]. Let x, y, z be three points in . In the case where the three points are collinear (that is, if they are on a common hyperbolic geodesic), we know that the triangle inequality is satisfied, and that it is an equality, that is, we have F .x; y/ C F .y; z/ D F .x; z/ if x, y, z lie in that order on the line. Assume now the three points are not collinear and let a, b, c, d , e, f the intersections with @ of the lines xz, yx and zy, with the orders of the various quadruples of collinear points represented in Figure 7. p

f a

a0

c y xg

d

z

b0

b

e Figure 7

By the cross ratio invariance in hyperbolic geometry (Proposition 13.3), we have sinh xb 0 sinh a0 g sinh xc sinh dy D sinh yc sinh dx sinh gb 0 sinh a0 x and

sinh gb 0 sinh a0 z sinh ye sinh f z D : sinh ze sinh f y sinh zb 0 sinh a0 g

Multiplying both sides of these two equations, we get sinh xb 0 sinh a0 z sinh dx sinh f y sinh xc sinh ye D : sinh yc sinh ze sinh zb 0 sinh a0 x sinh dy sinh f z

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By the Theorem of Menelaus (Theorem 13.4) applied to the triangle f a0 z, we have sinh a0 x sinh dx sinh f y : D sinh dy sinh f z sinh a0 z This gives

sinh xb 0 sinh xb sinh xc sinh ye D ; 0 sinh yc sinh ze sinh zb sinh zb

and the inequality is strict unless b D b 0 . From this the triangle inequality follows and at the same time we deduce that the inequality is strict for all x, y, z unless @ contains a hyperbolic segment. This gives the desired result. Proposition 13.5. A diffeomorphism W Hn ! Hn which sends each geodesic to a geodesic is an isometry of the hyperbolic space Hn . We thank Yukio Matsumoto for discussions which led to the following proof. Proof. When n D 2, by using the upper half-plane model of H2 , given a diffeomorphism sending a circle centered at a point on the x-axis to another circle centered on the x-axis, we claim that the map is an element of SL.2; R/, which is the isometry group of H2 . Consider a diffeomorphism F of the upper half-plane, fixing 1, 1 and 1. It is easy to see that in order to prove the claim, it suffices to show that the restriction of F to the x-axis, which we denote by f W R ! R, is the identity map. We first show that f is symmetric with respect to the origin of the x-axis, i.e. f .x/ D f .x/. Let us denote the semi-circle in the upper half-plane meeting the x-axis perpendicularly at a and b (a < b) by Ca;b . When a < b < c < d , we also denote the x-component of the intersection point of Ca;c and Cb;d by P .a; b; c; d /. By elementary algebraic calculations, we have P .1; a; 1; b/ D

1 C ab : aCb

It follows that 1 1 and P .b; 1; 0; 1/ D b b for b > 1. Because F maps the intersection of a pair of semi-circles to the intersection of the F -images of those semi-circles, we have P .1; 0; 1; b/ D

f .P .a; b; c; d // D P .f .a/; f .b/; f .c/; f .d //: Hence f .P .b; 1; 0; 1// D f

1 b

D P .f .b/; f .1/; f .0/; f .1// D

1 C f .b/f .0/ f .b/ C f .0/

and 1 C f . b1 /f .b/ D f .P .1; b1 ; 1; b// D f .0/ P f .1/; f b1 ; f .1/; f .b/ D f . b1 / C f .b/

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as we have P .1; b1 ; 1; b/ D 0. By eliminating f . b1 / in the two equations, we obtain f .b/ D f .b/ for b > 1, namely f W R ! R is symmetric outside Œ1; 1. Next, we show that for b > 1, f . b1 / D f . b1 /, implying the symmetry of f on .1; 1/. Choose R so that 1 < Œ1; 1,

R2 b

< R < b. Then from the symmetry of f outside

2 f .P .R; 0; R; b// D f Rb D P .f .R/; f .0/; f .R/; f .b// D

f .R/2 C f .0/f .b/ f .0/ C f .b/

is equal to 1 times

2 f .P .b; R; 0; R// D f Rb D P .f .b/; f .R/; f .0/; f .R// D

f .R/2 f .0/f .b/ ; f .0/ f .b/

concluding that f .0/ D 0. This in turn says that f .P .b; 1; 0; 1// D

1 1 1 C f .b/f .0/ D D f .b/ C f .0/ f .b/ f .b/

while f .P .1; 0; 1; b// D

1 C f .b/f .0/ 1 D ; f .b/ C f .0/ f .b/

thus implying f . b1 / D f . b1 /, proving the symmetry of f over the x-axis with respect to the origin. Utilizing the previous equation f

R2 b

D

f .R/2 C f .0/f .b/ f .0/ C f .b/

together with f .0/ D 0 and by letting R ! b, we have f .b 2 / D f .b/2 for b > 1 by the continuity of f , which implies that f .x/ D x ˛ for some ˛ > 0. Similarly one can show that f .x/ D x ˛ for 0 < b < 1. Now the equation

f

1 C ab aCb

D

1 C f .a/f .b/ f .a/ C f .b/

forces ˛ to be one. We conclude that f .x/ D x. In the following, we will consider the higher dimensional case. Let n 3. Given a point p in Hn and two distinct arc-length parameterized geodesics 1 ; 2 through p with their respective tangent vectors v1 ; v2 2 Tp Hn , the

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images . 1 / and . 2 / are geodesics distinct from each other and both passing through .p/. Let 12 be the two-dimensional subspace in Tp Hn spanned by v1 and v2 , and …12 be the image of 12 by the exponential map expp W Tp Hn ! Hn , which is a totally geodesically embedded copy of H2 . Consider a geodesic triangle whose vertices are p,

1 .1/ and 2 .1/. By the property that sends a geodesic to a geodesic, the geodesic triangle 4.p; 1 .1/; 2 .1// is sent to the geodesic triangle 4..p/; . 1 .1//; . 2 .1//, and the interior points of 4.p; 1 .1/; 2 .1// …12 are sent to the interior points of 4..p/; . 1 .1//; . 2 .1// (this follows from the fact that the convex hull of a geodesic triangle in Hn lies in the hyperbolic plane containing the three vertices). From this observation, we can deduce that the image of …12 by is identified with the image of d.12 / T.p/ Hn by the exponential map exp.p/ W T.p/ Hn ! Hn , once again a totally geodesically embedded copy of H2 . In short, the diffeomorphism sends a hyperbolic plane to a hyperbolic plane in the hyperbolic space Hn . Thus the restriction of to the hyperbolic plane …12 can be regarded as a diffeomorphism Q W H2 ! H2 which sends a geodesic to a geodesic. Now we know that such a map is a hyperbolic isometry of H2 , as was explained in the beginning of the proof. Hence the linear map d restricted to 12 Tp Hn preserves the norms and the angles. As the point p and v1 , v2 were chosen arbitrarily, we conclude that the map is a local isometry, which is also a global isometry in the current setting. Acknowledgement. This work is partially supported by the French ANR project FINSLER (A.P.) and by JSPS Grant-in-aid for Scientific Research No.24340009 (S.Y.).

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[22] A. Papadopoulos and S. Yamada, The Funk and Hilbert geometries for spaces of constant curvature. Monatsh. Math. 172 (2013), no. 1, 97–120. [23] B. B. Phadke. A triangular world with hexagonal circles. Geom. Dedicata 3 (1975), 511–520. [24] R. Rashed and A. Papadopoulos, Menelaus’ Spherics. Critical edition from the Arabic manuscript of al-HarawNı, with an English translation and a mathematical commentary, in preparation. [25] P. Stäckel and F. Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung zur Vorgeschichte der nicht-euklidischen Geometrie. An edition with introduction, comments and German translations of works on the parallel postulate by Wallis, Saccheri, Lambert, Gauss (notes and correspondence), Schweikart and Taurinus, B. G. Teubner, Leipzig 1895. [26] J. Stillwell, Sources of hyperbolic geometry. Hist. Math. 10, Amer. Math. Soc., Providence, RI; London Mathematical Society, London 1996. [27] W. E. Story, On the non-Euclidean trigonometry. Amer. J. Math. IV (1881), 332–335. [28] W. P. Thurston, The geometry and topology of three-manifolds. Princeton Lecture Notes, 1976 ca., available at MSRI, library.msri.org/books/gt3m/. [29] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, ed. by Silvio Levy, Princeton Math. Ser. 35, Princeton University Press, Princeton, NJ, 1997. [30] J. E. Valentine, An analogue of Ptolemy’s theorem and its converse in hyperbolic geometry. Pacific J. Math. 34 (1970) 817–825. [31] J. E. Valentine, An analogue of Ptolemy’s theorem in spherical geometry. Amer. Math. Monthly 77 (1970) 47–51. [32] S.Yamada. Convex bodies in Euclidean and Weil–Petersson geometries, Proc. Amer. Math. Soc. 142 (2014), no. 2, 603–616. [33] E. M. Zaustinsky, Spaces with nonsymmetric distance. Mem. Amer. Math. Soc. No. 34, 1959.

Part IV

History of the subject

Chapter 14

On the origin of Hilbert geometry Marc Troyanov

Hilbert geometry, which is the main topic of this Handbook, was born twelve decades ago, in the summer of 1894. Its birth certificate is a letter written by David Hilbert to Felix Klein, on August 14, 1894, sent from the village of Kleinteich near the city of Rauschen (today called Svetlogorsk), at the Baltic sea. The place was a fashionable vacation resort, situated some forty kilometers north of Hilbert’s hometown Königsberg (now Kaliningrad), where he was a Privatdozent.1 The mathematical part of this letter was published one year later in the Mathematische Annalen2 under the title Ueber die gerade Linie als kürzeste Verbindung zweier Punkte (On the straight line as shortest connection between two points) [6]. Before discussing the content of this letter, let us say a few words on the general situation of geometry in the XIXth century. At the dawn of the century, and despite numerous criticisms concerning the axioms, Euclid’s Elements were still considered as a model of mathematical rigor and an exposition of the “true” geometry. The 1820s saw on one hand the creation of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky, and on the other hand, a revival and a rapid development of projective geometry by Poncelet, Gergonne, Steiner, Möbius, Plücker and von Staudt. During this period, a controversy developed among supporters of the synthetic versus the analytic methods and techniques in the subject. In this context, the analytic methods are based on coordinates and algebraic relations for the description and the analysis of geometric figures, while the synthetic methods are based on the incidence relations between points, lines, planes and other loci. The key to unify both sides of the controversy came from the work of von Staudt [18], [19] who proved that numerical coordinates can be assigned to points on a line in a projective space through the use of purely synthetic methods and without using distances. Von Staudt’s theorem implies that the cross ratio of four aligned points can be defined synthetically, and thus it is invariant under collineations (that is, transformations preserving lines and incidence). Recall that the cross ratio of four points X, Y , Z, T is analytically defined as ŒX; Y; Z; T D 1 Hilbert 2 The

.x z/ .y t / ; .x t / .y z/

was appointed professor at Göttingen one year later. chief editor of this journal was Klein himself.

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where x; y; z; t 2 R [ f1g are coordinates on the given line, corresponding to the points X, Y , Z, T . The synthetic definition of the cross ratio is more elaborate and it is based on an iteration of the construction of the harmonic conjugate of a point with respect to a given pair of points (the harmonic conjugate corresponds to a cross ratio equal to 1 and can easily be constructed synthetically using complete quadrangles), together with ordering and density arguments. In fact, von Staudt was not able to fully prove his theorem since at his time the topological structure of the real numbers, in particular the completeness axiom, had yet to be clarified. This was done later in Dedekind’s short monograph Was sind und was sollen die Zahlen? (1888). A full exposition of von Staudt’s construction is given in Chapter VII of Veblen and Young’s book [16], see also Chapter VII of [15]. We refer to [10] and [17] for additional comments on von Staudt’s work. By the middle of the XIXth century, it was seen as a natural problem to try to define metric notions from projective ones, thus reversing the analytic way of seeing geometry. Von Staudt’s theorem allowed the use of the cross ratio in this adventure. The first step in this direction is usually attributed to Laguerre, who proved in 1853 that the Euclidean angle ' between two lines `, m through the origin in a plane is given by ˇ ˇ ˇ1 ˇ ˇ (1) ' D ˇ logŒX; Y; U; V ˇˇ; 2i where X 2 `, Y 2 m and U , V are points representing the “isotropic lines” of the plane3 . These are lines which are “orthogonal to themselves”, meaning that if U D .u1 ; u2 / and V D .v1 ; v2 /, then u21 C u22 D v12 C v22 D 0; or, equivalently, u2 D iu1 and v2 D iv1 . To explain Laguerre’s formula, one may assume x1 D y1 D u1 D v1 D 1; then x2 D tan.˛/, y2 D tan.ˇ/, u2 D i and v2 D i . We then have ŒX; Y; U; V D

.tan.˛/ i/ .tan.ˇ/ C i / D e2i.˛ˇ / : .tan.˛/ C i/ .tan.ˇ/ i/

Klein in [8], inspired by a formula published by Cayley4 in [3], observed later that Equation (1) also gives a projective definition of the distance in elliptic geometry, that is, the metric on RP n for which the standard projection Sn ! RP n is a local isometry. Here X and Y are arbitrary distinct points in RP n and U and V are points in CP n which are aligned with X and Y and belong to the quadric n X

u2i D 0:

(2)

iD0 3 In fact the formula in Laguerre’s paper does not explicitly involve the cross ratio, the present formulation seems to be due to F. Klein, see e.g. [9], p. 158. 4 Cayley’s formula did not involve the cross ratio; it is written in terms of homogenous coordinates.

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Klein also considered similar geometries where the quadric (2) is replaced by an arbitrary quadric, which, following Cayley’s terminology, he called In Pnthe absolute. 2 2 1871, Klein realized that choosing as the absolute the standard cone iD1 ui u0 D 0, the metric restricted to the interior of the cone is a model of Lobatchevsky’s geometry. See [2] for a historical discussion. In 1888, M. Pasch proposed a new axiomatic foundation of geometry based on the primitive notion of segment (or betweenness) rather than the projective notion of lines. Pasch’s geometry is often called ordered geometry and it greatly influenced Hilbert. Let us now return to Hilbert’s letter. The letter starts with a discussion of how geometry should be founded on three “elements” (in the sense of primitive notions) named points, lines and planes (Hilbert’s aim is a discussion of three-dimensional geometry). These elements should satisfy three groups of axioms, which he only shortly discusses. I. The first groups are axioms of incidence describing the mutual relations among points, lines and planes. These axioms state that two distinct points determine a unique line, that three non aligned points determine a unique plane, that a plane containing two distinct points contains the full line determined by those points, and that two planes cannot meet at only one point. Furthermore, every line contains at least two points, every plane contains at least three non aligned points and the space contains at least four non coplanar points. II. The second groups of axioms deals with ordering points on a line and these axioms have been proposed by M. Pasch in [12]. They state: between two distinct points A and B on a line, there exists at least a third point C , and given three points on a line, one and only one of them lies between the two others. Given two distinct points A and B, there is another point C on the same line such that B is between A and C . Given four points on a line, we can order them as A1 , A2 , A3 , A4 is such a way that if h < i < k, then Ai lies between Ah and Ak . If a plane ˛ contains a line a, then the plane is separated into two half-planes such that A and B belong to the same half-plane if and only if there is no point of the line a between A and B. III. The axiom of continuity. Let Ai be an infinite sequence of points on a line a. If there exists a point B such that Ai lies between Ak and B as soon as k i , then there exists a point C on the line a such that Ai lies between A1 and C for any i > 1 and any point C 0 with the same property lies between C and B. (We recognize an axiom stating that any decreasing sequence on a line that is bounded below has a unique greatest lower bound). Hilbert then claims that the von Staudt theory of harmonic conjugates can be fully developed on this axiomatic basis (in a “similar way” as in Lindemann’s book Vorlesungen über Geometrie). It therefore follows that coordinates can be introduced. More precisely, to each point, one can associate three real numbers x, y, z and each plane corresponds to a linear relation among the coordinates. Furthermore, if one interprets x, y, z as rectangular coordinates in the usual Euclidean space, then the

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points in our initial space correspond to a certain convex5 body in Euclidean space. Conversely, the points in an arbitrary convex domain represent points in our initial space: our initial space is built from the points in a convex region of the Euclidean space. The axiomatic framework set here by Hilbert is interesting in several aspects. The first group of axioms belongs to what is now called incidence geometry. Observe that Hilbert formulates them for a three-dimensional space – at that time it was unusual to discuss geometry in an arbitrary dimension n – but note also that this restriction reduces the complexity of the theory, because it is not so simple to define and discuss a general notion of dimension in pure incidence geometry. Also Desargues’ theorem (which plays a key role in the introduction of coordinates) need not be added as an axiom since it can be proved in this framework. The second group of axioms describes ordered geometry, see [15] and [14] for readable accounts of this topics and [13] for an impressive historical compilation of the literature since Pasch’s original work. Notice that the notion of line (and hence plane) can be defined on the sole base of the primitive notion of segment, but Hilbert found it convenient to have lines and planes as primitive notions also. Again, this reduces the complexity of the theory (and furthermore it is in Euclid’s spirit). In a more contemporary language, what Hilbert is claiming here is that an abstract space M consisting of points, together with a structure made of lines, planes and the betweenness relation (for points on a line) satisfying the said axioms is isomorphic to a convex region in R3 with the usual notions of lines and planes and the condition that C lies between A and B if and only if C 2 ŒA; B. It is not clear whether the axiomatics proposed in Hilbert’s letter is sufficient to rigorously prove this result, and the question is also not addressed in his Grundlagen [7]. However it is indeed the case that a synthetic axiomatic characterization of convex domains in Rn can be based on ordered geometry. A theorem of this type is given in W. A. Coppel’s book, who calls it the fundamental theorem of ordered geometry, see [4], p. 173. At this point of his letter, Hilbert stresses that arbitrary convex bodies also appear in Minkowski’s work on number theory. He then proceeds to define a notion of length for the segment AB in his general geometry, which he assumes is now realized as the interior of a convex region in R3 . He follows Klein’s construction and defines the distance between two distinct points A and B to be6

YA XB d.A; B/ D logŒX; Y; B; A D log ; YB XA

(3)

where X; Y are the two points on the boundary of the convex domain aligned with YA XB ; XA > 1 (if A and B in the order X; A; B; Y . Observe that d.A; B/ > 0 since AB A D B it is understood that d.A; B/ D 0). 5 Hilbert, following Minkowski, uses the expression nirgend concaven Körper, which means nowhere concave. In other words, the domain is convex, but not necessarily strictly convex. 6 The usual convention now is to divide this quantity by 2.

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Hilbert observes that the distance (3) depends on the given convex domain: if X approaches A and Y approaches B, then the distance d.A; B/ increases. In other words, a smaller convex domain gives rise to larger Hilbert distances. Hilbert then proceeds to give a proof of the triangle inequality. Here is the argument with Hilbert’s notation: From the invariance of the cross ratio with respect to the perspective at W , we have ŒU; V; C; A D ŒX 0 ; Y 0 ; D; A and ŒZ; T; B; C D ŒX 0 ; Y 0 ; B; D. Multiplying these identities gives ŒU; V; C; A ŒZ; T; B; C D ŒX 0 ; Y 0 ; D; A ŒX 0 ; Y 0 ; B; D D ŒX 0 ; Y 0 ; B; A ŒX; Y; B; A; which is equivalent to d.A; C / C d.C; B/ d.A; B/. W

Z V C X

X

A D

B

Y Y

U

T

Hilbert also observes that the triangle inequality degenerates to an equality when the three points are aligned with C between A and B. Furthermore, he discusses necessary and sufficient conditions for the equality in the triangle inequality. He shows that a non-degenerate triangle exists for which the sum of two sides is equal to the third if and only if there exists a plane which meets the boundary of the convex domain in two segments not on the same line (these would be the segments ŒV; T and ŒZ; U on the figure). He quotes, as a particularly interesting example, the case where the convex domain is bounded by a tetrahedron. As a final word, Hilbert points out that he always assumed the given convex body to be bounded, and this hypothesis implies that his geometry does not satisfy the Euclidean parallel postulate. Let us conclude with two remarks. We first mention that Hilbert came back at least at two occasions on the mathematical subjects discussed in his letter. First in his book

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on the foundation of geometry [7], in which he systematically develops the axiomatics of Euclidean 3-space, starting with the same axioms of incidence, order and continuity, to which he adds some axioms on congruence and the parallel postulate. The second occasion was the 1900 International Congress in Paris where he states the famous Hilbert problems. Problem IV concerns the construction and systematic treatment of the geometries in convex domains for which straight lines are the shortest, that is, the case where the triangle inequality is an equality in the case of three aligned points. Minkowski and Hilbert geometries are natural examples of such geometries. We refer to the discussion in the last chapter of this Handbook [11]. The second and last remark is about how Hilbert understood the word geometry. For Hilbert, a geometry was not conceived as an abstract metric space7 satisfying some specific axioms, but rather, as we saw, a geometry is a system made of points, lines and planes subject to a system of consistent interrelations wich are accepted as axioms. A notion of distance between points in a given geometry has then to be constructed from the given data and axioms and the properties of the distance, including the triangle inequality, is then a theorem that needs a proof rather than an axiom or a part of the initial definition. This is also in the spirit of Euclid’s Elements, where the triangle inequality is proved in Book 1, Proposition 20. The subject of metric geometry has been developed since the 1920s by a long list of mathematicians including Hausdorff, Menger, Blumenthal, Urysohn, Birkhoff, Busemann, Alexandrov, and others. Since the work of Gromov in the 1980s, metric geometry is seen as a topic of major importance in geometry. Acknowledgement. The author thanks A. Papadopoulos and K.-D. Semmler for carefully reading the manuscript and providing useful comments.

References [1]

H. Busemann, The geometry of geodesics. Academic Press, New York 1955, reprinted by Dover, Mineola, NY, 2005.

[2]

N. A’Campo and A. Papadopoulos, On Klein’s so-called non-Euclidean geometry. In Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics (ed. L. Ji andA. Papadopoulos), European Mathematical Society, Zürich, to appear.

[3]

A. Cayley, A sixth Memoir upon Quantics. Philosophical Transactions of the Royal Society of London 149 (1859), 61–90.

[4]

W. A. Coppel, Foundations of convex geometry. Austral. Math. Soc. Lect. Ser. 12, Cambridge University Press, Cambridge 1998

[5]

M. Fréchet, Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22 (1906), 1–72.

[6]

D. Hilbert, Über die gerade Linie als kürzeste Verbindung zweier Punkte. Math. Ann. 46 (1895), 91–96.

7 The

notion of abstract metric space first appeared in the 1906 thesis of M. Fréchet [5].

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[7]

D. Hilbert, Grundlagen der Geometrie. B.G. Teubner, Leipzig, 1899, various editions and translations.

[8]

F. Klein, Über die sogenannte Nicht-Euklidische Geometrie. Math. Ann. IV (1871), 573–625.

[9]

F. Klein, Vorlesungen über höhere Geometrie. Dritte Auflage, Grundlehren Math. Wiss. 22, Springer-Verlag, Berlin 1926.

[10] P. Nabonnand, La théorie des “Würfe” de von Staudt – une irruption de l’algèbre dans la géométrie pure. Arch. Hist. Exact Sci. 62 (2008), no. 3, 201–242. [11] A. Papadopoulos, Hilbert’s fourth problem. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 391–431. [12] M. Pasch, Vorlesungen über neuere Geometrie. Teubner, Leipzig 1882. [13] V. Pambuccian, The axiomatics of ordered geometry: I. Ordered incidence spaces. Expo. Math. 29 (2011), no. 1, 24–66. [14] W. Prenowitz, and M. Jordan, Basic concepts of geometry. Blaisdell Publishing Company, London 1965. [15] G. Robinson, The foundations of geometry. Math. Expositions 1, University of Toronto Press, Toronto, Ont., 1940. [16] O. Veblen and J. W. Young, Projective geometry. Vol. 1, Ginn and Co., London 1910. [17] J.-D. Voelke, Le théorème fondamental de la géométrie projective: évolution de sa preuve entre 1847 et 1900. Arch. Hist. Exact Sci. 62 (2008), no. 3, 243–296. [18] K. von Staudt Geometrie der Lage. Nürenberg 1847. [19] K. von Staudt Beiträge zur Geometrie der Lage. Nürenberg 1857.

Chapter 15

Hilbert’s fourth problem Athanase Papadopoulos

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 Precursors . . . . . . . . . . . . . . . . . . . 3 Hilbert’s Problem IV . . . . . . . . . . . . . . 4 Some early works on Hilbert’s Problem IV . . 5 Busemann’s approach and Pogorelov’s solution 6 Other works and other solutions . . . . . . . . 7 Further developments and perspectives . . . . Appendix I: Hilbert’s Problem IV . . . . . . . . . . Appendix II: Extract from the French version . . . References . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction On the 8th of August 1900, at the Second International Congress of Mathematicians held in Paris, David Hilbert delivered a lecture titled “The future problems of mathematics”, in which he presented a collection of open problems. Hilbert was forty-eight years old and was considered as one of the leading and most versatile mathematicians of his time. His list of problems turned out to be a working document for a large part of the mathematical research that was conducted in the twentieth century and it had a great influence on several generations of mathematicians. A first short paper based on that lecture appeared in 1900 in the newly founded Swiss journal L’Enseignement Mathématique [79], and after that several written versions of unequal lengths were published (see [68] for a commentary on these versions). This includes a paper published in the Bulletin of the American Mathematical Society ([80], 1901), containing a commented set of twenty-three problems.1 Today, several of these problems are 1 In the mid 1990s, a twenty-fourth problem was discovered in Hilbert’s massive files in Göttingen. This problem asks for “simpler proofs and criteria for simplicity”. The interested reader can refer to the reports in [118] and [117] on that intriguing problem that had remained hidden for almost a century. We also note that at the Paris 1900 congress, Hilbert presented only 10 problems. C. Reid, who recounts the story in [106], says (on p. 81): “In an effort to shorten his talk as Minkowski and Hurwitz had urged, Hilbert presented only ten problems out of a total of 23 which he had listed in his manuscript”.

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solved, but there is still very active research around them (the solved ones as well as the unsolved). As a general introduction to the Hilbert problems, we refer the reader to the two semi-popular books that appeared on the occasion of the anniversary of these problems: J. Gray’s The Hilbert Challenge [69] and B. H. Yandell’s The Honors Class: Hilbert’s Problems and Their Solvers [124]. The books [33] and [4] contain collections of more technical articles on the developments of these problems. Hilbert geometry, which is the topic of this Handbook, is closely connected with Hilbert’s Problem IV. The problem is titled “The problem of the straight line as the shortest distance between two points”. There are several interpretations of this problem and we shall discuss them below. Soon after Hilbert formulated it, the problem was reduced by G. Hamel – who was one of Hilbert’s student – to the case of metrics defined on convex subsets of Euclidean (or affine) spaces. If is a convex subset of a Euclidean space, then the restriction to of the ambient Euclidean metric is a metric on satisfying Hilbert’s requirement, that is, the (restriction to of the) Euclidean straight lines are geodesics for the restriction to of the Euclidean metric. One form of Hilbert’s problem asks for a characterization of all metrics on for which the Euclidean lines in are geodesics. Another natural question is to find a characterization of metrics on for which the Euclidean lines are the unique geodesics. One may put additional requirements on the metrics and try to characterize some special classes of metrics satisfying Hilbert’s requirements, such as geodesically complete metrics, that is, metrics whose geodesics (parametrized by arclength) can be extended infinitely from both sides. The Hilbert metrics are examples of geodesically complete metrics satisfying Hilbert’s problem requirement. In what follows, I shall present some ideas and make comments, several of them having a historical character, about Hilbert’s Problem IV and its ramifications, and I will concentrate on the ideas that arose from the work of Busemann. In fact, I generally followed the rule of not stating in the form Theorem: etc. anything that is not found in Busemann’s work. I have included in the footnotes some biographical notes on some of the mathematicians that appear in this paper, but not all of them. There are several available biographies of Hilbert and of Klein. For the latter, we refer the reader to [84]. The story of Hilbert’s Problem IV, of its various reformulations and of all the work done around it is an illustration of how mathematical problems can grow, transform and lead to new problems. And it also shows that mathematicians are persistent.

2 Precursors Hilbert’s problems did not come out of nowhere, and several of them have a history. Problem IV, like most of the others, arose as a natural question at that time. In the French version of the statement (see the appendix at the end of the present chapter), Hilbert mentions related work of Darboux. Darboux, in the work that Hilbert refers to,

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mentions Beltrami.2 We shall review some of Beltrami’s work related to the subject, but before that, we report on some works of Cayley and Klein. In a 1859 paper called A sixth Memoir upon Quantics [55], Cayley3 gave an interesting definition of a projective metric on the plane, that is, a metric on the projective plane such that the projective lines are geodesics for that metric. More specifically, Cayley started with a conic (which he called the absolute) in the projective plane, and he associated to it a metric d such that for any three points P1 , P2 , P3 in that order on a projective line, we have d.P1 ; P2 / C d.P2 ; P3 / D d.P1 ; P3 /. A few years later, Klein, in his two papers [86] and [87], re-interpreted Cayley’s ideas using the notion of cross-ratio, and he showed that by choosing the absolute to be a real conic in the projective plane and using Cayley’s construction, the interior of that conic is a model of hyperbolic geometry. Beltrami, in his paper [21] had already noticed this model, namely, he showed that the unit disc in the plane, with the Euclidean chords taken as 2 Eugenio Beltrami (1835–1900) was one of the major figures in differential geometry who continued the work of Riemann. He was born in a family of artists. He spent his childhood in a period of political turbulence: the Italian revolutions and the independence wars which led eventually to the unification of the country. He studied mathematics at the University of Pavia between 1853 and 1856, where he followed the lectures of Francesco Brioschi, but because of lack of money, his studies had to be interrupted. (Loria reports that Beltrami was expelled from the university because he was accused of promoting disorders against the rector [89], see also [72]). Beltrami worked during the next four years as a secretary of the director of the railway company of the Lombardo-Veneto in Verona. He undertook all over again his mathematical studies in Milan, between 1860 and 1862, where he followed the courses of Brioschi and of Cremona. Both men had a decisive influence on the future of Beltrami. Brioschi, who had moved to Milano, was very much involved in politics, and he played a major role in the formation of the new state of Italy. Beltrami published a mathematical paper in 1862, and he obtained in the same year a position at the University of Bologna. After that, he moved between several universities, partly because of the political situation in Italy. He spent his last years at the university of Rome. A stay in Pisa, between 1863 and 1866, where he was appointed professor of geodesy, was probably decisive for his mathematical future research. He was called there by Betti, and he also met there Riemann (who was in Italy for health reasons). Beltrami’s Saggio di Interpretazione della geometria non-Euclidea [21] and his Teoria fondamentale degli spazi di curvatura costante [22] which are mentioned in this survey were written during a second stay at the University of Bologna, where he held the chair of rational mechanics. The name of Beltrami is attached to the Beltrami differential, a basic notion in the theory of quasiconformal mappings, and with the Laplace–Beltrami operator. Besides mathematics, he cultivated physics, in particular thermodynamics, fluid dynamics, electricity and magnetism. He translated into Italian the work of Gauss on conformal representations. 3Arthur Cayley (1821–1895) grew up in a family of English merchants settled in Saint-Petersburg, and he lived in Russia until the age eight, when his family returned to England. Cayley is one of the main inventors of the theory of invariants. These include invariants of algebraic forms (the determinant being an example), but also algebraic invariants of geometric structures and the relations they satisfy (“syzygies”). He studied both law and mathematics at Trinity College, Cambridge, from where he graduated in 1842. He was talented in both subjects and as an undergraduate he wrote several papers, three of which were published in the Cambridge Mathematical Journal. The subject of these papers included determinants, which became later on one of his favorite subjects. After a four-year position at Cambridge university, during which he wrote 28 papers for the Cambridge journal, Cayley did not find any job to continue in academics. Looking for a new profession, law was naturally his second choice. He worked as a lawyer for 14 years, but he remained interested in mathematics, and he wrote about 250 mathematical papers. Cayley was appointed professor of mathematics at Cambridge in 1863. His list of publications includes about 900 items, on all the fields of mathematics of his epoch. The first definition of abstract group is attributed to him, cf. his paper [54]. Cayley proved that every finite group is a subgroup of some symmetric group. He is also one of the first discoverers of geometry in n dimensions. In his review of Cayley’s Collected Mathematical Papers edition in 13 volumes, G. B. Halsted writes: “‘Cayley not only made additions to every important subject of pure mathematics, but whole new subjects, now of the most importance, owe their existence to him. It is said that he is actually now the author most frequently quoted in the living world of mathematicians” [71]. The reader is referred to the interesting biography by Crilly [57].

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geodesics, is a model of the Lobachevsky plane with its non-Euclidean geodesics, but he did not have the formula for the distance in terms of the cross ratio. Klein’s model, with the cross ratio formula, constitutes what is called today the Klein model, or the Beltrami–Klein model, or also the Beltrami–Cayley–Klein model of hyperbolic space. It is worth noting that Klein also found models of the elliptic and of the Euclidean plane within the projective plane that use the cross ratio and a conic at infinity. In the case of the Euclidean plane, the conic is imaginary and the construction involves complex numbers. We refer the reader to Klein’s paper [86], and also to the commentary [1]. It may also be worth recalling here that Cayley considered the use of the cross ratio by Klein as a sort of circular reasoning, because he thought that the definition of the cross ratio is dependent upon some underlying Euclidean geometry, whereas Klein, in his paper [86], was aiming for a definition of the hyperbolic metric (and also, Euclidean and spherical metrics) based only on projective geometry. We can quote here Cayley from his comments on his paper [55] in Volume II of his Collected mathematical papers edition [62] (p. 605): I may refer also to the memoir, Sir R. S. Ball “On the theory of content,” Trans. R. Irish Acad. vol. xxix (1889), pp. 123–182, where the same difficulty is discussed. The opening sentences are – “In that theory [Non-Euclidian geometry] it seems as if we try to replace our ordinary notion of distance between two points by the logarithm of a certain unharmonic ratio. But this ratio itself involves the notion of distance measured in the ordinary way. How then can we supersede the old notion of distance by the non-Euclidian notion, inasmuch as the very definition of the latter involves the former?”

This is just to show that some of the most prominent mathematicians were missing an important idea, namely that the cross ratio does not depend on the underlying Euclidean geometry. In the same line of thought we quote Genocchi, an influential and well-established Italian mathematician of the second half of the nineteenth century4 who wrote ([67], p. 385): From the geometric point of view, the spirit may be chocked by certain definitions adopted by Mr. Klein: the notions of distance and angle, which are so simple, are replaced by complicated definitions [...] The statements are extravagant.”

The second paper of Klein [87] contains an elaboration on some of the ideas contained in the first one. In the meanwhile Beltrami found two models of hyperbolic geometry, and Klein noticed that his model coincides with one of the models that Beltrami found. This is the origin of the so-called Klein–Beltrami model of hyperbolic space. Klein’s aim in the two papers that we mentioned was to include the three 4Angelo

Genocchi (1817–1889) was in Italian mathematician who was also a very active politician. Like Cayley, he worked for several years as a lawyer. He taught law at the University of Piacenza, the city where he was born, and at the same time he was cultivating mathematics with passion. In 1859, Genocchi was appointed professor of mathematics at the University of Torino, and he remained there until 1886. During the academic year 1881–82, Guiseppe Peano served as his assistant, and he helped him later on with his teaching, when Genocchi became disabled after an accident. During several years, Genocchi was the main specialist in number theory in Italy.

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constant-curvature geometries in the setting of projective geometry (Cayley had done this only for Euclidean geometry), and although Klein did not formulate it in these words, he obtained models for these three geometries in which lines in projective space are geodesics. This fact was highlighted by Beltrami, who obtained in [21] the discmodel of hyperbolic geometry, where the Euclidean straight lines are the geodesics, but Beltrami did not have Klein’s formula involving the cross-ratio. One may add here that two years before Klein’s work on the subject, Beltrami wrote the following to Houël5 (letter dated July 29, 1869 [23], p. 96–97): The second thing [I plan to do in addition to the memoir I sent you]6 will be the most important, if I succeed to give it a concrete form, because up to now it only exists in my head in the state of a vague conception, although without any doubt it is based on the truth. This is the conjecture of a close analogy, and may be of an identity, between pseudo-spherical geometry7 and the theory of Mr Cayley on the analytical origin of metric ratios, using an absolute conic (or quadric). I almost did not know anything about that theory, when I was taken by the identity of certain forms. However, since the theory of invariants plays there a rather significant role and because I lost sight of it since a few years now, I want to do it again after some preliminary studies, before I address this comparison.

In another letter to Hoüel, written on July 5, 1872, Beltrami regrets the fact that he let Klein outstrip him ([23], p. 165): The principle which has directed my analysis8 is precisely that which Mr Klein has just developed in his recent memoir9 on non-Euclidean geometry, for 2-dimensional spaces. In other words, from the analytic point of view, the geometry of spaces of constant curvature is nothing else than Cayley’s doctrine of the absolute. I regret very much to have let Mr Klein supersede me on that point, on which I had already assembled some material, and it was my mistake for not giving to this matter enough importance.

There are comments on this work of Klein in the paper [1]. 5 Guillaume Jules Hoüel (1823–1886) is a major figure in the history of non-Euclidean geometry. There is a very interesting correspondence between him and Beltrami, and sixty-five of these letters were edited by Boi, Giacardi and Tazzioli in 1998 [23]. We refer to that correspondence several times in the present chapter. Hoüel translated into French and published in French and Italian journals works by several authors on non-Euclidean geometry, including Bolyai, Beltrami, Helmholtz, Riemann and Battaglini. In his first and memorable letter to Hoüel, written on November 18, 1868, Beltrami (who was seven years younger than Hoüel), starts as follows: “Professor, I take the liberty of sending you, in a file, two memoirs, the object of the first one is a real construction of non-Euclidean geometry, and the other, much less recent, contains the proof of some analytic results upon which this construction relies, but which, apart from that, have no immediate relation with that question”. The two memoirs are the Saggio di Interpretazione della geometria non-Euclidea [21] and the Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette [20]. It seems that Hoüel was the first to notice that Beltrami’s work on the pseudo-sphere implies that the parallel postulate is not a consequence of the other Euclidean postulates. Beltrami, in a letter dated January 2, 1870 ([23], p. 114), writes: “Your conclusion on the impossibility of proving in the plane the eleventh axiom, as far as it follows from my work, is also mine”. See the discussion in [95], §1.3 of the Introduction. 6 Beltrami, in this letter, gave a favorable response to Hoüel who had proposed to translate his two memoirs Saggio di Interpretazione della geometria non-Euclidea [21] and Teoria fondamentale degli spazi di curvatura costante [22] into French and to publish the translations in the Annales de l’École Normale Supérieure. 7 This is the term used by Beltrami to denote hyperbolic geometry. 8 Beltrami refers here to a note [18] which he had just published in the Annali di Matematica. 9 Beltrami refers to Klein’s paper [86].

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We next talk about Beltrami’s work. In 1865, Beltrami wrote a paper [20] whose title is Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette (A solution of the problem: Transfer the points of a surface over a plane in such a way that the geodesic lines are represented by straight lines). From the title, it is clear that the problem bears a direct relation to Hilbert’s problem. Indeed, if such a transfer map exists, then, by pushing forward the metric of the given surface to a metric on the Euclidean plane, we get a metric on that plane for which the Euclidean straight lines are geodesics. Beltrami worked on this subject in the framework of the differential geometry of surfaces, in the style developed by Gauss. In the introduction to his paper [20], Beltrami says that most of the research done before him on similar questions was concerned with the question of conservation of angles or of area, and that even though these two properties are regarded as the simplest and most important properties of such transfer maps, there are other properties that one might want to preserve. Indeed, the kind of question in which Beltrami was interested was motivated by the theory of geographical maps. A geographical map is a representation in the Euclidean plane of a part of the globe, considered as a sphere. It was clear since the beginning of the art of geographical maps – which dates back, as a science, to Greek antiquity – that there is no such a representation which is faithful (that is, where the proportions are respected) regarding distances and angles at the same time. Beltrami declares in this paper that since the projection maps that are used in this science are primarily concerned with the measure of distances, one would like to exclude projection maps where the images of distance-minimizing curves are too remote from straight lines. He mentions in passing that the central projection of the sphere is the only map that transforms geodesics into straight lines. As a matter of fact, this is the question that motivated Beltrami’s research on the problem referred to in the title of his essay, “Transfer the points of a surface over a plane etc.” Beltrami then writes that beyond its applications to geographic map drawing, the solution of the problem may lead to “a new method of geodesic calculus, in which the questions concerning geodesic triangles on surfaces can all be reduced to simple questions of plane trigonometry”. He finally acknowledges that his investigations on this question did not lead him to any general solution, and that the case of the central projection of the sphere is the only one he found where the required condition is realized. Beltrami proved though an important local result in that paper, viz. he showed that the only surfaces that may be (locally) mapped to the Euclidean plane in such a way that the geodesics are sent to Euclidean lines must have constant Gaussian curvature. In fact, Beltrami starts with a surface whose line element is given, in the tradition of Gauss, by p ds D Edu2 C 2F dudv C Gdv 2 : The line element represents the distance between the two infinitesimally nearby points .u; v/ and .uCdu; vCdv/ on the surface. After several pages of calculations Beltrami

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ends up with the following values for the functions E, F and G: 8 R2 .v 2 Ca2 / ˆ ˆ ˆ E D .u2 Cv2 Ca2 /2 ; < R2 uv F D .u2 Cv 2 Ca2 /2 ; ˆ ˆ 2 2 ˆ .u Ca2 / : G D .uR2 Cv 2 Ca2 /2 :

397

(2.1)

He then uses the following formula for the curvature, which he derived in his Ricerche [19], art. XXIV: ! !! @E @E @G F 2 @F CF 1 @ @ @E 1 @u E @v @v @u E @u D p C p p R1 R2 @v 2 EG F 2 @u EG F 2 EG F 2 After a computation, Beltrami finds that the curvature is constant and equal to 1 1 D 2: R1 R2 R He concludes from this: “Therefore our surfaces are those of constant curvature. In particular, if the quantity R is real, the formulae (2.1) are responsible for all surfaces that are applicable on the sphere of radius R”. Beltrami summarizes his result in the following statement: Theorem 2.1. The only surfaces that can be represented over a plane, in such a way that to every point corresponds a point and to every geodesic line a straight line are those whose curvature is everywhere constant (positive, negative or zero). When this constant curvature is zero, the correspondence does not differ from the ordinary homography. When it is nonzero, this correspondence is reducible to the central projection over the sphere and to its homographic transformations. Beltrami adds: Since among all the surfaces of constant curvature the only one that can have applications in the theory of geographic maps and in geodesy is probably the spherical surface, in this way from the point of view of these applications, what we asserted confirms that the only solution to the problem is obtained essentially by the central projection.

In conclusion to his paper, Beltrami states the following: In order for the points of a surface to be transported over a surface of constant curvature in such a way that the geodesic lines of the former be represented by geodesic lines of the latter, it is necessary that the first surface be of constant curvature.

Thus, in his conclusion, Beltrami considered only the case where the curvature is positive, and therefore the only surface he found that has the required property (apart from the plane itself) is the sphere (of a certain radius). We refer the reader to the chapter [120] in this Handbook for an exposition with a detailed proof of this result of Beltrami. Two years later, Beltrami wrote another paper, Saggio di Interpretazione della geometria non-Euclidea [21], which soon became famous, in which he worked on

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surfaces of constant negative curvature, a context in which he gave an interpretation of Lobachevsky’s geometry. In that paper, Beltrami defined a family of metrics on the disc of radius a > 0 centered at the origin of the Euclidean plane, using the formula ds 2 D R2

.a2 v 2 /du2 C 2ududv C .a2 u2 /dv 2 ; .a2 u2 v 2 /2

(2.2)

where ds 2 denotes as usual the square of the infinitesimal length element. He showed that the Gaussian curvature of that surface is constant and equal to 1=R2 , and that its geodesics are the Euclidean straight lines. He also stated the following: It follows from what precedes that the geodesics of the surface are represented in their total (real) development by the chords of the limit circle, whereas the prolongation of these chords in the exterior of the same circle are devoid of any (real) representation.10

Figure 1. Extracted from Beltrami’s paper [21]. The figure represents the unit disc model of the hyperbolic plane with a triangle pqr having two vertices at infinity.

Beltrami gave in the paper [21] the first Euclidean model of hyperbolic plane (the socalled “Beltrami” model, or “Klein” model, or “Klein–Beltrami” model or “CayleyKlein–Beltrami” model11 ), and at the same time he gives a new example of a metric on the plane satisfying Hilbert’s problem. We mention by the way that this research by Beltrami was motivated by his reading of the works of Lobachevsky. In a letter to Hoüel, written on November 18, 1868, Beltrami writes: “That writing [The Saggio 10 The 11 The

English translation in [114] of this sentence from [21] is not very faithful. four names are reasonable, see the comments in [1].

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etc.] was written during last year’s fall, after thoughts that abounded in me at the epoch of the publication of your translation of Lobatschewsky. I think that the idea of constructing non-Euclidean geometry on a perfectly real surface was entirely new” ([28]), p. 65–66). In Note I at the end of the paper [21], Beltrami recalls the metric of constant positive curvature 1=R2 that he defined in his 1865 paper [20], whose square of the infinitesimal line element is given by the formula ds 2 D R2

.a2 C v 2 /du2 2ududv C .a2 C u2 /dv 2 .a2 C u2 C v 2 /2

(2.3)

and he makes an interesting remark on the relation between the two metrics (2.2) and (2.3), namely, that the formula for one pmetric canpbe obtained from the other one by replacing the constants R and a by R 1 and a 1, thus confirming the statement that hyperbolic geometry is in some sense spherical geometry worked out on a “sphere of imaginary radius”. A modern proof of some of Beltrami’s results are contained in Busemann’s book [40], Chapter II, §15. Busemann introduces this result as follows: Although the methods are quite foreign to the rest of the book, we prove this fundamental Theorem of Beltrami in Section 15, because it is by far the most striking example of a Riemannian theorem without a simple analogue in more general spaces.

Busemann states Beltrami’s result as follows (see [40], p. 85, where Busemann refers to Blaschke [24]): Theorem 2.2. Let C be a connected open set of the projective plane P 2 equipped with a Riemannian metric whose geodesics lie on projective lines. Then the Gauss curvature is constant. For dimension 3, Busemann refers to Cartan [52]. There exists an earlier proof by Schur [109] and there are several modern proofs, see e.g. [90] and [60]. Remark 2.3.12 It is not sufficiently known that Beltrami’s theorem holds with very weak assumptions on the regularity on the metric. This was first noticed by Hartman and Wintner in an interesting paper [76], in which they prove that if a Riemannian metric is continuous and projective, then it is real analytic. Pogorelov rediscovered the result of Hartman and Wintner several years later. Note that by Brickell [32], in dimension 3, a continuous mapping between two smooth affine spaces which sends geodesics to geodesics (possibly without preserving parameterization) is automatically smooth. Darboux, who is quoted by Hilbert in his formulation of his fourth problem, in his Leçons sur la théorie générale des surfaces (Troisième partie), following Beltrami’s work, considers, in the setting of the calculus of variations, a similar problem. He 12 I

owe this remark to J. C. Álvarez Paiva.

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writes: “Proposons-nous d’abord le problème de M. Beltrami: Trouver toutes les surfaces qui peuvent être représentées géodésiquement sur le plan.” ([58], p. 59). That is to say, Darboux considers Beltrami’s problem of sending a surface into the plane by a map that sends the surface geodesics to the plane straight lines. According to Busemann ([44], p. 56), Darboux proposed several questions that are related to Hilbert’s problem, some of them being more general than the one in hand.13 In fact, one can relate the above-mentioned “problem of geographical maps” to problems that were studied by Lambert, Euler, Lagrange, Gauss, Darboux, Liouville and Bonnet. In his paper [61], Euler studied the problem of mapping a sphere on a plane such that “figures are similar” at the infinitesimal level. Gauss formulated again the problem of “representing a surface on the plane so that the representation is similar, at the very small level, to the original surface”. See [66]. This means that we ask for the similarity at the infinitesimal level for lines and angles between them, see Gauss’s quotes and the comments in the paper [94]. Using a more modern language, this is the question of finding conformal coordinates. In 1816, H. C. Schumacher, a famous German–Danish astronomer at Copenhagen who was a friend of Gauss and had been his student in Göttingen, announced a prize to be awarded by the Royal Society of Copenhagen for an answer to the problem of obtaining a general method for mapping conformally a surface onto another one. Gauss won the prize for a solution he submitted in 1822. This solution is contained in his paper [66].

3 Hilbert’s Problem IV The best way in which one can have an idea on Hilbert’s Problem IV is to read Hilbert’s text, and this is why we reproduce it in the appendix at the end of this chapter. It is also good to remember and to meditate on Hilbert’s conclusion to his Mathematical Problems.14 : The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection become ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. 13 Darboux’s “solution” of Hilbert’s fourth problem (before this problem was stated) misses however one of the most important features of the problem: by considering the variational problem for integrands of the form f .x; y; y 0 / instead of integrands of the form L.x; y; x 0 ; y 0 / that are homogeneous of degree one in the velocities x 0 and y 0 , Darboux gives many variational problems whose extremals are straight lines, but which cannot define metrics because they are not even defined on the whole tangent bundle (minus the origin). Darboux’s approach can be modified and be made to yield yet another solution to Hilbert’s fourth problem in dimension two in the smooth asymmetric case. Berck and Álvarez-Paiva did this, and later found out that it is also implicit in Hamel’s paper [75]. I owe this information to Juan Carlos Álvarez Paiva. 14 The translation is from the article in the 1902 Bulletin of the AMS [80].

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For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself more clearly. But we ask, with the extension of mathematical knowledge, will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science. The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of the exact knowledge of natural phenomena. That it may completely fulfill his high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples.

We now make some comments on Hilbert’s Problem IV. The formulation by Hilbert and the scope of his fourth problem are very broad compared to other problems in his list,15 and this problem admits several interpretations and generalizations. In fact, the problem was considered several times in the past as being solved.16 We shall recall in later sections how several mathematicians who worked on this problem interpreted it in their own manner and contributed to a solution. We mention right away the names of G. Hamel and P. Funk who worked on it in the early years of the 20th century, the fundamental work of H. Busemann over a span of fifty years (1930–1980), and the works of A. V. Pogorelov, Z. I. Szabó, R. Ambartzumian, R. Alexander, I. M. Gelfand, M. Smirnov and J.-C. Álvarez Paiva. We shall elaborate on some of these works below. There are at least five natural settings in which one can approach Hilbert’s Problem IV: (1) The setting of Riemannian geometry. In this setting, the problem was settled by Beltrami. 15As examples of Hilbert problems that are more precisely stated, we mention Hilbert’s Problem III, asking whether two 3-dimensional Euclidean polyhedra of the same volume are scissors-equivalent. This problem was solved in the negative by M. Dehn soon after Hilbert formulated it. We shall elaborate on this in Footnote 21 below. Another well-known example of a precise problem in Hilbert’s list is Part 2 of Problem VII, which asks whether given any algebraic number a 6D 0; 1 and given any irrational algebraic number b, ab is transcendental. This question was solved by A. O. Gelfand in 1934 and the proof was refined by T. Schneider in 1935. The result is called now the Gelfand–Schneider Theorem. But there are also problems in Hilbert’s list which are vaguely stated (and some of them are so because of their nature). For instance, Problem VI asks for a “mathematical treatment of the axioms of physics”. 16 One can search for papers titled “A solution to Hilbert’s Problem IV” and similar titles; there are several such papers.

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(2) The setting of the calculus of variations, which was the one of Darboux in the work we mentioned. In this setting, Hilbert’s problem is often described as an “inverse problem”; that is, one looks for a Lagrangian (defining the metric) which is associated to a given set of extremals (which are the geodesic lines). As is well known, the methods of the calculus of variations started in the works of Johann and Jakob Bernoulli, of Euler and of Lagrange. (3) The setting of metric geometry, where one does not assume any differentiability. This setting is at basis of the methods of Busemann and of the large amount of interesting works that arose from them. (4) The setting of Finsler geometry, where the metric is associated to a norm on each tangent space of the manifold. A minimum of differentiability is required in this setting, at least to give a meaning to tangent vectors. This setting can be considered as lying at the border of metric and of differential geometry. (5) The setting of the foundations of geometry. This was probably the most important setting for Hilbert, and we shall elaborate on it below. In fact, this setting is also inherent in one way or another in all the different settings mentioned above. There are also other points of view and relations with other questions. For instance, R. Alexander established relations between Hilbert’s Problem IV and zonoid theory (see [3] and the paper [115] by Z. Szabó), S. Tabachnikov made relations with the theory of magnetic flows (see [116]). I. M. Gelfand, M. Smirnov and J.-C. Álvarez Paiva developed relations with symplectic geometry (see [9], [7] and [8]). The result in [7] can be stated as follows: There exists a (twistorial) correspondence between smooth, symmetric solutions of Hilbert’s fourth problem in 3-dimensional projective space and anti-self-dual symplectic forms in S 2 S 2 . In any case, Hilbert’s Problem IV and the various approaches to solve it had an important impact on the development of several fields of mathematics, including all the ones we mentioned, to which we can add convex geometry and integral geometry. The developments that arose from that problem brought together major ideas originating from the various fields mentioned. Let us return to Hilbert’s statement, “the problem of the straight line as the shortest distance between two points”. Today, when we talk about straight lines being the shortest distance, we think of a subset of a Euclidean space Rn equipped with a metric to which the term “shortest distance” refers, and where the term “shortest line” refers to a Euclidean line, or a Euclidean segment. One should remember however that the axioms for a metric space, as we intend them today, were formulated (by M. Fréchet) only in 1906, that is, seven years after Hilbert stated his problem. From Hilbert’s own elaboration on his problem [80], it appears that he was thinking, rather than of a metric space, of a geometrical system, like Euclid’s axiomatic system (or like his own axiomatic system); that is, a system consisting of undefined objects (points, lines, congruence, etc.) and axioms that make connections between these objects. In fact, Hilbert’s problem can be included in a perspective that was dear to him on

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the axioms of geometry, and, more generally, on the foundations of mathematics.17 Indeed, in his comments about the problem, Hilbert gave the following formulation of the question18 : In analogy with the way Lobachevsky (hyperbolic) geometry sits next to Euclidean geometry, by denying the axiom of parallels and retaining all the other axioms of Euclidean geometry, explore other suggestive viewpoints from where geometries may be devised which stand, from such a point of view, next to Euclidean geometry.

This sentence clearly shows that Hilbert suggested to work out a theory in which one negates one of the axioms (other than the axiom of parallels) of Euclidean geometry, while keeping the other axioms untouched. Such an interpretation of Hilbert’s Problem IV led to interesting developments which we shall mention in the next section, but of course no geometry whose importance is comparable to the Lobachevsky geometry was discovered by keeping, among the axioms of Euclidean geometry, the parallel axiom untouched and denying another one. In fact, Hilbert gave a more precise formulation of that problem in the axiomatic framework, by adding that he asks for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms, except the one of the congruence of triangles and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.19

Hilbert’s problem, in this setting, can be formulated as follows: Consider the axiom system of Euclidean geometry, drop from it all the congruence axioms that contain the notion of angle20 and replace them by an axiom saying that in any triangle the sums of the lengths of any two sides is not smaller than the length of the third side (that is, the triangle inequality). Then: (1) characterize the geometries satisfying these conditions; (2) study individually such geometries. We shall elaborate more on the axiomatic point of view in the next section. 17 We can recall here Hilbert’s own words on a system of axioms for a geometry: “It must be possible to replace in all geometric statements the words point, line, plane by table, chair, beer, mug”, cf. Hilbert’s On the axiomatic method in mathematics (recalled in [106], p. 57). One may also recall here that others, before Hilbert (and probably since Euclid’s epoch), had the same point of view on the axiomatic method, even if they did not express it with the same words. 18 The translation is in [80]. 19 We note that the word “congruence”, in Hilbert’s point of view, designates a relation between primary objects like segments, angles, triangles, etc. satisfying certain properties (some of which are axioms), and that this word does not necessarily refer to an equivalence relation in a metric sense, that is, an isometry. In fact, as in Euclid’s axioms, there is no distance or length function involved in Hilbert’s axioms. Congruence can be thought of as an undefined notion. 20 For instance, in Hilbert’s axioms for Euclidean geometry, the following axiom is part of the set of congruence axioms for angles:

b 1 b

Given an angle ABC and given a point B 0 and a ray B 0 C 0 starting at B 0 , and given a choice of a side on the line B 0 C 0 , there exists a ray B 0 A0 starting at B 0 , with A0 being on the chosen side of B 0 C 0 , such that A0 B 0 C 0 ABC . Note that in Euclid’s Elements, this is a theorem (Proposition 23 of Book 1).

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4 Some early works on Hilbert’s Problem IV Among the major mathematicians who worked on Hilbert’s Problem IV, there are Hilbert himself and his students Dehn21 , Hamel22 and Funk23 . The title of Dehn’s dissertation is Die Legendre’schen Sätze über die Winkelsumme im Dreieck (1900) [59] (Legendre’s theorems on the sum of the angles in a triangle), the title of the one of Hamel is Über die Geometrien in denen die Geraden die kürzesten sind (1901) (On the geometries where the straight lines are the shortest), and the title of the one of Funk is Über Flächen mit lauter geschlossenen geodätischen Linien (1911) (On surfaces with none but closed geodesic lines). All these works are related to Problem IV. Among the other early works that are related to Hilbert’s problem, one also has to mention those of Finsler24 and of Berwald.25 We shall report on some of their works. 21 Max Dehn (1878-1952) is one of the main founders of combinatorial topology and combinatorial group theory. His thesis, defended in 1900, concerns questions on the foundations of geometry, and it is related to Hilbert’s Problem IV, in an extended form discussed by Hilbert. Dehn constructed a geometry in which there are infinitely many lines passing through a point and disjoint from another line and in which Legendre’s theorem stating that the sum of the angles in any triangle is at most fails. (Therefore this geometry is different from hyperbolic geometry.) Dehn’s geometry is non-Archimedean, and in this work Dehn constructed a nonArchimedean Pythagorean field. The same year, Dehn, who was only 22 years old, solved Hilbert’s Problem III which we already mentioned in Footnote 15. The problem asks whether two 3-dimensional (Euclidean) polyhedra that have the same volume are scissors-equivalent, that is, whether they can be obtained from each other by cutting along planes and re-assembling. The corresponding result in dimension 2 is true and is known as the Bolyai–Gerwien Theorem. Dehn showed that in dimension 3 the answer is no, which was Hilbert’s guess. In his work on that problem, Dehn introduced a scissors-equivalence invariant for 3-dimensional polyhedra, which is now called the Dehn invariant, and he showed that this invariant does not give the same value to the tetrahedron and the cube. In 1965, J.-P. Sydler – an amateur mathematician – completed Dehn’s theory by showing that two polyhedra are scissors-equivalent if and only if they have the same volume and the same Dehn invariant. 22 Georg Hamel (1877–1954) entered Göttingen University in 1900. In 1901 he was awarded his doctorate, under the supervision of Hilbert. Hamel is probably best known for his contribution on the so-called Hamel basis, a work published in 1905, in which he gave a construction of a basis of the real numbers as a vector space over the rationals. This construction is important because it involves one of the earliest applications of the axiom of choice. Hamel later made a name for himself in the field of mechanics. 23 Paul Funk (1886–1969) is mostly known for his work on the calculus of variations and for his introduction of the Funk transform (also called the Minkowski-Funk transform), an integral transform obtained by integrating a function on great circles of the spheres. The subject of his dissertation work (1911), surfaces with only closed geodesics, which he wrote under Hilbert, was further developed by Carathéodory. We owe him the Funk metric, an asymmetric metric defined on convex subsets of Rn which is a non-symmetric version of the Hilbert metric satisfying the requirements of Hilbert’s Problem IV. This metric is studied in detail in Chapter 2 of this Handbook [100]. 24 Paul Finsler (1894–1970) wrote his doctoral dissertation in Göttingen in 1919 under the supervision of Carathéodory. The title of the thesis is Über Kurven and Flächen in allgemeinen Räumen (On Curves and surfaces in general spaces). According to Busemann [38], Finsler was not the first to study Finsler spaces. These spaces had been discovered by Riemann, who mentioned them in his famous lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypothesis which lie at the foundations of geometry) (1854) [107]. Busemann recalls in this respect that the goal that Riemann set for himself was the definition and discussion of the most general finite-dimensional space in which every curve has a length derived from an infinitesimal length or line element. The name Finsler space is due to É. Cartan, after his paper Les espaces de Finsler [53] published in 1934. In their paper [50], Busemann and Phadke write the following: “Finsler was the first who investigated non-Riemannian spaces (under strong differentiability hypotheses) guided by Carathéodory, whose methods in the calculus of variations form the basis of Finsler’s thesis, which has no relation to our G-spaces. The misnomer was caused by the fact that Finsler’s thesis (1918) was inaccessible until 1951, when Birkhäuser reissued it unchanged”.

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Let us now say a few words on the work of Hilbert on the foundations on mathematics, in order to explain the axiomatic approach that he alludes to in the comments on his fourth problem. In the various revised editions of his Grundlagen der Geometrie26 (Foundations of geometry) [81], Hilbert worked out several sorts of geometries which he termed as “non-Euclidean”,27 a word he used in a very broad sense, in the formulation of his fourth problem. This includes first the well-known non-Archimedean geometry,28 in which all the axioms of Euclidean geometry are kept untouched except the Archimedean axiom. We already mentioned that a non-Archimedean geometry was also the subject of the doctoral thesis of Dehn (1902). Then, Hilbert constructed a nonArguesian geometry, which is a plane geometry in which the theorem of Desargues29 of projective geometry fails,30 and a non-Pascalian geometry, in which the theorem of Pascal fails.31 To define these geometries, one has to construct new number fields, in particular a field of non-Archimedean numbers and a field of non-Pascalian num25 Ludwig Berwald (1883–1942) obtained his doctorate at the Royal Ludwig-Maximilians-University of Munich under the guidance of Auriel Voss. The title of his dissertation was: Über die Krümmungseigenschaften der Brennflächen eines geradlinigen Strahlsystems und der in ihm enthaltenen Regelflächen (On the curvature properties of the caustics of a system of rays and the ruled surfaces contained in it). Berwald is considered as one of the founders of Finsler geometry. 26 The first edition appeared in 1899, that is, one year before Hilbert formulated his Problems. 27 The term non-Euclidean is used here in a broad sense, and not only in its classical sense which usually includes only hyperbolic and spherical geometry. 28 Guiseppe Veronese (1854–1917) constructed, in 1889 (that is, a few years before Hilbert did), a nonArchimedean geometry. Hilbert mentions the work of Veronese in the statement of his problem [80]. 29 Since this theorem plays a central role in the work of Hilbert and in the work of Busemann on axioms, we recall its statement. Consider in the projective plane two triangles abc and ABC . We say that they are in axial perspectivity if the three intersection points ab \ AB, ac \ AC , bc \ BC are on a common line. We say that the three triangles are in central perspectivity if the three lines Aa; Bb; C c meet in a common point. Desargues’ theorem says that for any two triangles, being in axial perspectivity is equivalent to being in central perspectivity. This theorem was published for the first time by A. Bosse, in his Manière universelle de M. Desargues pour manier la perspective par petit pied comme le géométral (Paris, 1648, p. 340). Bosse’s memoir is reproduced in Desargues’ Œuvres (ed. N. G. Poudra, Paris 1884, p. 413–415). Desargues’ proof of the theorem uses Menelaus’ Theorem. Von Staudt, in his Geometrie der Lage (Nuremberg, 1847) gave a proof of this theorem that uses only projective geometry notions, namely, notions of incidence between points and lines, etc., but in three-space, and not only in the plane. Klein, in his famous paper [87], noted that this theorem cannot be proved using only twodimensional projective geometry. Hilbert, in his Grundlagen der Geometrie (2nd. ed., §23) exhibited a plane geometry in which all the axioms of projective geometry are satisfied, but where the theorem of Desargues does not hold. Thus, the theorem of Desargues follows from the axioms of three-dimensional projective geometry, but not of those of the two-dimensional. It can be proved in two-dimensional projective geometry if one introduces metric notions. Conversely, Hilbert proved that for three-dimensional projective geometry, it is sufficient to have the axioms of two-dimensional geometry together with the plane theorem of Desargues. 30A model of non-Arguesian geometry, described by Hilbert in the first editions of his Grundlagen, is obtained by taking an ellipse in the Euclidean plane and replacing the Euclidean segments intersecting it by arcs of circles that pass by a common fixed point P . Thus, in this geometry, a line is taken in the usual sense if it does not intersect the (interior of the) ellipse, and in the case where it intersects it, it is made out of three pieces: two half-lines, outside the ellipse, connected by an arc of a circle inside the ellipse. The group of Euclidean axioms which Hilbert calls the “projective axioms” are satisfied by such a geometry provided the point P is not too close of the ellipse. In later editions, Hilbert replaced this model by a model found by F. R. Moulton, cf. [93]. 31 Pascal’s theorem in projective geometry says that given a hexagon in the projective plane, there is an equivalence between the following two statements: (1) the hexagon is inscribed in a conic; (2) The intersections of the lines containing pairs of opposite sides are aligned.

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bers. A non-Pascalian geometry is also non-Archimedean. Hilbert also considered a geometry he called non-Legendrian, in which it can be drawn from a point infinitely many parallels to a line that does not contain it, and where, like in spherical geometry, the angle sum in a triangle is greater than two right angles. He then considered a geometry he called semi-Euclidean, in which one can draw from a point infinitely many parallels to a line not containing it, and where the angle sum in a triangle is equal to two right angles. In fact, both non-Legendrian and semi-Euclidean geometries are considered by Dehn in his 1900 dissertation [59], as two distinct non-Archimedean geometries. Hilbert also considered a non-Pythagorean geometry. In his paper Über eine neue Begründung der Bolyai–Lobatschefskyschen Geometrie [82], he constructed a non-Archimedean Lobachevsky geometry. Another “non-Euclidean” geometry that Hilbert mentioned is the Minkowski geometry, which he described as a geometry in which all the axioms of Euclidean geometry are satisfied except a “triangle congruence” axiom. In this geometry, the theorem stating that the angles at the basis of an isosceles triangle are equal is not satisfied and the theorem stating that in any triangle the sum of the lengths of two sides is less than the length of the third side is taken as an axiom. In his comments on his fourth problem, Hilbert mentioned a geometry he described in his paper [78] where the parallel axiom is not satisfied but all the other axioms of a Minkowski geometry are satisfied. In this paraphernalia of geometries, only a few satisfy the requirements of Problem IV, but it is clear from the statement of the problem and the work that Hilbert did at the same period that he was thinking about new geometries in which a variety of other properties could replace the fact that “shortest distances are realized by straight line”. Poincaré commented on these geometries and on the number fields to which they give rise in his review of Hilbert’s Grundlagen der Geometrie, [105]. He writes (p. 2): Many contemporary geometers [...] in recognizing the claims of the two new geometries [the hyperbolic and the spherical, which Poincaré calls the Lobachevsky and the Riemann geometries] feel doubtless that they have gone to the extreme limit of possible concessions. It is for this reason that they have conceived what they call general geometry, which includes as special cases the three systems of Euclid, Lobachevsky, and Riemann, and does not include any other. And this term, general indicates clearly that, in their minds, so other geometry is conceivable. They will loose this illusion if they read the work of Professor Hilbert. In it they will find the barriers behind which they have wished to confine us broken down at every point.

In the conclusion to his comments on Problem IV, Hilbert declares that it would be desirable to make a complete and systematic study of all geometries in which shortest distances are realized by straight lines. The conclusion of the problem in its French version is different from that of the English one: in the former, Hilbert includes the problem in the setting of the calculus of variations, and he refers to the work of Darboux that we already alluded to. The reader can find an English translation of the conclusion in the Appendix to the present chapter. It may be worth mentioning that in order to make the relation between the axiomatic abstract setting and the metric setting, one may refer to Euclid. We already recalled

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that there is no distance function involved in Euclid’s Elements, but if we introduce this metric language, it is a consequence of Euclid’s axioms (in particular Axiom 2) that on a line, distance is additive, and it follows from Proposition 20 of Book I that if between three points distance is additive then the three points are on a line (see [77]). This implies that the Euclidean lines in the sense of the axioms of geometry are the (geodesic) lines in the metric sense. We now say a few words on other early works on Problem IV. Hamel considered in [75] metric spaces .; d / where is an open subset of a projective space P n for some n > 1 and where d satisfies the following strong version of Hilbert’s requirements (stated here in modern terms, following Busemann [44]): (1) .; d / is a geodesic metric space and the projective lines are geodesics; (2) the closed balls are compact; (3) if x; y; z do not lie on a projective line, then d.x; y/ C d.y; z/ > d.x; z/. Note that Conditions (1) and (3) imply that the metric d is uniquely geodesic. Under these hypotheses, Hamel proved that the set is necessarily convex. More precisely, he proved the following: Theorem 4.1. If the metric space .; d / satisfies (1) to (3), then is of one of the following two types: (1) can be mapped homeomorphically onto the projective space P n in such a way that the maximal geodesics of are mapped to great circles and all great circles have equal lengths; (2) can be mapped homeomorphically onto an open convex subset of affine space An in such a way that the maximal geodesics of have infinite length and are mapped to the intersection of with affine lines in An . In particular, closed geodesics and maximal open geodesics cannot coexist in .; d / . Let us note that the two geometries of constant curvature +1 and -1 are examples of metric spaces described in Theorem 4.1, namely, spherical geometry is an example of type (1), and hyperbolic geometry, through the Klein–Beltrami model, is an example of type (2). Hamel considered only smooth metrics, and his methods are variational. In his thesis [74] (1901), he considered non-necessarily symmetric metrics. In his later paper [75] (1903) he also considered non-symmetric metrics although he was not very explicit about that. He refers to symmetric metrics by the terms “Starke Monodromieaxiome” and to non-symmetric metrics by “Schwache Monodromieaxiome”.32 The main 32 The meaning is “strong monodromy hypothesis” and “weak monodromy hypothesis” respectively. The reason for this terminology comes from dimension two, where Hamel (and Funk after him) use polar coordinates. In this case, the Lagrangian depends on r and . The periodicity of the angle variable imposes that its value is invariant by adding integer multiples of 2 (this is the weak monodromy hypothesis). In the case of a symmetric metric, the value of must be invariant by adding integer multiple of (this is the strong monodromy hypothesis). I owe this remark to Marc Troyanov.

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result he obtained in that paper is a characterization of the differential of arclength, for a general smooth (symmetric and non-symmetric) metric satisfying Hilbert’s requirement, in dimensions 2 and 3. Hamel’s result is presented in Chapter 3 [120] in this Handbook. At its publication, Hamel’s paper [75] was considered as a solution to Hilbert’s problem, under the given restrictive assumptions. Concerning Hamel’s method, in his 1974 report [46] on Hilbert’s Problem IV, Busemann writes (p. 137): We do not reproduce it here, since it is not based on general ideas like integral geometry and the expression given by Hamel is not illuminating. Also, inevitably in view of the time it was written, the discussion of completeness does not satisfy our modern requirements; the necessary concepts did not yet exist. However, the immense number of possibilities becomes quite clear from [75].

Busemann, in [40], gave a proof of the same result, putting it in a modern setting and using completely different methods, which are purely metrical. He gave a nonsymmetric version which he attributed to Hamel. A proof of Hamel’s result is contained in Busemann [44], p. 37, which Busemann states as follows in the setting of nonsymmetric metrics: Theorem 4.2. Under the hypotheses of Theorem 4.1 but without the assumption that the metric is symmetric and where Condition (2) is replaced by the requirement that the right closed balls are compact,33 the space .; d / is of one of the following two types: • D P n , and each segment of a projective line in P n , traversed in either direction, is a geodesic. Furthermore, all such geodesic are projective lines which, traversed in either direction, have the same length. • is a convex set and contained in some affine n-dimensional subspace An of P n . Finally, we mention a result of Berwald related to Hilbert’s Problem IV. Berwald proved in [27] (with a simpler proof given by Funk in [64]) that 2-dimensional Finsler spaces of constant flag curvature34 that satisfy the requirements of Hilbert’s Problem IV are characterized by the property that any isometry of a geodesic onto another (or onto itself) is the restriction of a projectivity. This result generalizes Beltrami’s result which he obtained in the setting of Riemannian geometry. A proof of this result is given in Chapter 3 [120] of this Handbook. The works of Hamel, Funk and Berwald had a great influence on Busemann, who made a new definite breakthrough on the problem, which we describe in the next section. a non-symmetric metric d , one has to distinguish between the right open ball of center x and radius r, D fy j d.x; y < rg and the left open ball of center x and radius r, B C .x; r/ D fy j d.y; x/ < rg. The right and left closed balls are defined by replacing the strict inequalities by large inequalities. 34 There are several notions of curvature in the setting of Finsler spaces, and an important one is the notion of flag curvature, which is a kind of Finsler analogue of the sectional curvature of Riemannian geometry. We refer the reader to §41 of Busemann’s Geometry of geodesics [40] for a discussion of curvature in Finsler spaces and to Chapter 3 of this Handbook [120]. 33 For

B C .x; r/

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5 Busemann’s approach and Pogorelov’s solution We start with a reformulation of the problem. We already mentioned that, in its original formulation, Hilbert’s Problem IV is sometimes considered as too broad, and several mathematicians in their work on it made it more precise. Busemann, who spent a large part of his intellectual activity thinking and working on that problem reformulated it as follows (see [46]): The fourth problem concerns the geometries in which the ordinary lines, i.e. lines of an n-dimensional (real) projective space P n or pieces of them, are the shortest curves or geodesics. Specifically, Hilbert asks for the construction of these metrics and the study of the individual geometries. It is clear from Hilbert’s comments that he was not aware of the immense number of these metrics, so that the second part of the problem is not at all well posed and has inevitably been replaced by the investigation of special, or special classes of, interesting geometries.

The second part of the problem is indeed very wide, and because of that it is likely to remain open forever. Busemann writes in [47]: The discovery of the great variety of solutions showed that Part (2) of Problem 4 is not feasible. It is therefore no longer considered as part of the problem. But many interesting special cases have been studied since 1929.

Part (2) of the problem is nevertheless fascinating, in many cases of the individual geometries. There are various examples where new phenomena can be found, even in the following classes of classical metric spaces: (1) Minkowski metrics (that is, translation-invariant metrics on Rn defined by norms, which may be non-symmetric);35 (2) Funk metrics;36 (3) Hilbert metrics. Each of the geometries for which the Euclidean straight lines are geodesics defines a new world, and one may try to answer in that world numerous questions regarding triangles, boundary structure, trigonometry, parallelism, perpendicularity, horocycles, compass and straightedge constructions, and there are many others. The two- and three-dimensional cases are particularly worth investigating in detail. According to Busemann [47], at the time where Hilbert proposed his problem, the only classes of metrics (“besides the elementary ones”) that were known to satisfy the requirements of that problem were the Minkowski metrics, introduced around 1890 by H. Minkowski for their use in number theory, and the Hilbert metrics. The latter were discovered by Hilbert in 1894 [78], as a generalization of the Klein–Beltrami model of hyperbolic geometry. We note that both classes of metrics are Finsler and almost 35 The Minkowski metrics are the translation-invariant metrics associated to finite-dimensional normed spaces, except that the underlying norm function (and therefore the distance function) is not necessarily symmetric. See Chapter 1 [99] in this Handbook. 36 For a survey on Funk geometries, see Chapter 2 [100] in this Handbook.

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never Riemannian.37 Indeed, it is a consequence of Beltrami’s work that we already mentioned, that the only Riemannian metrics that satisfy Hilbert’s requirements are the metrics of constant curvature. The Minkowski and the Hilbert metrics are mentioned in Hilbert’s comments on Problem IV. Busemann notes furthermore that Minkowski geometries satisfy the parallel axiom and that Hilbert geometries generalize the hyperbolic one.38 Busemann adds that probably Hilbert did not think of “mixed situations”, which Busemann himself considered in his various books and papers and which we recall now. Busemann introduced new classes of metrics which satisfy Hilbert’s requirements and which are combinations of Minkowski and Hilbert metrics. In his book [40] (p. 111 ff.), he first makes the simple remark that the metric L.p; q/Ce.p; q/ where L is the Hilbert metric and e is the Euclidean metric has the required properties, and then he defines several variations on this metric. In particular, he considers a combination of the Euclidean metric with the Hilbert metric restricted to the intersection of the convex set with a flat subspace. He then defines (in dimension 2) several classes of combinations which depend on parameters and which satisfy interesting parallelism properties (in the sense of hyperbolic geometry). In [46], returning to these examples, he considers the explicit metric e.x; y/ C je xn e yn j on affine space with affine coordinates x D .x1 ; : : : xn /, and he notes that there are variations on this example where instead of the exponential function one can take any monotone function. He then adds the following: Many such functions can be added, and watching convergence even infinitely many. Also, the addition can be replaced by an integration over a continuous family. Of course, it is very easy to generate nonsymmetric distances by this method, for example by modifying nonsymmetric Minkowski distances. It seems clear that Hilbert was not aware of these simple possibilities.

For arbitrary open convex subsets of An , Busemann also exhibits “infinitely many essentially different Desarguesian metrics”, namely, he takes the metric e.x; y/ C j tan xn tan yn j defined on the strip =2 < xn < =2, and again, he notes that the only property used is that tan t increases monotonically from 1 to C1 in .=2; =2/. One of the major achievements of Busemann was to put Hilbert’s Problem IV in an adequate metric setting. In particular, he introduced the terminology of a G-space and of a Desarguesian space which we recall now, and over several decades he did extensive work on the geometry of these spaces. At several places, he gave slightly different but overall equivalent definitions of G-spaces and of Desarguesian. In [36] and [37], Busemann gave the following definition, see also Busemann’s book The geometry of geodesics [40]. 37 The exceptions in both cases are represented by ellipsoids: the case where the convex set on which the Hilbert metric is defined is an ellipsoid, and the case where the unit ball of the Minkowski metric is an ellipsoid centered at the origin. 38 Recall that the Hilbert metric of the disc is the hyperbolic metric.

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Definition 5.1 (G-space). A G-space39 is a metric space .X; d / satisfying the following properties: (1) (Finite compactness) Every bounded infinite set has an accumulation point. (2) (Menger convexity) For any x and z in X , there exists a point y which is between x and z, that is, y different from x and z and satisfies d.x; y/Cd.y; z/ D d.x; z/. (3) (Local extendability) For every point p of X there exists an open ball B centered at p such that for every x and y in B there exists a point z in B such that y is between x and z. (4) (Uniqueness of extension) If y is between x and z1 and between x and z2 and if d.y; z1 / D d.y; z2 / then z1 D z2 . In particular, in a G-space, geodesics are extendable, and the extension is unique. Busemann gave then the following definition of a Desarguesian space (see [40] and [47]): Definition 5.2 (Desarguesian space). A Desarguesian space is a metric space .R; d / satisfying the following properties: (1) R is an open nonempty subset of projective space P n equipped with a metric .x; y/ whose associated topology is the one induced from its inclusion in P n . (2) The closed balls in R are compact. (3) Any two points in R can be joined by a geodesic segment, that is, a curve of length .x; y/. (Here, the length of a curve is defined as usual as the supremum over all subdivisions of the image set of a segment of the set of lengths of polygonal curves joining the vertices of the subdivision; in modern terms, a metric where any two points can be joined by a geodesic segment is said to be geodesic). (4) For any three points x, y, z in P n that are not collinear, the strict triangle inequality .x; y/ C .y; z/ < .x; z/ holds between them; that is, the three points do not belong to a geodesic. Remark 5.3 (Non-symmetric metrics). In the later generalizations of G-spaces and of Desarguesian space to non-symmetric metrics that Busemann considered (see e.g. [44]), the condition that the closed balls are compact is replaced by the condition that the right closed balls are compact. 39 The expression G-space stands for “geodesic space”. The concept was introduced quite early by Busemann. In their paper [50], p. 181, Busemann and Petty recall that “The G-spaces were first introduced without this name by Busemann in his thesis in 1931.” This notion turned out to be of paramount importance, and a lot of work has been done, by Busemann and others, especially around the conjecture of Busemann saying that every finite-dimensional G-space is a topological manifold. Busemann himself proved the conjecture in dimensions 1 and 2 ([40] §9 and §10), Krakus [88] proved it in dimension 3 using a theorem of Borsuk [30] and P. Thurston [119] in dimension 4. See the survey [73] by Halverson and Repovš. It may be useful to recall in this respect that G-spaces are metric and separable, and therefore the usual concept of dimension in the sense of Meyer-Urysohn applies to them, cf. [83].

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Thus, whereas the definition of a G-space concerns general metric spaces, the one of a Desarguesian space concerns metrics on subsets of projective space. A Desarguesian space is a particular case of a G-space and it is also a particular case of a space satisfying the requirement of Hilbert’s Problem IV. In a Desarguesian space, the geodesic joining two points is unique and is necessarily the affine or projective line. The terminology chosen by Busemann comes from the fact that in such a space, Desargues’ theorem40 holds. Beltrami’s theorem, which we already quoted, implies that in the Riemannian case, the only Desarguesian spaces are the spaces of constant curvature, that is, the Euclidean, hyperbolic or elliptic spaces. The terminology used today for spaces satisfying Hilbert’s problem requirements, especially in the context of Finsler geometry, is that of a projective metric, or projectively flat metric (cf. also Busemann in [39]). Let us note that Busemann was suspicious about the Finsler geometry viewpoint, in the sense in which this word was used in the early 1950s. His feeling about it is expressed in his survey paper [38] which starts as follows: The term “Finsler space” evokes in most mathematicians the picture of an impenetrable forest whose entire vegetation consists of tensors. The purpose of the present lecture is to show that the association of tensors (or differential forms) with Finsler spaces is due to an historical accident, and that, at least at the present time, the fruitful and relevant problems lie in a different direction.

The paper ends with the following words: This confirms that in spite of all the work on Finsler spaces we are now at a stage which corresponds to the very beginning in the development of ordinary differential geometry. Therefore the mathematician who likes special problems has the field. After sufficiently many special results have been accumulated someone will create the appropriate tools. At the present time it is difficult to guess what they will be beyond a vague feeling that some theory of integro-differential invariants will be essential.

Busemann interpreted Hamel’s work in his setting of Desarguesian spaces; let us quote him from [40] (p. 66): Whereas Hamel [75] has given a method for constructing all Desarguesian, sufficiently differentiable G-spaces, no entirely satisfactory infinitesimal characterization of these spaces, in terms of analogues to curvature tensors say, has ever been given. The freedom in the choice of a metric with given geodesics is for non-Riemannian metrics so great, that it may be doubted, whether there really exists a convincing characterization of all Desarguesian spaces.

Busemann also considered the axiomatic viewpoint. In §13 of his book The geometry of geodesics [40], he made relations between the metrical point of view on Problem IV and the foundations of geometry point of view. The Section starts with the following introduction: This section is closely related to the classical results of the foundations of geometry, and the methods of this field are partly used here. We outline briefly the analogies as well as the differences of that work with the present. 40 See

Footnote 29.

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A. V. Pogorelov,41 in his monograph [104] that accounts for his solution of Hilbert’s Problem IV, notes that the Minkowski geometries (respectively the Hilbert geometries) realize the system of axioms of Euclidean geometry (respectively of Lobachevsky geometry) after we drop all the axioms of congruence involving the concept of angle and adding to them the triangle inequality axiom (which is one of Hilbert’s requirements at the level of axioms). He adds (without further elaboration) that Hilbert’s problem can be formulated in a similar way for elliptic geometries. Álvarez Paiva in [5] makes further comments on the elliptic geometry case. The idea of Pogorelov’s proof of Hilbert’s Problem IV came from Busemann, who, at the Moscow ICM (1966), gave a construction of a Desarguesian space, which introduced ideas from integral geometry in the approach to Hilbert’s problem.42 The method provided a metric on any bounded open convex set B in a projective space P n . We already noted that (by a result of Hamel) a solution of Hilbert’s problem is necessarily a metric defined either on P n or on an open convex subset of affine space (sitting as the complement of a hyperplane in P n ). We briefly recall Busemann’s definition in case where B D P n . Let H be the set of hyperplanes in P n . For any subset X of P n , we let X be the set of hyperplanes that intersect X. On the set H , we define a nonnegative measure m satisfying the following properties: (1) For any point p in P n , m.fpg/ D 0. (2) For every nonempty open subset X of P n , m.X / > 0. (3) m.P n / D 2k < 1. Then we notice that for every line L in P n we have L D P n , which shows that m.L/ D 2k. Now for every pair of distinct points x and y in P n , consider the line L that contains them. The two points divide L into two arcs, A and B, and m.A/ C m.B/ D 41Aleksei Vasil’evich Pogorelov (1919–2001) spent most of his career in Kharkiv. He belonged to the famous Russian mathematical school, and his two advisors wereAlexandrov and Efimov. He first studied at the Zhukowski Air ForceAcademy in Moscow, and he worked as an engineer before he started his graduate studies in mathematics in Moscow. Before his work on Hilbert’s Problem IV, Pogorelov had published several important papers, including solutions of some famous problems. In 1949, he gave a substantial generalization of Cauchy’s rigidity theorem from convex polyhedra to convex surfaces. We recall that Cauchy’s theorem (1813) states that any two convex polytopes in R3 which have congruent faces are congruent. Pogorelov’s theorem states that any two closed isometric convex surfaces in R3 are congruent. The proof used a famous result of Alexandrov on gluing convex surfaces, which was used by the latter to prove that any surface homeomorphic to a sphere and equipped with a metric of positive curvature is isometric to a convex surface, answering a famous problem which was asked by Weyl. The history of Cauchy’s theorem is interesting. A mistake in Cauchy’s proof was found by E. Steinitz in 1920, and corrected by him in 1928. Other improvements and extensions of the theorem are due to several people, among them Dehn (1916) and Alexandrov (1950). In 1952, Pogorelov published a solution of the socalled multi-dimensional regularity problem of the Minkowski problem, stating that any convex surface whose Gaussian curvature is positive and is a C m function of the outer normal, for m 3, is C mC1 -regular. (This problem has several later developments and generalizations, by Nirenberg, Cheng–Yau and others.) 42 There is no text by Busemann in the proceedings of this ICM, and his name is not mentioned in the list of participants. But there is a paper by him in Russian, in the Russian journal Uspehi of the same year, [43], in which he presents Hilbert’s problem and his results related to the solution of that problem.

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m.P n / D 2k. Therefore, one of the two values m.A/ or m.B/, say m.A/, is k. Setting .x; y/ D m.A/ defines a Desarguesian metric on P n . Explicit formulae can be obtained for such measures since the set of hyperplanes in projective space can easily be parametrized. One must remember here that Busemann, at the time of the Moscow 1966 ICM, was working in relative isolation, and that the subject of metric geometry was not fashionable in the West. On the contrary, in the Soviet Union, the subject, which flourished independently, was considered as important, and it was represented by a host of very good and influential mathematicians including A. D. Alexandrov, V. A. Zalgaller, V. A. Toponogov, N. V. Efimov, Yu. G. Rechetnjak, and A. V. Pogorelov himself. The interaction between the two schools, before Busemann’s visit to Moscow, was poor. One should also note that Busemann, during his college years, followed the lectures of the topologist P. S. Alexandrov and of other Russian mathematicians. In her biography of Courant (who was Busemann’s advisor in Göttingen), Constance Reid writes ([106], p. 106): Since 1923 Alexandrov had returned each year, either alone or accompanied by countrymen. From 1926 through 1930, Courant always arranged for him to give courses in topology, each for a quite different audience of mathematicians. The summer that Courant was trying to keep up some semblance of attendance at Wiener’s lectures, Alexandrov’s were crowded. “Alexandroff and the other Russian visitors were very important, very influential,” I was told by Herbert Busemann, who came in Göttingen in 1925 to study mathematics after “wasting” – as he said – several years of his life in business to please his father, one of the directors of Krupp. “The Russians filled a gap because they were familiar with certain more abstract tendencies which were not well represented in Göttingen. Courant, as probably many have told you, was rather reactionary in his mathematical outlook. He didn’t see the importance of many of the modern things”.

let us also recall that in 1985, Busemann received the Lobachevsky prize, which is the most prestigious reward in geometry that was given the Soviet Union (and, before that, in Russia).43 In last paper [50], written in collaboration with Phadke and published in 1993, Busemann recalls his beginning in metric geometry. He writes the following (p. 181): Busemann has read the beginning of Minkowski’s Geometrie der Zahlen in 1926 which convinced him of the importance of non-Riemannian metrics. At the same time he heard a course on point set topology and learned Fréchet’s concept of metric spaces. The older generation ridiculed the idea of using these spaces as a way to obtain results of higher differential geometry. But it turned out that a few simple axioms on distance suffice to obtain many non-trivial results of Riemannian geometry and, in addition, many which are quite inaccessible to the classical methods.

In the concluding remarks of the same paper, Busemann writes: 43 The prize was given to Busemann “for his innovative book The geometry of geodesics” which he had written 30 years before. We also recall that the first recipients of this prize were Sophus Lie in 1897, Wilhelm Killing in 1900 and David Hilbert in 1903. The prize was awarded to A. D. Alexandrov in 1951.

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The acceptance of our theory by others was slow in coming, but A. D. Alexandrov and A. M. Gleason were among the first who evinced interest and appreciated the results.

Let us now quote Pogorelov, from the introduction to his monograph [104]: The occasion for the present investigation is a remarkable idea due to Herbert Busemann, which I learned from his report to the International Congress of Mathematicians at Moscow in 1966. Busemann gave an extremely simple and very general method of constructing Desarguesian metrics by using a nonnegative completely additive set function on the set of planes and defining the length of a segment as the value of the function of the set of planes intersecting the segment. I suspected that all continuous Desarguesian metrics could be obtained by this method. The proof of this in the 2-dimensional case strengthened my belief in this conjecture and I announced a general theorem in [103]. However it turned out later, on making a detailed investigation of the three-dimensional case, that the completely additive set function figuring in Busemann’s construction may not satisfy the non-negativity condition. Therefore, the result given here, while preserving its original form, assumes that other conditions are satisfied.

Busemann’s construction mentioned by Pogorelov is based on the integral formula for distances defined as a measure on the set of planes that we mentioned above. The formula is contained in earlier work of Blaschke [25], based on an integral formula which is due to Crofton44 in the setting of geometric probability. We shall say a few more words on this formula below, since it was at the basis of most of the developments that followed. As already mentioned, Pogorelov’s solution of Hilbert’s Problem IV consisted in showing that every Desarguesian metric is given by Busemann’s construction, but with a slight modification. Pogorelov obtained the result, first in dimension n D 2 for a general continuous metric, and then for n D 3, with a smoothness assumption. He also proved that in dimension 3, every metric that is a solution of Hilbert’s Problem IV is a limit (in the sense of uniform convergence on compact sets) of a sequence of such metrics that are of class C 1 . It turned out that the proof for n D 3 needs more work than the one for n D 2; in particular, a new definition of the measure m, allowing it to take negative values (but with still positive values on every set of the form T , where T is a segment) is needed. In fact, Pogorelov’s solution (even in dimension 2) is variational and it uses the smoothness condition. For dimension 2, Pogorelov obtained the result for continuous metrics by using an approximation argument (which does not apply in dimensions > 2). One should also note that the fact that in dimension > 2 the measure m can take negative values is also mentioned in Busemann’s earlier paper [41] in which he showed that all Minkowski metrics which are smooth enough can be obtained by this construction. 44A formula for a metric on a subset of Euclidean space that uses a measure on the set of Euclidean hyperplanes, where the distance between two points x and y is equal to the measure of the set of Euclidean hyperplanes that meet the segment Œx; y, is sometimes referred to as a Crofton (or sometimes Cauchy–Crofton) formula. The name is after Morgan Crofton (1826–1915) who proved a result of integral geometry relating the length of a curve to the expected number of times a random line intersects it, see [108], p. 12–13.

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Pogorelov’s exposition in the book [104] is elementary and pleasant. Busemann wrote a review on that book in [47] in which he confirmed that Pogorelov’s method of proof for n D 3 works, with more technicalities, for n > 3. It is fair to recall here that Hilbert, in the spirit of his time, restricted himself to dimensions 2 and 3. In their paper [48], Busemann and Phadke work out a general and global45 version of the 1865 Beltrami Theorem [20] in the setting of Desarguesian spaces. They define a general class of metric spaces which they call chord spaces. These are metric spaces possessing a distinguished class of extremals (that is, locally isometric maps from an interval into the metric space), called chords, which behave locally like the affine lines, but which are not the unique extremals of the metric. Chord spaces are the subject of the monograph [49]. Busemann and Phadke obtain in [48] the following results, which they “believe to be the most general meaningful version of that theorem”: Theorem 5.4. A locally Desarguesian simply connected chord space is either defined in all of S n or is an arbitrary open convex set of an open hemisphere of S n (considered as the projective space An ). Theorem 5.5. A simply connected locally Desarguesian and locally symmetric Gspace is Minkowskian, hyperbolic or spherical. At the end of their paper [48], Busemann and Phadke write: Surprisingly, one can clearly deduce from a paragraph (which is too long to be produced here in total) in Beltrami’s paper [21] that he would have welcomed our generalization of his theorem. He says that a theorem which holds under weaker hypotheses than stated has not been fully understood and that the proper generalization may involve the disappearance of some of the original concepts (in our case the Riemannian metric).

6 Other works and other solutions More general versions of Pogorelov’s solution in the 2-dimensional case were later on obtained by completely different methods by Ambartzumian [16] and by Alexander [2]. Ambartzumian’s method can be used only in dimension 2 and it gives the result of Pogorelov directly for continuous metrics, without passing by smooth metrics and without using any approximation argument. The method has a combinatorial character, and it uses the scissors-congruence of triangles which is the subject of Hilbert’s Problem III which we already mentioned in Footnote 15. The fact that Ambartzumian’s method does not work in dimension > 2 can be compared to the fact that the solution to Hilbert’s Problem III says that in dimension > 2, two polyhedra with the same volume are not necessarily scissors-equivalent. 45 Let

us recall that Beltrami’s problem is indeed local.

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Szabó, in a paper [115] containing several new ideas on the subject, gave two new proofs of Pogorelov’s theorem in a Finslerian setting, valid in all dimensions, one of them being elementary and based on ideas originating in Ambartzumian’s work. Szabó’s result is given in terms of a partial differential equation satisfied by the indicatrix of the Finsler structure, which in dimension n is assumed to be of class C nC2 . The proof uses Fourier transform theory, and precisely, a relation between the problem of characterizing (symmetric) projective metrics of low regularity and that of characterizing the (distributional) Fourier transform of norms on finite-dimensional vector spaces. For the relation between the Fourier transform and integral geometry on normed and projective Finsler spaces, see the paper [6] by Álvarez Paiva and Fernandes. A new point of view on Hilbert’s problem, based on an integral formula in dimension 2, is given in the paper [2] by Alexander, whose result can be considered as a general solution which does not even assume the continuity of the metric. In that paper, the author considers an indefinite metric (that is, a metric which does not necessarily separates points) on a 2-dimensional space homeomorphic to a Euclidean plane, equipped with a set of “lines” (in an abstract sense), which are homeomorphic to the real line. He shows that under very mild conditions (the metric is not assumed to be continuous and the set of lines are only assumed to satisfy Desargues theorem) there exists a unique Borel measure on the set of lines such that the distance between two points is given by integrating this measure. The methods are based on the author’s combinatorial interpretation of the Crofton formula from integral geometry. This work includes Busemann’s integral construction in the abstract setting of the axioms of geometry which was one of Hilbert’s favorite settings. The relation of Hilbert’s Problem IV with the foundations of geometry is highlighted again in the paper [3] by Alexander in which this author extends his previous result to higher dimensions, using the notion of zonoids from convexity theory.46 Alexander works in the settings of hypermetrics, that is, metrics d such that for any points n P1 ; : : : ; P Pm in R and for any integers N1 ; : : : ; Nm satisfying N1 C C Nm D 1, one has i <j d.Pi ; Pj /Ni Nj 0 (see [85]). This is a strong version of the triangle inequality (and for m D 3, the property boils down to the triangle inequality). Alexander notes in [3] that in dimension 2, all solutions of Hilbert’s problem are hypermetrics, and the main result of this paper is that there is a natural linear isomorphism between the cone of hypermetrics on Rn that are additive along lines (that is, hypermetrics that satisfy the requirement of Hilbert’s Problem IV) and the cone of positive Borel measures m on the set of hypersurfaces of Rn satisfying m.P / D 0 and 0 < m.ŒP; Q/ D 0 < 1 for any distinct P and Q in Rn . Here, as before, for X Rn , X denotes the set of hyperplanes that intersect X . The isomorphism is given by a generalized Crofton-type formula d.P; Q/ D m.ŒP; Q/. Alexander’s 46 One first defines a zonotope as a finite Minkowski sum of line segments. Equivalently, a zonotope is the affine projection of a high-dimensional cube. This can be seen as a generalization of the notion of polytope. A zonoid is then a limit of zonotopes in the sense of Blaschke–Hausdorff convergence. Equivalently, a zonoid is the limit of sums of segments. We refer the reader to the survey [29] by E. D. Bolker.

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methods using zonoids were further generalized by Schneider (see [112] and [113]), with a result valid for a higher-dimensional analogue of Hilbert’s problem involving area, volume, etc. and not only length. In the last section of this chapter, we recall Busemann’s formulation of this higherdimensional analogue. In the paper [15], Ambartzumian and Oganian give (Theorem 1) a necessary and sufficient condition for a Finsler metric in dimension 2 to be a solution of Hilbert’s problem. In the same paper, they study a parametric version of the problem, where the convex domain which is the support of the metric (which is assumed to be centrally symmetric) depends continuously on parameters. They also consider the question of when the family of Finsler metrics obtained is a Minkowski metric. They obtain a complete solution in the case of a dependence on a single (1-dimensional) parameter. The smoothness condition is essential in their work. Among the other recent works on the Finsler smooth case, we mention the works of Z. Shen and of R. L. Bryant. We recall that Funk, in his papers [63] and [64] gave a classification of Finsler metrics on convex domains in the plane which have constant flag curvature. In his paper [111], Shen described a class of smooth projective Finsler metrics of constant flag curvature using algebraic equations. In the paper [65], Funk proved that the standard metric of the sphere S 2 is the only flat projective Finsler metric, satisfying some additional conditions. Bryant, in [31] and [34] generalized the result of Funk; he described a 2-parameter family of projective Finsler metrics on S 2 and he showed that this family gives all such metrics. In his work, Bryant made use of the symplectic geometry of the space of geodesics (the quotient of the unit tangent bundle by the geodesic flow). Finally, we mention that Álvarez Paiva gave in [8] another point of view on Hilbert’s Problem IV in the framework of symplectic and contact topology. He obtained a characterization of Finsler metrics which are solutions of Hilbert’s problem in terms of a class of symplectic forms on the space of lines in Rn . At the same time, the result of Álvarez Paiva makes a relation between Hilbert’s problem and the “black and white” theory of Guillemin and Sternberg [70]. The results of Álvarez Paiva in his paper [8] were first announced in the paper [9] by himself, Gelfand and Smirnov, The paper [56] by M. Crampin contains reformulations and new versions of the works of Hamel and of Álvarez Paiva.

7 Further developments and perspectives A problem like Hilbert’s Problem IV never dies, and as we tried to show, it admits several generalizations and formulations in various settings. We mention that there are classical works on problems that are more general than Hilbert’s problem, namely, problems where the metrics on subsets of the plane are to be found, with prescribed geodesics which are curves that are not necessarily straight lines, and which satisfy

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certain conditions, e.g. they foliate the plane and each of them is homeomorphic to a real line that separates the plane into two components. Blaschke and Bol consider such problems in [26]. Before that, Darboux had already considered such a general problem, in the setting of the calculus of variations, see [58] and he obtained local existence results using theorems on partial differential equations. Busemann and Salzmann also considered such general problems in [51]. The case of nonsymmetric metrics is still wide open although, talking about this case, Busemann mentions in [44] (p. 36): “most contributors to Hilbert’s Problem IV have treated this general case”. In fact, Hilbert, when he formulated his problem, certainly had this case in mind, since he knew about the interesting examples of non-symmetric metrics discovered by Minkowski, and he was aware of the fact that extremal metrics obtained by the methods of the calculus of variation may lead to non-symmetric length functions. We already mentioned that Hamel, who proposed in his thesis [74] the first proof of Hilbert’s problem for dimensions 2 and 3, considered the case of nonsymmetric metrics. Szabó, in his paper [115], considered only the symmetric case and in the final section of that paper he outlined a sequel to that work which would treat the non-symmetric case; the sequel never appeared. Alexander, in the paper [2] that we mentioned above discusses the nonsymmetric case but without being able to settle it. Busemann also asked for the generalization of Hilbert’s problem (see [41] and [42]) to the study of k-dimensional area in n-dimensional affine spaces for which the affine k-flats minimize area. We quote Busemann, from the introduction to [42]: In his Problem IV, Hilbert [Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 1900, 253–297] suggested studying all geometries in which the straight lines are the shortest connections or minimize length. Although the problem in this form proved too general to be fruitful, it has given rise to many investigations dealing either with special geometries or with special questions. The higher-dimensional analogue to Problem IV concerns a-dimensional metrics in subsets of the n-dimensional projective space in which the a-flats minimize area. Such geometries apparently have never been treated, at least in the large. The present paper represents a first step in this direction.

In the same paper, Busemann asks for the study of k-dimensional areas that are given by a Crofton-type formula, similar to the one he gave for distances. We already mentioned advances on this problem, cf. the papers [6] byÁlvarez Paiva and Fernandes, in which the authors prove the Crofton formulas for smooth projective metrics, and [112] by Schneider, in which the author establishes again the Crofton-type formula for the Holmes–Thompson area for Finsler projective hypermetrics. Besides the two questions in Hilbert’s Problem, viz. the characterization of the metrics with the required properties and the study of such metrics individually, there is a third question, that lies in between, which is the question of finding general properties of such metrics. Busemann considered this question in various papers and books, and we mention as an example the following embedding result (part of it is obtained in [40], p. 81, and it was completed in [44], p. 32):

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Theorem 7.1. For any n-dimensional Desarguesian G-space R there exists an .n C 1/-dimensional Desarguesian space R such that R is a hyperplane in R and the restriction of R to R is the given metric on R. Another general result on Desarguesian spaces, also due to Busemann, is the following characterization of Hilbert and Minkowski geometries among all Desarguesian spaces ([44], p. 38): Theorem 7.2. Let .; d / be a non-compact Desarguesian space whose distance function d is non-necessarily symmetric. Assume that any isometry between two lines in is a projectivity. Then .; d / is of one of the following two sorts: • is a convex subset of projective space and d is its Hilbert metric; • is affine space and d is a Minkowski metric. This is a special case of a theorem that Berwald obtained under an additional differentiability hypothesis and which we mentioned at the end of Section 4. The following result of Busemann gives a characterization of Funk and Minkowski geometries among Desarguesian spaces [45]: Theorem 7.3. Any Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space. We also mention a result by Busemann which is more in the axiomatic spirit of Hilbert. Busemann develops in [40], §23 and 24, a theory of parallelism between geodesics in metric spaces which generalizes the Euclidean theory. He introduces in particular a parallel axiom (see [40], p. 141) and he proves the following (Theorem 24.1 of [40]): Theorem 7.4. A Desarguesian space in which the parallel axiom holds and where the spheres are convex is Minkowskian. Several of Busemann’s students and collaborators worked on general properties of Desarguesian spaces. We mention in particular the following result of E. M. Zaustinsky (one of Busemann’s students) [125]. Theorem 7.5. A Desarguesian metric defined on affine space and whose right and left open balls are all compact is necessarily a Minkowski metric. B. B. Phadke [102] (who was also a student of Busemann), proved the following. Theorem 7.6. A Desarguesian space in which the equidistant loci to hyperplanes are hyperplanes is necessarily a Minkowski space. We mention the following problems:

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Problem 7.7. Modify Hilbert’s fourth problem so that geometric circles replace lines, in spaces of constant curvature and on more general metric spaces. Problem 7.8. Generalize Hilbert’s Problem IV to the setting of metric spaces or convex subsets of metric spaces other than Euclidean spaces; that is, study the metrics on subspaces such that the geodesics of the ambient metric are geodesics for the new metric. (We recall that Busemann developed a theory of convexity in general metric spaces, see [40] and the exposition in [101].) Problem 7.9. Study in particular the preceding problem on Teichmüller space equipped with its various metrics. Problem 7.10. Study a complex analogue of Hilbert’s problem. Problem 7.11. In the paper [12], the authors used the great variety of explicit constructions of projective metrics to show that totally geodesic submanifolds (in that case, planes) are not minimal surfaces for the Hausdorff measure. A nice question mentioned in that paper is to see whether the following holds: Suppose that for a given projective metric, planes (or hyperplanes) are minimal for the Hausdorff measure; is the metric then either flat or Riemannian? Problem 7.12. The set of projective pseudo-metrics (pseudo distance functions) on Rn is a closed convex cone. Determine the extreme rays of this cone, for n 3. Problem 7.13. (This is an open version of Beltrami’s theorem): Consider a Riemannian metric defined on the real projective space such that all projective planes are minimal surfaces. Is the metric of constant curvature? Let us make a few comments on these problems. Problem 7.7 was solved by Álvarez Paiva and Berck in [13], for Finsler metrics and in the case of circles on the two-sphere. The 2-dimensional smooth version was solved by Segre [110]. Regarding Problem 7.8, in the case where the metric is Riemannian or semiRiemannian, this problem was extensively studied in the literature, see for example [91]. See also Chapter 12 of [35] for a list of open problems related to this topic. We also mention the work in [97] and the survey in Chapter 13 of this Handbook [98], which are related to a version of Hilbert’s Problem IV in the setting of constant curvature geometries. We study there two metrics (namely a Funk-type metric and a Hilbert-type metric) on open subsets of the sphere and of hyperbolic space such that the geodesics for the restriction of the ambient metric are geodesics for our two metrics. We can also mention the so-called Apollonian weak metric, which is a metric defined on open subsets of Rn . In the case where the convex set is the open unit disc or

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the upper half-plane (considered as the ambient spaces of the Poincaré models of hyperbolic geometry), the non-Euclidean geodesics are also geodesics of the Apollonian metric; see [96], and the recent paper [123]. The reason for the mention of Teichmüller space in Problem 7.9 is the amount of interest and of profound work that was done around it in the last six or seven decades. We note that there is a so-called Weil–Petersson Funk metric metric on Teichmüller space, which was first introduced by Yamada in [122], defined in analogy with the Funk metric on convex subsets of Rn (see Chapter 12 [92] in this Handbook and see also the survey [121]). There are several ways to formulate precisely Problem 7.10, and we propose the use of complex geodesics (holomorphic maps from the unit disc to a complex manifold). The Kobayashi metric could play the role played by the Euclidean metric in the real case. Problem 7.12 is due to Álvarez Paiva. It is solved for n D 2 in [14]. Problem 7.13 is due to Bryant, and it was communicated to me by Álvarez Paiva who also noted that there are examples of such metrics with non-constant curvature in R3 , but there are no known examples defined on the whole projective space. Let us also mention that Álvarez Paiva studied the problem of extending Hilbert’s problem to rank one symmetric spaces. In a recent preprint, [10], he showed that if M is a manifold diffeomorphic to a compact rank-one symmetric space equipped with a reversible (i.e. symmetric) Finsler metric F all of whose geodesics are closed and of the same length and if L is a Finsler metric on M whose geodesics coincide as unparametrized geodesics with those of F , then L is the sum of a reversible Finsler metric and an exact 1-form. Finally, Álvarez Paiva and Berck obtained recently a complete characterization of projective Finsler (not necessarily symmetric) metrics. The preprint [11] contains the details of the result for metrics defined on the whole projective n-space.

Appendix I: Hilbert’s Problem IV (The translation is from the article in the 1902 Bulletin of the AMS.) Problem 4: Problem of the straight line as the shortest distance between two points. Another problem relating to the foundations of geometry is this: If from among the axioms necessary to establish ordinary Euclidean geometry, we exclude the axiom of parallels, or assume it as not satisfied, but retain all the other axioms, we obtain, as is well known, the geometry of Lobachevsky (hyperbolic geometry). We may therefore say that this is a geometry standing next to Euclidean geometry. If we require further that that axiom be not satisfied whereby, of three points on a straight line, one and only one lies between the other two, we obtain Riemann’s (elliptic) geometry, so that this geometry appears to be the next after Lobachevsky’s. If we

Chapter 15. Hilbert’s fourth problem wish to carry out a similar investigation with respect to the axiom of Archimedes, we must look upon this as not satisfied, and we arrive thereby at the non-Archimedean geometries which have been investigated by Veronese and myself. The more general question now arises: Whether from other suggestive standpoints geometries may not be devised which, with equal right, stand next to Euclidean geometry. Here I should like to direct your attention to a theorem which has, indeed, been employed by many authors as a definition of a straight line, viz., that the straight line is the shortest distance between two points. The essential content of this statement reduces to the theorem of Euclid that in a triangle the sum of two sides is always greater than the third side – a theorem which, as is easily seen, deals solely with elementary concepts, i.e., with such as are derived directly from the axioms, and is therefore more accessible to logical investigation. Euclid proved this theorem, with the help of the theorem of the exterior angle, on the basis of the congruence theorems. Now it is readily shown that this theorem of Euclid cannot be proved solely on the basis of those congruence theorems which relate to the application of segments and angles, but that one of the theorems on the congruence of triangles is necessary. We are asking then, for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom. One finds that such a geometry really exists and is no other than that which Minkowski constructed in his book, Geometrie der Zahlen,47 and made the basis of his arithmetical investigations. Minkowski’s is therefore also a geometry standing next to the ordinary Euclidean geometry; it is essentially characterized by the following stipulations: 1. The points which are at equal distances from a fixed point O lie on a convex closed surface of the ordinary Euclidean space with O as a center. 2. Two segments are said to be equal when one can be carried to the other by a translation of the ordinary Euclidean space. In Minkowski’s geometry the axiom of parallels also holds. By studying the theorem of the straight line as the shortest distance between two points, I arrived48 at a geometry in which the parallel axiom does not hold, while all other axioms of Minkowski’s geometry are satisfied. The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e.g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seems to me desirable.

47 Leipzig, 48 Math.

1896. Annalen, vol. 46, p. 91.

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Appendix II: Extract from the French version Sur les problèmes futurs des mathématiques, 1902 The conclusion of the French version is different from the English one: In the plane, and assuming the continuity axiom, the problem in hand leads to a question treated by M. Darboux49 : To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines; this question seems to me worth of and interesting generalizations.50 [Dans le plan et en admettant l’axiome de la continuité, le problème dont il s’agit conduit à la question traitée par M. Darboux : Déterminer tous les problèmes du calcul des variations dans le plan où les solutions sont toutes des droites du plan ; questions qui me semble digne de généralisations fécondes et intéressantes.]

Acknowledgements. I would like to thank Juan Carlos Álvarez Paiva, Charalampos Charitos, Jeremy Gray, François Laudenbach, Darshana Prapajati, Vladimir Matveev, Marc Troyanov and Sumio Yamada for comments on a preliminary version of this chapter. This work was supported by the French ANR project FINSLER. The paper was completed during a stay at the Erwin Schrödinger International Institute for Mathematical Physics (Vienna).

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49 Leçons

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50 Compare

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Open problems

We present nineteen open problems on Hilbert geometry. Other open problems are contained in various chapters of this Handbook. In particular Chapter 8 and Chapter 15 contain several problems.

Problem 1 (B. Lemmens) Let (X, dX ) be a Hilbert geometry and let f : X → X be a nonexpansive map on X, that is, a map satisfying dX (f (x), f (y)) ≤ dX (x, y) for all x, y ∈ X. Beardon proved in [3] that if X is a strictly convex domain and f has no fixed point in X, then there exists η ∈ ∂X such that limk→∞ f k (x) = η for all x ∈ X, and the convergence is uniform on compact subsets of X. What about general Hilbert geometries? It was conjectured independently by Anders Karlsson and Roger Nussbaum, see [26], that if (X, dX ) is a general Hilbert geometry and f is a fixed point free nonexpansive map on X, then there exists a convex subset ⊂ ∂X such that for all x ∈ X, each limit point of the orbit Ox = {f k (x) : k = 0, 1, 2, . . .} lies inside . Besides strictly convex Hilbert geometries, the conjecture has only been confirmed for polyhedral domains, see the paper [22] by Brian Lins. Further details can be found in the articles by Karlsson and Lemmens and Nussbaum in this volume [15], [20].

Problem 2 (B. Lemmens) It is well known that the Hilbert geometry of an n-simplex is isometric to (Rn , · H ), where xH = (max xi ) ∨ 0 − (min xi ) ∧ 0 i

i

is a polyhedral norm. In general, if f : → is a nonexpansive map under a norm with polyhedral unit ball, and f has a fixed point, then every orbit of f converges to a periodic orbit of f , i.e., for each x there exists an integer p such that f kp (x) → ξx as k → ∞, where ξx is a periodic point of f with period p. Moreover, there exists an a priori upper bound for the maximum period of periodic points of f in terms of the number of facets of the unit ball of the norm. Thus, if f is a Hilbert metric nonexpansive map on the n-simplex and f has a fixed point, then every orbit converges to a periodic orbit of f , and there exists an a priori upper bound for the maximum period of its periodic points in terms of n. What is the optimal upper bound? For the 2-simplex, it was shown by Lemmens in [19] that 6 is the optimal upper bound. The Rn

Rn

434

Open problems

current best upper bound is:

m max 2 , k k k

where m = n(n + 1)/2. It is believed, however, that there exists c > 2 such that the periods of the periodic points do not exceed cn , but no proof is known.

Problem 3 (B. Lemmens) The following conjecture was posed by Bletz-Siebert and Foertsch in [9]: It is impossible to isometrically embed a Euclidean plane into any Hilbert geometry (X, dX ), or, as they say the Euclidean rank of any Hilbert geometry is 1. They proved the conjecture for all strictly convex domains X, for all domains whose boundary is C 1 , and for all 3-dimensional domains. The conjecture is also known to be true for polyhedral domains X, as it is impossible to have a nonexpansive map f : S → S, with S ⊆ X, which has a fixed point and whose orbits do not converge to periodic orbits. (For further details see the contribution of Lemmens and Nussbaum in this volume [20].) A variant of this problem was posed by Cormac Walsh. Walsh defined the Minkowski rank of a Hilbert geometry (X, dX ) as the largest dimension of a finitedimensional normed space that can be isometrically embedded into (X, dX ). As the Hilbert metric on the n-simplex is isometric to an n-dimensional normed space, its Minkowski rank is n. It also easy to show that the Minkowski rank of any strictly convex Hilbert geometry is 1. Walsh conjectured that in general the Minkowski rank of a Hilbert geometry (X, dX ) is the maximum dimension of a affine subspace H such that H ∩ X is a simplex.

Problem 4 (B. Lemmens) In a recent paper [35], Cormac Walsh showed that the collineation group of each finite-dimensional Hilbert geometry is a subgroup of index at most 2 in the group of isometries. Moreover, it has index 2 if and only if the domain is not the projection of a symmetric cone that is not a Lorentz cone. Is there an extension to infinite-dimensional Hilbert geometries of these results? Some results in this direction have been obtained by Lemmens, Roelands and Wortel in [21].

Problem 5 (A. Papadopoulos) The Hilbert metric is a generalization of Klein’s model of hyperbolic space, where the underlying set is the interior of an n-dimensional ellipsoid and where the distance between two distinct points x and y is the logarithm of the cross ratio of the quadruple

Open problems

435

consisting of x and y and the two intersection points of the Euclidean line that joins x and y with the boundary of the ellipsoid, taken in the natural order. A fact which is not so well known is that Klein also obtained models for spherical and Euclidean geometries using the cross ratio and taking instead of the ellipsoid other kinds of conic sections. (In the case of Euclidean geometry, the conic is degenerate.) The distance again is defined as a multiple of a (complex) logarithm of a cross ratio. The methods and the formulae that define the three geometries of constant curvature are formally very similar, and the constructions of the three geometries are hereby done in a unified way in the realm of projective geometry. This is the main theme of Klein’s two papers On the so-called non-Euclidean geometry, I and II ([17], [18]). An exposition together with a mathematical commentary of these two papers are contained in [1]. We propose the following problems: (1) Develop generalizations of the two other geometries that are parallel to the generalization of hyperbolic geometry to Hilbert geometry. (2) Are there non-symmetric versions of such metrics similar to the Funk metric? (3) Under some regularity conditions to be specified, are such metrics Finslerian with constant flag curvature? In this case the possible geometries would be limited by the results stated in Appendix A of Chapter 3 of this volume [30] (in particular Theorem A3).

Problem 6 (A. Papadopoulos) Study families of Hilbert and of Funk metrics parametrized by convex sets.

Problem 7 (I. Kim) In a recent paper [16], Kim and Zhang showed that the space of convex real projective structures admits a natural L2 Kähler metric, where the Weil–Petersson metric on Teichmüller space is totally geodesic. Is this Kähler metric non-positively curved?

Problem 8 (I. Kim) In the preprint [13], Foulon and Kim exhibited an example of a sequence of Hilbert metrics on a surface where the entropy of the geodesic flow goes to zero with respect to the Sinai–Ruelle–Bowen measure, yet remains bounded below with respect to the Bowen–Margulis measure. Is the entropy with respect to the Sinai–Ruelle–Bowen measure zero if and only if the Hilbert area of the surface is infinite?

436

Open problems

Problem 9 (M. Troyanov) Cheng and Yau proved the existence of a natural Riemannian metric on a sharp convex domain of Rn that is invariant under the group of projective transformation. This metric is obtained by solving the Monge–Ampère equation. It is related to the theory of affine spheres and is sometimes called the affine metric on the convex domain, see Theorem 3.2 in Chapter 11 of this Handbook or the book [25] by Nomizu and Sasaki. Benoist and Hulin recently proved that the affine metric is bi-Lipschitz equivalent to the Hilbert metric. A general construction of a Riemannian metric from a Finsler one is given by the Binet–Legendre metric, see [23]. In the case of the Hilbert metric, this metric is also known to be bi-Lipschitz equivalent to the Hilbert metric (see [24]). We ask the following: (1) What are the relations between the Binet–Legendre and the affine metric in a sharp convex domain? (2) Study these metrics from a Riemannian viewpoint: their curvature, etc. These metrics are obtained by rather indirect constructions, and computing some explicit examples should be a useful (and non trivial) first step. Similar questions could be asked about the metric constructed by Vinberg as the Hessian of the characteristic function of the cone over a convex domain, see Section 4.3 in Chapter 5 of this Handbook.

Problem 10 (S. Yamada) In a sense, modern differential geometry started with the Gauss equation; namely, given a codimension-1 submanifold in Rn , the canonical affine connection that is both torsion-free and metric compatible is singled out from the canonical Euclidean connection and the nearest point projection operators on the tangent spaces of the submanifold. Although there have been some studies on connections and parallel transport in the framework of Finsler geometry, little has been investigated in terms of the theory of submanifolds. Hence, we propose to study the issue of canonical connection(s) defined on a submanifold within an ambient space where some Hilbert/Funk geometry is defined. In particular, the case of the finite-dimensional simplex as the ambient space is already interesting, where the Hilbert metric is known to be isometric to a normed space, hence the space is equipped with a Berwald connection. On the other hand, infinite-dimensional simplices naturally appear as ambient spaces for Teichmüller spaces as submanifolds [27], [36] and in that framework, the Teichmüller metric, the Thurston asymmetric metrics and the length spectrum metrics are manifested.

437

Open problems

Problem 11 (L. Marquis) In [5], Benoist exhibits an example of a strongly irreducible subgroup of SL(4, R) and a properly convex open set of RP4 preserved by such that / is compact, but the convex set is not strictly convex nor with C 1 -boundary. Benoist suggests in the same paper that this construction can be easily generalized to dimensions 4, 5, 6, 7. Is it possible to find such a pair (, ) in any dimension? We recall that Benzécri shows in [6] that one cannot find such a pair in dimension 2.

Problem 12 (L. Marquis) Kapovich in [14] and Benoist in [4] gave examples of subgroups of SL(4, R) and of strictly convex open sets of RP4 preserved by such that / is compact but equipped with its Hilbert metric is not quasi-isometric to the real hyperbolic space H4 of dimension 4. Can one find infinitely many pairs (, ) with / compact and strictly convex up to quasi-isometry? In such a pair, the group has to be Gromov-hyperbolic. A nice invariant of quasiisometry of a Gromov-hyperbolic group is the conformal dimension of the boundary ∂ of . Can one compute the conformal dimension of ∂ in Kapovich’s or Benoist’s examples using Hilbert geometry?

Problem 13 (C. Vernicos) Using the Raleigh quotient, and regardless of any definition of a Laplacian, it is possible to define the bottom of the spectrum, λ1 , of a Hilbert geometry (see [11], [10], [31]). I proved in [31] that for any n-dimensional Hilbert geometry one has λ1 ≤

(n − 1)2 . 4

(∗)

Colbois conjectured that the equality should characterize hyperbolic geometry. In the case of a C 1 divisible convex set, the conjecture is true. To see this, we can use a classical method in Riemannian geometry which relates λ1 to the volume entropy h. The method consists in using the so-called test functions in Rayleigh quotients. For instance, for any h > h, one can consider the functions

fr (t) = e−h t − e−h r for t ≤ r and 0 for t > r. Then, for a given Hilbert geometry and ω ∈ , we compute the limit of the Rayleigh quotients of the functions fr d (ω, ·) as r → +∞.

438

Open problems

From that we can deduce the inequality λ1 ≤ (h )2 /4 which finally gives λ1 ≤

h2 . 4

In the case of the Hilbert geometry of a C 1 divisible convex set, Crampon [12] proved that h ≤ n−1 with equality if and only if the Hilbert geometry is the hyperbolic one. Hence the equality in (∗) implies the equality h = n − 1 and by Crampon’s theorem we easily conclude.

Problem 14 (C. Vernicos) A definition of amenability is possible in the realm of Hilbert geometry and was introduced in [31]. One possible definition is that a Hilbert geometry is amenable if and only if the bottom of its spectrum λ1 is equal to zero. We would like to have a necessary and sufficient condition in terms of properties of the convex set for its Hilbert geometry to be amenable. For instance, one can consider the action of the group PGL(n, R) of projective transformations on the set of open convex sets in Rn endowed with the Hausdorff distance. The Hilbert geometries arising from elements in the orbit of a given convex body under this action are all isometric. Again in [31], it was proved that if the closure of that orbit with respect to the Hausdorff distance contains an open convex polytope, then the initial Hilbert geometry is amenable. We conjecture that the Hilbert geometry of a convex set is amenable if and only if the closure of its orbit PGL(n, R) · contains an open convex polytope.

Problem 15 (C. Vernicos) I proved in [33] that there exists a universal constant an such that the volume of any ball of radius r in any n-dimensional Hilbert geometry is ≥ an r n . We ask: Is this constant an equal to the volume of the unit ball of the n-dimensional simplex? Another way of looking at this problem is the following: One can associate to any pointed open convex set (, ω), a function V (ω, ·) : R+ → R+ which maps r ∈ R+ to the Holmes–Thompson (or Busemann) volume of the ball of radius r centred at ω, with respect to the Hilbert metric of . Let us call this function the volume function. Is it true that the volume function is bounded from below by the volume function of the n-simplexes Sn and from above by the volume function of the hyperbolic space Hn ? That is, do we have, for all r ≥ 0, VSn (p, r) ≤ V (ω, r) ≤ VHn (q, r) ?

Open problems

439

As the group of projective transformations fixing a simplex (resp. an ellipsoid) acts transitively, the volume function of the simplex (resp. of hyperbolic geometry) does not depend on the centre p (resp. q) of the balls. Such an equality would prove the upper entropy bound conjecture which can be stated as follows: log V (ω, r) ≤ n − 1. h() := lim inf r→+∞ r

Problem 16 (C. Vernicos) The volume function can be seen as a function from the set of pointed open convex sets into the set of real functions (see Problem 15). Is this function 1-to-1?

Problem 17 (C. Vernicos) The pseudo-Gaussian curvature of a convex set was defined by G. Berck, A. Bernig and C. Vernicos in [7] as a function on the boundary ∂ of a convex body as follows. Fix a Euclidean metric in Rn . Following A. D. Alexandrov [2] at almost every point p of ∂ there is a well-defined normal. Such points are called smooth. For such a point p ∈ ∂, let n(p) be the outward normal of ∂ at p. For each unit vector e ∈ Tp ∂, let He (p) be the affine plane containing p and directed by the vectors e and n(p). We then define Re as the radius of the biggest disc containing p inside ¯ e := ∩ He (p). The pseudo-Gaussian curvature k(p) of ∂ at p is the minimum of the numbers n−1 Rei (p)−1 , i=1

where e1 , . . . , en−1 range over all orthonormal bases of Tp ∂. The fact that the square of this function is in L1 for any plane convex body is a key ingredient in the proof of the upper entropy bound conjecture in the plane. We ask: Is the square root of the pseudo-Gaussian curvature always in L1 ? A positive answer would imply, by Theorem 3.1 in [7], the entropy upper bound conjecture and an asymptotic of the ratio of the volume function by the function r → sinh(r)n−1 , introduced as the centro-projective area in [7]. We also ask: Is there a Bishop volume comparison type theorem in Hilbert geometry, i.e., for any and ω ∈ , is there a ρ0 > 0 such that for any R > r > ρ0 , VHn (o, R) V (ω, R) ≤ ? V (ω, r) VHn (o, r)

440

Open problems

In other words, is it true that the function V (ω, r)/VHn (o, r) is a non-increasing function for r > ρ0 . Note that it suffices to prove this statement for a dense subset of the set of convex sets, and that this would also yield the entropy upper bound conjecture.

Problem 18 (C. Vernicos) The approximability of a convex body, a concept introduced by Schneider and Wieacker [28], is a constant which somehow measures how well a convex set can be approximated by polytopes. More precisely, let N(ε, ) be the smallest number of vertices of a polytope whose Hausdorff distance to the convex body is less than ε > 0. Then the approximability of is a() := lim inf ε→0

log N(ε, ) . − log ε

We proved in [34] that in dimensions 2 and 3 the volume entropy and twice the approximability coincide, and that 2a() ≤ h() in all dimension. Does the equality between the entropy and twice the approximability always occur? Again, this would prove the entropy upper bound conjecture. What metric spaces can be embedded, in general, into a Hilbert geometry and, in particular, in a Hilbert geometry of the same dimension? For example, we believe that it is not possible to embed the three-dimensional Heisenberg group in a three-dimensional Hilbert geometry.

Problem 19 (C. Vernicos) Consider in a Hilbert geometry (with C 2 boundary to start with) a point p, and for any R > 0 a polytope PR with minimal vertices which contains the ball of radius R − 1 and is contained in the ball of radius R + 1, both centred at p. Compute the difference (resp. the ratio) between the Busemann volume of the ball B(p, r) and the Busemann volume of PR . Study the behaviour as R → +∞. Does this yield a functional related to the centro-projective area? (Note that this function has to be a projective invariant because of the projective invariance of Hilbert geometries.)

Open problems

441

References [1]

N. A’Campo and A. Papadopoulos, On Klein’s So-called Non-Euclidean geometry. In Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics, (L. Ji and A. Papadopoulos, eds.), Eurpoean Mathematical Society, Zürich, to appear.

[2]

A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 (1939), 3–35.

[3]

A. F. Beardon, The dynamics of contractions. Ergodic Theory Dynam. Systems 17 (1997), no. 6, 1257–1266.

[4]

Y. Benoist, Convexes hyperboliques et quasiisométries. Geom. Dedicata 122 (2006), 109–134.

[5]

Y. Benoist, Convexes divisibles. IV. Structure du bord en dimension 3. Invent. Math. 164 (2006), no. 2, 249–278.

[6]

J.-P. Benzécri, Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88 (1960), 229–332.

[7]

G. Berck, A. Bernig and C. Vernicos, Volume entropy of Hilbert geometries. Pacific J. Math. 245 (2010), no. 2, 201–225.

[8]

A. Bernig, Hilbert geometry of polytopes. Archiv Math. 92 (2009), 314–324.

[9]

O. Bletz-Siebert and T. Foertsch, The Euclidean rank of Hilbert geometries. Pacific J. Math. 231 (2007), no. 2, 257–278.

[10] B. Colbois and C. Vernicos, Les géométries de Hilbert sont à géométrie locale bornée. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 4, 1359–1375. [11] B. Colbois and C. Vernicos, Bas du spectre et delta hyperbolicité en géométrie de Hilbert plane Bull. Soc. Math. France 134 (2006), no. 3, 357–381. [12] M. Crampon, Entropies of compact strictly convex projective manifolds. J. Mod. Dyn. 3 (2009), no. 4, 511–547. [13] P. Foulon and I. Kim, Entropy and deformation of real projective structures on surface. In preparation. [14] M. Kapovich, Convex projective structures on Gromov–Thurston manifolds. Geom. Topol. 11 (2007), 1777–1830. [15] A. Karlsson, Dynamics of Hilbert nonexpansive maps. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 263–273. [16] I. Kim and G. Zhang, Kähler metric on Hitchin component. Preprint, arXiv:1312.1965. [17] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie (erster Aufsatz). Math. Ann. IV (1871), 573–625; English translation by J. Stillwell, [29], 69–111. [18] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie (zweiter Aufsatz). Math. Ann. VI (1873), 112–145. [19] B. Lemmens, Nonexpansive mappings on Hilbert’s metric spaces. Topol. Methods Nonlinear Anal. 38 (2011), no. 1, 45–58.

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Open problems

[20] B. Lemmens and R. Nussbaum, Birkhoff’s version of Hilbert’s metric and its applications in analysisIn Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 275–303. [21] B. Lemmens, M. Roelands, and M. Wortel, Isometries of infinite dimensional Hilbert geometries. Preprint, arXiv:1405.4147. [22] B. Lins, A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on a polyhedral domain. Math. Proc. Cambridge. Philos. Soc., 143 ( 2007) no. 1, 157–164. [23] V. Matveev and M. Troyanov, The Binet–Legendre metric in Finsler geometry. Geom. Topol. 16 (2012), 2135–2170. [24] V. Matveev and M. Troyanov, Completeness and incompleteness of the Binet–Legendre metric. Preprint, arXiv:1408.6401. [25] K. Nomizu and T. Sasaki, Affine differential geometry. Cambridge Tracts in Math. 111, Cambridge University Press, Cambridge 1994. [26] R. D. Nussbaum, Fixed point theorems and Denjoy–Wolff theorems for Hilbert’s projective metric in infinite dimensions. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 199–250. [27] K. Ohshika, H. Miyachi and S.Yamada, Weil–Petersson Funk metric on Teichmüller space. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 339–352. [28] R. Schneider and J. A. Wieacker, Approximation of convex bodies by polytopes. Bull. London Math. Soc. 13 (1981) 149–156. [29] J. Stillwell, Sources of hyperbolic geometry. History of Mathematics Series 10, Amer. Math. Soc., Providence, RI; London Math. Soc., London 1996. [30] M. Troyanov, Funk and Hilbert geometries from the Finslerian viewpoint. In Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, eds.), European Mathematical Society, Zürich 2014, 69–110. [31] C. Vernicos, Spectral Radius and amenability in Hilbert geometries. Houston J. Math. 35 (2009), no. 4, 1143–1169. [32] C. Vernicos, Lipschitz caracterisation of polytopal Hilbert geometries to appear in Osaka J. Math., arXiv :0812.1032v1. [33] C. Vernicos, Asymptotic volume in Hilbert geometries. Indiana Univ. Math. J. 62 (2013), no. 5, 1431–1441. [34] C. Vernicos, Approximability of convex bodies and volume entropy of Hilbert geometries. Preprint, arXiv:1207.1342. [35] C. Walsh, Gauge-reversing maps on cones, and Hilbert and Thompson isometries. Preprint, arXiv:1312.7871. [36] S. Yamada, Local and global aspects of Weil–Petersson geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zürich 2014, 43–112.

List of Contributors

Mickaël Crampon, Universidad de Santiago de Chile, Av. El Belloto 3580, Estación Central, Santiago, Chile email: [email protected] Ren Guo, Department of Mathematics, Oregon State University, Corvallis, OR 973314605, U.S.A. email: [email protected] Anders Karlsson, Section de mathématiques, Université de Genève, 2-4 Rue du Lièvre, Case Postale 64, 211 Genève 4, Switzerland email: [email protected] Inkang Kim, School of Mathematics, KIAS, Heogiro 85, Dongdaemun-gu Seoul, 130-722, Republic of Korea email: [email protected] Bas Lemmens, School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, UK email: [email protected] Ludovic Marquis, Institut de Recherche Mathématique de Rennes, Rue Sine 190, 31062 Toulouse Cedex 4, France email: [email protected] Hideki Miyachi, Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan email: [email protected] Roger Nussbaum, Department of Mathematics, Rutgers, The State University Of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, U.S.A. email: [email protected] Ken’ichi Ohshika, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, 560-0043, Osaka, Japan email: [email protected] Athanase Papadopoulos, Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France email: [email protected] Marc Troyanov, Section de mathématiques, École Polytechnique Fédérale de Lausanne, SMA–Station 8, 1015 Lausanne, Switzerland email: [email protected]

444

List of Contributors

Constantin Vernicos, Institut de mathématique et de modélisation de Montpellier, Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34395 Montpellier Cedex, France email: [email protected] Cormac Walsh, INRIA Saclay & Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France e-mail: [email protected] Sumio Yamada, Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima, Tokyo 171-8588, Japan email: [email protected]

Index absolutely continuous measure, 193 area centro-projective, 439 affine chart, 209 affine metric, 316, 318 affine midpoint, 21 affine normal, 316 affine patch, 58, 310 affine ratio, 37, 61 affine sphere, 316 affinely convex, 52 Alexandrov’s theorem, 202 algebraic horosphere, 216 almost-geodesic, 130 ˛-conformal density, 332 amenable Hilbert metric, 438 Anosov decomposition, 189 Anosov flow, 189, 327 approximability convex body, 440 arithmetic symmetrization metric, 34 asymptotic convex hypersurface, 231 asymptotically harmonic, 195 attractor, 298 automorphism group, 166 axioms continuity, 385 Hilbert, 385 incidence, 385 order, 385

forward, 43 left, 43 right, 43 Beltrami Theorem, 99 Beltrami, Eugenio, 393 Beltrami–Cayley–Klein model, 59,308, 310, 394 Benzécri, Jean-Paul, 308, 312 Berwald metric, 74 Berwald, Ludwig, 405 ˇ-convexity, 186 bi-, 213 Binet–Legendre metric, 436 biproximal, 213 Birkhoff theorem, 282 Birkhoff version of Hilbert’s metric, 276 Blaschke connection, 316 bottom of the spectrum, 437 boundary horofunction, 130 Bowen–Margulis measure, 194, 332 bulging deformation, 324 Busemann G-space, 27 Busemann cocycle, 330 Busemann function, 127, 178, 215, 266, 330 Busemann point, 131 Busemann volume, 210, 311, 438 Busemann zero curvature, 28 Busemann, Herbert, 392, 401, 408, 409 Busemann–Hausdorff volume, 166

backward complete Finsler manifold, 73 backward complete metric, 47 backward open ball, 43 backward proper metric, 47 backward sphere, 43 ball backward, 43

Cayley, Arthur, 393 centered John ellipsoid, 20 centro-projective area, 439 character variety, 321 characteristic function, 143 closed orbit, 170 collineation, 115, 140

446 cometric, 172 compactification horofunction, 130 complete Finsler manifold, 73 cone, 277, 317 above , 212 ice-cream, 143 Lorentz, 143 normal, 277 symmetric, 143 total, 277 cone linear mapping, 283 conformal density, 332 conformal dimension, 437 conical face, 113 conical flag, 114 connection, 173 convex, 209 with C 1 boundary, 211 affinely, 52 ˇ-convex, 246 class C 1C" , 246 cone, 222 divisible, 208 geodesically, 52 Menger, 411 projective structure, 167, 198 properly, 163, 209 quasi-divisible, 208 round, 211 strictly, 211 strongly, 246 convex body, 209 approximability, 440 convex cone, 222, 317 decomposable, 222 indecomposable, 222 sharp, 217 convex domain proper, 35 sharp, 60 convex hull, 218 convex hypersurface, 231

Index

asymptotic, 231 convex polytope, 42 convex projective structure, 308 deformation space, 321 Goldman parameters, 321 marked, 313 convex real projective manifold, 313 convex set divisible, 313 hyperbolic space, 358 projective automorphism group, 310 projective space, 310 properly, 310 sphere, 358 symmetric, 223 convex structure dual, 326 convex structures deformation space, 313 moduli space, 313 C 1C regularity, 186 Crofton formula, 115 cross ratio Euclidean, 364 hyperbolic, 364 spherical, 365 current, 329 curvature of a Finsler metric, 175 of Hilbert geometry, 177 of the boundary, 202 pseudo-Gaussian, 439 curvature zero (Busemann), 28 decomposable convex cone, 222 Dehn invariant, 404 Dehn, Max, 404 Denjoy–Wolff theorem, 298 density function, 154 Desargues’ property, 27 Desarguesian space, 27, 411 detour cost, 131 detour metric, 132

Index

developing map, 198 divisible, 167, 226, 308 convex set, 313 divisible convex, 208 division ratio, 37, 61 dominate, 277 dual, 217 dual convex structure, 326 dual wedge, 277 dyadic number, 21 earthquake deformation, 324 Ehresmann, Charles, 308, 311 eigenvalues, 293 ellipsoid, 220 centered John, 20 John, 20 elliptic, 214 energy of a curve, 84 entropy measure-theoretical, 193 topological, 193, 198, 328 volume, 197, 331 epigraph topology, 134 equiaffine metric, 316 ergodic measure, 192 Euclidean Jordan algebra, 288 Euler lemma, 25 Euler–Lagrange equations, 85 exposed face, 49 exposed point, 49 extreme set, 135 face, 113, 214, 278 dimension, 214 open, 214 support, 214 face (of a convex set), 48 Fenchel–Nielsen parameters, 320 Finsler cometric, 172 Finsler metric, 163, 310 for Hilbert geometry, 165 regular, 164 Ricci curvature, 96

447

Finsler structure, 72, 151 Euclidean tautological, 362 hyperbolic tautological, 362 spherical tautological, 363 tautological, 34 Finsler, Paul, 404 fixed point, 296 fixed point theorem, 264 flag curvature, 96 Hilbert metric, 100 flip map, 198 flow Anosov, 327 topologically mixing, 328 foot, 53 formal Christoffel symbols, 86 forward bounded metric, 47 forward boundedly compact metric, 47 forward Cauchy sequence, 47 forward open ball, 43 forward proper metric, 47 forward sphere, 43 functional Minkowski, 19 fundamental tensor (of a Finsler metric), 83 Funk metric, 36, 76, 128, 280, 340 hyperbolic, 359 relative, 58 reverse, 41 spherical, 360 Funk, Paul, 404 gauge, 136 gauge-preserving map, 144 gauge-reversing map, 142 Genocchi, Angelo, 394 geodesic, 85 geodesic current, 329

448

Index

geodesic flow, 327 for regular Finsler metrics, 164 for Hilbert geometries, 169 geodesic metric space, 210, 328 Gromov-hyperbolic, 237 geodesically convex, 51, 52 geodesics, 164 in Hilbert geometry, 166 geometry Hilbert, 148 non-Archimedean, 405 non-Arguesian, 405 non-Euclidean, 405 non-Legendrian, 406 non-Pascalian, 405 semi-Euclidean, 406 Goldman parameters, 321 Gromov product, 269, 292 Gromov-hyperbolic, 237 group, 168 space, 168 G-space, 27, 410

Hilbert, David, 383 Hitchin component, 308, 325 Hoüel, Guillaume Jules, 395 holonomy map, 198 homogeneous Hilbert geometry, 166 homogeneous tangent bundle, 165 homography, 115 horoball, 129, 132, 178, 266 horofunction, 127, 130, 266 horofunction boundary, 130 horofunction compactification, 130 horosphere, 178, 215 algebraic, 216 hyperbolic, 214 isometry, 168 orbit, 186 space, 165 hyperbolic metric, 307 hyperbolic metric space, 328 hypermetric, 417 hyperplane supporting, 211

Hölder cocycle, 331 Hölder regularity, 186 Hamel potential, 93 Hamel, Georg, 404 harmonic measure, 195 harmonic symmetrization, 81 Hessian, 164, 202 Hilbert 1-form, 171 Hilbert form, 92 Hilbert measure, 155 Hilbert metric, 58, 128, 147, 148, 163, 209, 276, 310 amenable, 438 Birkhoff version, 278 Finsler norm, 310 flag curvature, 100 hyperbolic, 366 spherical, 367 Hilbert Problem IV, 422 Hilbert volume, 311

ice-cream cone, 143 ideal triangle, 155 idempotents, 293 complete system of orthogonal, 293 primitive, 293 indecomposable, 222 indecomposable convex cone, 222 indicatrix, 17, 73 invariant measure, 192 isometry group, 166 Jacobi field, 175 Hilbert geometry, 177 Jacobi operator, 175 Hilbert geometry, 177 John ellipsoid, 20, 218, 247 Klein model, 59, 81, 394 Krein–Milman theorem, 288 Laguerre, Edmond, 384

Index

metric left open ball, 43 affine, 316, 318 Legendre transform, 172 arithmetic symmetrization, 34 on HM , 172 backward complete, 47 Legendre–Clebsch condition, 83 backward proper, 47 Legendre–Fenchel transform, 137 Binet–Legendre, 436 Lemma Blaschke, 316 Euler, 25 detour, 132 Schur, 97 equiaffine, 316 Selberg, 239 forward bounded, 47 Zassenhaus–Kazhdan–Margulis, 251 forward boundedly compact, 47 length forward proper, 47 translation, 330 Funk, 36, 128, 340 length spectrum, 326 hyperbolic, 359 length spectrum metric, 61 spherical, 360 limit set, 296 geodesically convex, 51 Liouville measure, 171, 193 Gromov hyperbolic, 328 locally finite fundamental domain, 233 Hilbert, 128, 147, 209 locally Ptolemaic, 150 hyperbolic, 366 Lorentz cone, 143, 287 spherical, 367 Lyapunov decomposition, 188 hyperbolic, 307 Lyapunov exponents, 188 length spectrum, 61 of a periodic orbit, 188 max-symmetrization, 61 marked convex projective structure, 242, projective, 36, 70, 249 313 projectively flat, 13 marked length spectrum, 326 relative Funk, 58 marked projective structure, 242 reverse Funk, 41, 61 convex, 242, 313 reverse-Funk, 129 equivalent, 242 Sasaki, 176 marking, 313 Thurston, 33, 349 max-symmetrization variational formulation of metric, 61 Teichmüller, 348 Mazur–Ulam Theorem, 20 weak, 12, 33 measure weak Minkowski, 13 Bowen–Margulis, 332 Weil–Petersson, 341 Hilbert, 155 completion, 342 Patterson–Sullivan, 331 Weil–Petersson Funk, 60, 344 Sinai–Ruelle–Bowen, 435 metric space measure of maximal entropy, 194 geodesic, 210 measure-theoretical entropy, 193 proper, 210, 266 Menelaus, 372 uniquely geodesic, 210 Menelaus Theorem, 373 midpoint property, 21 Menger convexity, 411 minimal displacement, 265

449

450

Index

Minkowski functional, 19, 38 Funk metric Euclidean, 362 hyperbolic, 362 spherical, 363 Minkowski norm strictly convex, 25 strongly convex, 25 Minkowski rank, 434 mixing measure, 192 topologically, 190 natural, 249 nearest point, 53 non-Archimedian field, 404, 405 non-Arguesian geometry, 405 non-Euclidean geometry, 405 non-Legendrian geometry, 406 non-Pascalian field, 406 non-Pascalian geometry, 405, 406 nonexpansive map, 265, 433 orbit, 170 ordered geometry, 385 orthogonal representation, 226 osculating Riemannian metric, 94 Painlevé–Kuratowski topology, 132 pair of pants, 314 decomposition geodesic, 315 topological, 315 parabolic, 214 parallel transport, 174 parts, 278 parts of boundary, 132 Pasch, Moritz, 385 patch affine, 310 Patterson–Sullivan geodesic current, 333 Patterson–Sullivan measure, 331 period, 296 cocycle, 331

periodic point, 296 perpendicular, 55 Perron–Frobenius operator, 284 Perron–Frobenius theory, 264 perspectivity, 148 physical measure, 200 planar automorphism, 219 Pogorelov, Alexei Vasil’evich, 399, 409, 413 polytope, 113 positive hyperbolic transformation, 322 positively biproximal, 213 positively proximal, 213 positively semi-proximal, 213 power, 213 primitive element, 170 Problem Hilbert fourth, 422 product, 222 projective metric, 249 natural, 249 volume, 249 projective center, 152 projective diameter, 282 projective line, 309 projective manifold, 242, 311 convex, 313 strictly, 313 isomorphism, 311 projective metric, 36, 70, 249 natural, 249 projective structure, 311 convex, 308, 312 flat, 242 strictly convex, 308 projective volume, 249 natural, 249 projective weak metric, 13 projectively equivalent, 242 projectively flat metric, 13

Index

projectively flat Finsler manifold, 88 proper convex, 310 proper convex domain, 35 proper metric space, 210 properly convex, 163, 209 indecomposable, 222 product, 222 properly convex set, 310 proximal, 213, 225, 233 action, 233 bi, 213 group, 233 positively, 213 semi, 213 pseudo-Gaussian curvature, 439 Ptolemaic, 150 Ptolemaic inequality, 150 quadratic representation, 294 quasi-divisible, 226, 208 quasi-hyperbolic, 214 rank Hilbert geometry, 434 Randers metric, 73 Rayleigh quotient, 437 reductive, 234 regular orbit, 188 regular Finsler metric, 164 relative Funk metric, 58 representation equal mod 2, 226 highest restricted weight, 225 highest weight, 225 lattice of restricted weights, 225 orthogonal, 226 proximal, 225 restricted weight, 225 symplectic, 226 weight, 225 reverse Funk metric, 41, 61, 129 reversible norm, 310 reversible tautological

451

Finsler structure, 34 Ricci curvature of a Finsler metric, 96 right open ball, 43 round, 211 Ruelle’s inequality, 198, 201 Sasaki metric, 176 Schur Lemma, 97 Schwarzian derivative, 106 Selberg lemma, 239 semi-proximal, 213 semi-simple k-semi-simple, 213 S1 -semi-simple, 213 semicontraction, 265 sharp convex cone, 217 sharp convex domain, 60 simplicial cone, 289 Sinai–Ruelle–Bowen measures, 200 space G-, 410 Busemann G-, 27 Desarguesian, 27, 411 spectral radius, 213 spectrum, 293 sphere backward, 43 forward, 43 spherical metric projective model, 88 spray, 87 stable set, 179 standard positive cone, 287 -map, 143 strictly convex, 211 strictly convex Minkowski norm, 25 strictly convex projective structure, 308 strictly convex set, 49 strongly convex, 246 strongly convex Finsler metric, 83 strongly convex Minkowski norm, 25 subhomogeneous map, 282 support hyperplane, 38, 211

452 support of a face, 214 supporting functional, 38 symmetric cone, 143 symmetric convex set, 223 symmetric Hilbert geometry, 166 symplectic representation, 226 tautological Finsler structure, 34, 56 tautological Finsler structure Euclidean, 362 hyperbolic, 362 spherical, 363 Teichmüller component, 308 Teichmüller space, 341 tensor, 83 Theorem Beltrami, 99 Birkhoff, 282 Ceva, 63 Denjoy–Wolff, 298 Desargues, 405 Funk–Berwald, 70 Krein–Milman, 49, 288 Mazur–Ulam, 20 Menelaus, 61, 373 Pascal, 405 sector figure, 372 Thurston metric, 33, 349 topological entropy, 193, 198, 328 topologically mixing, 190, 328 transitive, 190 topology epigraph, 134 Painlevé–Kuratowski, 132 transitive flow, 190 translation distance, 168 translation length, 214, 265,

Index

330 translation number, 265 triangle ı-thin, 328 unipotent, 213 uniquely geodesic metric space, 210 unstable set, 179 upper asymptotic volume, 116 variation norm, 289 variational formulation of Teichmüller metric, 348 variational principle, 194 Veronese, Guiseppe, 405 vertex, 113 Vinberg’s -map, 143 volume, 165 Busemann, 210, 438 Busemann–Hausdorff, 166 projective, 249 volume entropy, 197, 331 von Staudt, Karl Georg Christian, 383 weak Finsler structure, 72 weak metric, 12, 33 geodesic, 51 projective, 13 weak Minkowski metric, 13 wedge, 277 Weil–Petersson Funk metric, 60, 344 Weil–Petersson metric, 341 Zariski-dense, 168 Zermelo transform, 75 zero curvature Busemann, 28 zonoid, 417 zonotope, 417

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