Recent Developments in Pseudo-Riemannian Geometry (Esl Lectures in Mathematics and Physics)

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Author: Dmitri V. Alekseevsky and Helga Baum

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ESI Lectures in Mathematics and Physics Editors Joachim Schwermer (Institute for Mathematics, University of Vienna) Jakob Yngvason (Institute for Theoretical Physics, University of Vienna) Erwin Schrödinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien Austria The Erwin Schrödinger International Institute for Mathematical Phyiscs is a meeting place for leading experts in mathematical physics and mathematics, nurturing the development and exchange of ideas in the international community, particularly stimulating intellectual exchange between scientists from Eastern Europe and the rest of the world. The purpose of the series ESI Lectures in Mathematics and Physics is to make selected texts arising from its research programme better known to a wider community and easily available to a worldwide audience. It publishes lecture notes on courses given by internationally renowned experts on highly active research topics. In order to make the series attractive to graduate students as well as researchers, special emphasis is given to concise and lively presentations with a level and focus appropriate to a student's background and at prices commensurate with a student's means. Previously published in this series: Arkady L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations Werner Ballmann, Lectures on Kähler Manifolds Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization


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Recent Developments in Pseudo-Riemannian Geometry Dmitri V. Alekseevsky Helga Baum Editors

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Editors: Prof. Dr. Dmitri V. Alekseevsky School of Mathematics King’s Buildings Edinburgh University Mayfield Road EH9 3JZ, Edinburgh UK

Prof. Dr. Helga Baum Institut für Mathematik Humboldt-Universität zu Berlin Rudower Chaussee 25 Johann von Neumann-Haus 12489 Berlin Germany

2000 Mathematics Subject Classification (primary; secondary): 53-00; 53C50 Key words: Pseudo-Riemannian manifold, Lorentzian manifold, spacetime, neutral signature, holonomy group, pseudo-Riemannian symmetric space, Cahen–Wallach space, hypersymplectic manifold, anti-self-dual conformal structure, integrable system, neutral Kähler surfaces, Einstein metric, Einstein universe, essential conformal structure, conformal transformation, causal hierarchy, geodesic, supergravity, generalized geometry, generalized G-structure, Killing spinor

ISBN 978-3-03719-051-7 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.

© 2008 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 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 Printed in Germany 987654321

Preface

In differential geometry one studies local and global properties of smooth manifolds M equipped with a metric tensor g – that is, a smooth field of symmetric bilinear forms of fixed signature .p; q/ on the tangent spaces of M – which encodes the geometry. If the metric tensor g is positive definite, the pair .M; g/ is called a Riemannian manifold. If g is indefinite, .M; g/ is referred to as a pseudo-Riemannian manifold. The difference in the signature of the metric g has essential consequences for the geometric structures as well as for the methods of their investigation. In Riemannian geometry, important progress has been made over the past thirty years in understanding the relations between the local and global structure of Riemannian manifolds. Many classification results for different classes of Riemannian manifolds were obtained: manifolds with additional geometric structures, manifolds satisfying curvature conditions, symmetric and homogeneous Riemannian spaces, etc. Similar results for pseudo-Riemannian manifolds are rare, and many problems are still open. For a long time, the main source of problems in pseudo-Riemannian geometry was general relativity, which deals with 4-dimensional Lorentzian manifolds (space-times) where the signature of the metric is .1; 3/. However, the developments in theoretical physics (supergravity, string theory) require a deeper understanding of the geometric structure of higher dimensional manifolds with indefinite metrics of Lorentzian and other signatures. Moreover, pseudo-Riemannian metrics naturally appear in different geometric problems, e.g. in CR geometry or on moduli spaces of geometric structures. Sometimes, one can use a special Ansatz or “Wick-rotations” to transform problems of pseudo-Riemannian geometry into questions of Riemannian geometry. But in many aspects, pseudo-Riemannian and Riemannian geometry differ essentially, and many specific, highly nontrivial and interesting new questions appear in the pseudo-Riemannian setting. There has been substantial progress over the past few years in solving some of these problems. In order to stimulate cooperation between different groups of researchers working in this field, we organized a scientific programme Geometry of Pseudo-Riemannian Manifolds with Applications in Physics, which was held at the Erwin Schrödinger International Institute for Mathematical Physics (ESI) in Vienna between September and December of 2005. In the course of this programme, the idea of this volume was born. It aims to introduce a broader circle of mathematicians and physicists to recent developments of pseudo-Riemannian geometry, in particular to those developments which were discussed during the ESI Special Research Semester. We now briefly sketch the contents of this book. A basic problem of differential geometry, which is completely solved for Riemannian manifolds, but becomes quite complicated in the pseudo-Riemannian setting, is to determine all possible holonomy groups of pseudo-Riemannian manifolds. Contrary to the Riemannian case, the holonomy representation of an indefinite metric must not decompose into irreducible rep-

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resentations. In the indefinite case, additional holonomy representations occur, which have an isotropic holonomy invariant subspace without an invariant complement. Such representations are difficult to classify. The first two contributions to this book describe recent progress concerning this question. The article of Ines Kath and Martin Olbrich deals with pseudo-Riemannian symmetric spaces. The classification of symmetric spaces with completely reducible holonomy representation, i.e., any invariant subspace has an invariant complement, was established long ago. In this case the transvection group is semi-simple. For symmetric spaces with non-reductive holonomy representation, the transvection group is more complicated. It has a proper Levi decomposition equipped with a biinvariant metric. Kath and Olbrich give a survey of new approaches to the classification of this type of pseudo-Riemannian symmetric spaces and explain applications to the classification of pseudo-hermitian, quaternionic Kähler and hyper-Kähler symmetric spaces. Furthermore, they describe the complete classification of Lorentzian symmetric spaces and of symmetric spaces with metrics of index 2. The article of Anton Galaev and Thomas Leistner is about the classification of holonomy groups of Lorentzian manifolds, which was completed only recently. They describe the list of all possible Lorentzian holonomy groups, outline the proof of this result, and explain a method to construct local metrics for all possible holonomy groups. Furthermore, they give a brief outlook on the classification problem for metrics with higher signature. Besides Lorentzian manifolds, which are basic for general relativity, pseudo-Riemannian manifolds of split signature .m; m/ are of special interest. The article of Andrew Dancer and Andrew Swann, and that of Maciej Dunajski and Simon West, deal with this class of pseudo-Riemannian manifolds. Dancer and Swann give a survey of so-called hypersymplectic manifolds, which were introduced by Hitchin as a cousin of hyper-Kähler manifolds. They are based on the algebra of split quaternions rather than the usual quaternions. Hypersymplectic manifolds are also Ricci-flat and Kähler, but of split signature .2n; 2n/. The article describes construction methods for hypersymplectic manifolds using ideas from symplectic and toric geometry. Dunajski and West cover the special case of dimension 4. More generally, they survey the geometry of 4-dimensional anti-selfdual manifolds of signature .2; 2/, local and global construction methods for these, and relations to integrable systems. The article of Brendan Guilfoyle and Wilhelm Klingenberg links special geometry in signature .2; 2/ to classical surface theory and geometric optics. Here a neutral Kähler metric appears on the space L of oriented lines in Euclidean 3-space. The fundamental objects in geometric optics are 2-parameter families of oriented lines, hence immersed surfaces in the pseudo-Kähler space L. These surveys of the geometry of manifolds with split signature are followed by a group of articles dealing with special properties of conformal transformations in the pseudo-Riemannian setting. Some of the main differences between Riemannian and pseudo-Riemannian conformal geometry are already manifested in the flat model of

Preface

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conformal Lorentzian geometry – the conformal compactification of Minkowski space, or equivalently, the ideal boundary of anti-de Sitter space – often referred to as the Einstein Universe in the physics literature. The article by Thierry Barbot, Virginie Charette, Todd Drumm, William M. Goldman, and Karin Melnick gives a detailed introduction to the geometric and causal structure of the Einstein Universe. In particular, the article analyzes the 3-dimensional case, where the group of conformal transformations is locally isomorphic to the group of linear symplectomorphisms of R4 . The authors explain the dynamics of actions of discrete subgroups of the symplectic (conformal) group on the 3-dimensional Einstein Universe. They also describe the actions of discrete subgroups of Lorentzian transformations, which act freely and properly on the 3-dimensional Minkowski space, and describe the construction of complete flat Lorentzian manifolds. The article of Charles Frances considers essential conformal structures. A group G of conformal transformations of a manifold .M; g/ is called essential if no metric in the conformal class of g is preserved by G. It is well known that the conformal group of a Riemannian manifold .M; g/ of dimension n 3 is essential if and only if .M; g/ is conformally diffeomorphic to Sn or Rn with the canonical flat metric. In the pseudo-Riemannian case the situation is quite different and much more complicated, as Frances’ contribution illustrates. Using the special dynamics of discrete subgroups of the conformal group acting on the Einstein Universe, he constructs a large class of conformally flat, compact Lorentzian manifolds with non-equivalent essential conformal structures. Wolfgang Kühnel and Hans-Bert Rademacher study other aspects of pseudo-Riemannian conformal geometry. In their article they elucidate pseudo-Riemannian manifolds with essential infinitesimal conformal transformations (conformal Killing fields), in particular gradient fields. Furthermore, they study manifolds which are conformally equivalent to Einstein spaces, describe the conformal group of plane wave metrics which occur as Penrose limits of arbitrary space-times, and finally, discuss manifolds with conformal Killing spinors, which induce special kinds of infinitesimal conformal transformations. The article of Ettore Minguzzi and Miguel Sanchez, and that of Anna Maria Candela and Miguel Sanchez, give an introduction to causality theory of Lorentzian manifolds and properties of geodesics in pseudo-Riemannian geometry. The former article describes the different causality notions for space-times and their relations to each other, from non-totally vicious to global hyperbolic space-times. In particular, recent results on the existence of smooth Cauchy surfaces and smooth time-functions of globally hyperbolic manifolds are discussed. One of the main differences between Riemannian and pseudo-Riemannian manifolds is in the behavior of geodesics. Whereas compact Riemannian manifolds are always geodesically complete, and geodesically complete Riemannian manifolds are always geodesically connected, both properties fail for indefinite metrics. In the article of Candela and Sanchez these differences are illustrated by means of special space-times. Moreover, conditions and properties for geodesically complete manifolds and for incompleteness (Singularity Theorems) are discussed. Furthermore,

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the authors give an introduction to and survey of variational approaches for studying geodesic connectedness for stationary and orthogonal splitting space-times. The volume concludes with three articles which describe applications of methods and results from pseudo-Riemanian geometry to mathematical physics. Jose Figueroa-O’Farrill shows how Lorentzian symmetric spaces arise as supersymmetric supergravity backgrounds. His article is devoted to the local classification of supergravity theories in dimension 11, 10, 6 and 5. Lorentzian symmetric spaces appear as relevant geometries for the so-called maximal supersymmetric backgrounds, and play a role for the determination of parallelisable supergravity backgrounds in type II supergravity. Applications of methods from the geometry of split signature metrics can be seen in the article of Frederik Witt. He studies the geometry of type II supergravity compactifications in terms of an oriented vector bundle E endowed with a bundle metric of split signature, which is associated with a so-called generalized G-structure, introduced by Hitchin. In particular, integrable generalized G-structures are considered. Finally, the article of Gaetano Vilasi is about Einstein metrics with 2-dimensional Killing leaves and their physical interpretations. Here, solutions of the vacuum Einstein field equations for metrics with a non-abelian 2-dimensional Lie algebra of Killing fields are explicitly described. Although pseudo-Riemannian geometry has experienced a rapid development in recent years and essential results were obtained, many fundamental questions are still open. We hope that this volume will stimulate interest for studying and solving geometric problems of pseudo-Riemannian geometry, which arise naturally in differential geometry and mathematical physics. We would like to thank the Erwin Schrödinger Institute in Vienna for the opportunity to organize the special programme Geometry of Pseudo-Riemannian Manifolds with Applications in Physics at ESI, for the financial and the organizational support, and for the stimulating scientific atmosphere, without which this research programme would not have been so successful. Moreover, we would like to thank Manfred Karbe and Irene Zimmermann from the EMS Publishing House for pleasant cooperation during the preparation of the volume. Berlin, May 2008

Dmitri Alekseevski and Helga Baum

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v The classification problem for pseudo-Riemannian symmetric spaces by Ines Kath and Martin Olbrich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Holonomy groups of Lorentzian manifolds: classification, examples, and applications by Anton Galaev and Thomas Leistner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Hypersymplectic manifolds by Andrew Dancer and Andrew Swann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Anti-self-dual conformal structures in neutral signature by Maciej Dunajski and Simon West . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A neutral Kähler surface with applications in geometric optics by Brendan Guilfoyle and Wilhelm Klingenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A primer on the .2 C 1/ Einstein universe by Thierry Barbot, Virginie Charette, Todd Drumm, William M. Goldman, and Karin Melnick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Essential conformal structures in Riemannian and Lorentzian geometry by Charles Frances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Conformal transformations of pseudo-Riemannian manifolds by Wolfgang Kühnel and Hans-Bert Rademacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 The causal hierarchy of spacetimes by Ettore Minguzzi and Miguel Sánchez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Geodesics in semi-Riemannian manifolds: geometric properties and variational tools by Anna Maria Candela and Miguel Sánchez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

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Lorentzian symmetric spaces in supergravity by José Figueroa-O’Farrill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Metric bundles of split signature and type II supergravity by Frederik Witt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455 Einstein metrics with 2-dimensional Killing leaves and their physical interpretation by Gaetano Vilasi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

The classification problem for pseudo-Riemannian symmetric spaces Ines Kath and Martin Olbrich

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Metric Lie algebras . . . . . . . . . . . . . . . . . 2.1 Examples of metric Lie algebras . . . . . . . 2.2 Metric Lie algebras and quadratic extensions 2.3 Quadratic cohomology . . . . . . . . . . . . 2.4 A classification scheme . . . . . . . . . . . . 2.5 Classification results for metric Lie algebras .

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Symmetric spaces . . . . . . . . . . . . . . . . . 3.1 Symmetric triples and quadratic extensions 3.2 The geometry of quadratic extensions . . . 3.3 Symmetric spaces of index one and two . .

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Special geometric structures on symmetric spaces . . . . . . . 4.1 Examples of geometric structures . . . . . . . . . . . . . 4.2 Pseudo-Hermitian symmetric spaces . . . . . . . . . . . 4.3 Quaternionic Kähler and hyper-Kähler symmetric spaces

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Further applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Extrinsic symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Manin triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Appendix: Some lemmas and proofs . . . 6.1 Implications of .h; K/-equivariance 6.2 Proof of Proposition 2.12 . . . . . . 6.3 Proof of Proposition 3.8 . . . . . . .

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1 Introduction There are many basic problems in differential geometry that are completely solved for Riemannian manifolds, but that become really complicated in the pseudo-Riemannian situation. One of these problems is the determination of all possible holonomy groups of pseudo-Riemannian manifolds. While holonomy groups of Riemannian manifolds are classified the problem is open for general pseudo-Riemannian manifolds, only the

Supported by Heisenberg program, DFG.

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I. Kath and M. Olbrich

Lorentzian case is solved. The difficulty is that in general the holonomy representation of a pseudo-Riemannian manifold does not decompose into irreducible summands. Of course, we can decompose the representation into indecomposable ones, i.e., into subrepresentations that do not have proper non-degenerate invariant subspaces. By de Rham’s theorem the indecomposable summands are again holonomy representations. This reduces the problem to the classification of indecomposable holonomy representations. Indecomposable holonomy represesentations that are not irreducible have isotropic invariant subspaces. Such representations are especially difficult to handle if these invariant subspaces do not have an invariant complement. Manifolds that have an indecomposable holonomy representation are called indecomposable. Manifolds that are not indecomposable are at least locally a product of pseudo-Riemannian manifolds. Hence, we can speak of local factors of such a manifold. Many open questions in pseudo-Riemannian differential geometry are directly related to the unsolved holonomy problem. One of these open questions is the classification problem for symmetric spaces. Pseudo-Riemannian symmetric spaces are in some sense the most simple pseudo-Riemannian manifolds. Locally they are characterised by parallelity of the curvature tensor. As global manifolds they are defined as follows. A connected pseudo-Riemannian manifold M is called a pseudo-Riemannian symmetric space if for any x 2 M there is an involutive isometry x of M that has x as an isolated fixed point. In other words, for any x 2 M the geodesic reflection at x extends to a globally defined isometry of M . The theory of Riemannian symmetric spaces was developed simultaneously with the theory of semisimple Lie groups and algebras by E. Cartan during the first decades of the twentieth century. It results in a complete classification of these spaces, see Helgason’s beautiful book [30] on the subject. The theory remains similar in spirit as long as one is interested in pseudo-Riemannian symmetric spaces whose holonomy representation is completely reducible as an algebraic representation, i.e., if any invariant subspace of the holonomy representation has an invariant complement. These so called reductive symmetric spaces were classified by Berger [9] in 1957. This classification is essentially the classification of involutions on real semisimple Lie algebras. In order to understand non-reductive pseudo-Riemannian symmetric spaces one has to consider more general Lie algebras, which are, moreover, equipped with an invariant inner product. Note that in contrast to the semisimple case this inner product is really an additional datum since it is not just a multiple of the Killing form. Such a pair .g; h ; i/ consisting of a finite-dimensional real Lie algebra and an ad-invariant non-degenerate symmetric bilinear form on it is called a metric Lie algebra. In the literature metric Lie algebras appear under various different names, e.g. as quadratic or orthogonal Lie algebras. The transition from symmetric spaces to metric Lie algebras proceeds as follows. Let .M; g/ be a pseudo-Riemannian symmetric space. The group G generated by compositions of geodesic reflections x ı y , x; y 2 M , is called the transvection group of .M; g/. It acts transitively on M . We fix a base point x0 2 M . The reflection x0 induces an involutive automorphism of the Lie algebra g of G, and thus a decomposition g D gC ˚ g . The natural identification Tx0 M Š g induces

The classification problem for pseudo-Riemannian symmetric spaces

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an ad.gC /-invariant bilinear form h ; i on g . It is an important observation (see [16]) that h ; i extends uniquely to an ad.g/- and -invariant inner product h ; i on g. Thus, starting with a pseudo-Riemannian symmetric space .M; g/, we obtain a metric Lie algebra .g; h ; i/ together with an isometric involution on it. The resulting triple .g; ; h ; i/ will be called a symmetric triple later on. Up to local isometry, .M; g/ can be recovered from this structure. We will call a symmetric space as well as its associated symmetric triple semisimple (reductive, solvable etc.) if its transvection group is semisimple (reductive, solvable etc.). The reader will find more details on the correspondence between symmetric spaces and symmetric triples in Section 3.1. For the general theory of symmetric spaces he may consult [10], [16], [30], [39], [40], [42]. The moral we want to stress at this point is that the understanding of metric Lie algebras is crucial for the understanding of symmetric spaces. The present paper focuses on the classification problem for pseudo-Riemannian symmetric spaces. The above discussion reduces this problem to the classification of symmetric triples. It is easy to see that we can decompose every symmetric triple into a direct sum of a semisimple one and one whose underlying Lie algebra does not have simple ideals. Of course, pseudo-Riemannian symmetric spaces that are associated with semisimple symmetric triples are reductive, and thus, as explained above, already classified. This is the reason for our decision to concentrate here on metric Lie algebras and symmetric triples without simple ideals. Thus the investigation of the geometry of semisimple symmetric spaces, which is still an active and interesting field, will be left almost untouched in this paper. The classification of metric Lie algebras appears to be very difficult. Most likely, one has to accept that one will not get a classification in the sense of a list that includes all metric Lie algebras for arbitrary index of the inner product. The same is true for symmetric triples, the existence of an involution neither decreases nor really increases the difficulties. Therefore the aim is to develop a structure theory for metric Lie algebras (and symmetric triples) that allows a systematic construction and that gives a “recipe” how to get an explicit classification under suitable additional conditions, e.g., for small index of the inner product. In [34], [35], [36] we developed a new strategy to reach this aim. The initial idea of this strategy is due to Bérard-Bergery who observed that every symmetric triple without simple ideals arises in a canonical way by an extension procedure from “simpler” Lie algebras with involution. We used this idea to give a cohomological description of isomorphism classes of metric Lie algebras (and symmetric triples), which gives a suitable classification scheme. Here we will present this method, called quadratic extension, and some of its applications. Moreover, we will survey earlier and related results due to Cahen, Parker, Medina, Revoy, Bordemann, Alekseevsky, Cortés, and others concerning metric Lie algebras and symmetric triples from this new point of view. We do not aim at a complete overview on the work on metric Lie algebras and symmetric spaces, for instance the basic results of Astrahancev (see e.g. [4]) will not be discussed. However, we try to present a quite complete up-to-date account for classification results for metric Lie algebras (Section 2.5), symmetric spaces (Section 3.3),

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and symmetric spaces with certain complex or quaternionic structures (Sections 4.2 and 4.3). Note that metric Lie algebras are of interest in their own right, not only in the context of symmetric spaces. They naturally appear in various contexts, e.g., in Mathematical physics or in Poisson geometry. As an illustration, we shortly discuss the notions of Manin triples and pairs and present a new construction method for Manin pairs based on the theory of quadratic extensions of metric Lie algebras (Section 5.2). As a further application we study pseudo-Riemannian extrinsic symmetric spaces by our method (Section 5.1). Though being a survey article the paper also contains some new results. A first group of new results appears in Section 2 and is due to the fact that we develop here a unified theory which works for metric Lie algebras, symmetric triples, and symmetric triples with additional structures at once. Most of these results are straightforward generalisations of the corresponding special results given in the original papers [35], [36], [37]. Proofs that really require new ideas will be given in the appendix. This generality makes Sections 2.2 and 2.3 a little bit more technical than usual for survey articles. However, having mastered these moderate technical difficulties the reader will see in the subsequent sections how quite different results follow easily from one general principle. The results in Section 3.2 concerning the geometric meaning of the quadratic extension procedure and the above mentioned construction method for Manin pairs appear here for the first time. In addition, we announce a new result on the structure of hyper-Kähler symmetric spaces (Theorem 4.4). The theory of metric Lie algebras and pseudo-Riemannian symmetric spaces is far from being complete. In fact, there is a huge amount of open problems. The difficulty is to find those questions, which really lead to new theoretical insight and not just to messy calculations. We hope that the questions raised at several places in this paper belong to the fist category. Some conventions. We denote by N the set of positive integers and we put N0 WD N [ f0g. Let .a; h ; ia / be a pseudo-Euclidean vector space. A subspace a0 a is called isotropic if h ; ia ja0 D 0. A basis A1 ; : : : ; Ap ; ApC1 ; : : : ; ApCq of a is called orthonormal if Ai ? Aj for i 6D j , hAi ; Ai ia D 1 for i D 1; : : : ; p and hAj ; Aj ia D 1 for j D p C 1; : : : ; p C q. In this case .p; q/ is called signature and p index of h ; ia (or of a). Let h ; ip;q be the inner product of signature .p; q/ on RpCq for which the standard basis of RpCq is an orthonormal basis. Then we call Rp;q WD .RpCq ; h ; ip;q / standard pseudo-Euclidean space. We will often describe a Lie algebra by giving a basis and some of the Lie bracket relations, e.g. we will write the three-dimensional Heisenberg algebra as h.1/ D fŒX; Y D Zg. In this case we always assume that all other brackets of basis vectors vanish. If we do not mention the basis explicitly, then we suppose that all basis vectors appear in one of the bracket relations (on the left- or the right-hand side). Let l be a Lie algebra and let a be an l-module. Then al denotes the space of invariants in a, i.e., al D fA 2 a j L.A/ D 0g for all L 2 l.


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2 Metric Lie algebras 2.1 Examples of metric Lie algebras. The easiest example of a metric Lie algebra is an abelian Lie algebra together with an arbitrary (non-degenerate) inner product. A further well-known example is a semisimple Lie algebra equipped with a non-zero multiple of its Killing form. Let H be a Lie group and h its Lie algebra. The cotangent bundle T H can be given a group structure such that the associated Lie algebra equals h ad Ë h . Now let h ; ih be any invariant symmetric bilinear form on h (which can degenerate). We can define on h ad Ë h a symmetric bilinear form h ; i by adding h ; ih to the dual pairing of h and h , that is by hH1 C Z1 ; H2 C Z2 i D Z1 .H2 / C Z2 .H1 / C hH1 ; H2 ih for H1 ; H2 2 h, Z1 ; Z2 2 h . It is not hard to prove that h ; i is invariant and nondegenerate, its signature equals .dim h; dim h/. Hence .h ad Ë h ; h ; i/ is a metric Lie algebra. In particular, h ; i induces a biinvariant metric on T H . The following construction is a generalisation of the previous example. It is due to Medina and Revoy [44]. Starting with an n-dimensional metric Lie algebra and an arbitrary m-dimensional Lie algebra it produces a metric Lie algebra of dimension n C 2m. Let .g; h ; ig / be a metric Lie algebra and let .h; h ; ih / be a Lie algebra with an invariant symmetric bilinear form (which can degenerate). Furthermore, let W h ! Der a .g; h ; ig / be a Lie algebra homomorphism from h into the Lie algebra V of all antisymmetric derivations of g. We denote by ˇ 2 C 2 .g; h / WD Hom. 2 g; h / the 2-cocycle (see 2.3 for this notion) ˇ.X; Y /.H / WD h.H /X; Y ig ;

X; Y 2 g; H 2 h:

On the vector space d WD h ˚ g ˚ h we define a Lie bracket Œ ; by Q XQ ; HQ / D Œ.Z; X; H /; .Z; Q C ad .H /ZQ ad .HQ /Z; ŒX; XQ g C .H /XQ .HQ /X; ŒH; HQ h / .ˇ.X; X/ h h and an inner product h ; i by Q XQ ; HQ /i D hX; XQ ig C hH; HQ ih C Z.HQ / C Z.H Q h.Z; X; H /; .Z; / for all Z; ZQ 2 h , X; XQ 2 g and H; HQ 2 h. Then d .g; h/ WD .d; h ; i/ is a metric Lie algebra. It is called double extension of g by h since it can be regarded as an extension of the semi-direct product g Ì h by the abelian Lie algebra h . If the signature of g equals .p; q/ and if dim h D m, then the signature of d .g; h/ equals .p C m; q C m/. The importance of this construction becomes clear from the following structure theorem by Medina and Revoy. It says that we can inductively produce all metric Lie algebras from simple and one-dimensional ones by taking direct sums and applying the double extension procedure.

6


Theorem 2.1 (Medina–Revoy [44]). If .g; h ; i/ is an indecomposable metric Lie algebra, then either g is simple or g is one-dimensional or g is a double extension d .g; Q h/ of a metric Lie algebra gQ by a one-dimensional or a simple Lie algebra h. We remark that for the special case of solvable metric Lie algebras this result can already be found in the form of exercises in Kac’s book [31], Exercises 2.10,11. For a complete proof in this case see also [23]. Using Theorem 2.1 it is not hard to see that any indecomposable non-simple Lorentzian metric Lie algebra is the double extension of an abelian Euclidean metric Lie algebra by a one-dimensional Lie algebra. This allows the classification of isomorphism classes of Lorentzian metric Lie algebras [43], compare Example 2.2 and Theorem 2.4. In principle one can try to use this method to classify also metric Lie algebras of higher index. This was done in [5] for index two. However, now the following difficulty arises. In general a metric Lie algebra of index greater than one can be obtained in many different ways by double extension from a lower-dimensional one. Thus in order to solve the classification problem we would have to decide under which conditions two metric Lie algebras arising in different ways by repeated application of the double extension construction (and direct sums) are isomorphic. This seems to be very complicated. Therefore we are now looking for a way that avoids this difficulty. In the following we will develop a structure theory for metric Lie algebras which is more adapted to classification problems. The basic idea of this theory goes back to Bérard-Bergery [7] who suggested to consider indecomposable non-semisimple pseudo-Riemannian symmetric spaces as the result of two subsequent extensions. Our starting point is the following construction. Let l be a Lie algebra and let W l ! so.a/ be an orthogonal representation V of l on a pseudo-Euclidean vector space .a; h ; ia /. Take ˛ 2 C 2 .l; a/ WD Hom. 2 l; a/ V and 2 C 3 .l/ WD Hom. 3 l; R/. We consider the vector space d WD l ˚ a ˚ l and define an inner product h ; i by hZ1 C A1 C L1 ; Z2 C A2 C L2 i WD hA1 ; A2 ia C Z1 .L2 / C Z2 .L1 / for Z1 ; Z2 2 l ; A1 ; A2 2 a and L1 ; L2 2 l. Moreover, we define an antisymmetric bilinear map Œ ; W d d ! d by Œl ; l ˚ a D 0 and ŒL1 ; L2 D .L1 ; L2 ; / C ˛.L1 ; L2 / C ŒL1 ; L2 l ; ŒL; A D hA; ˛.L; /i C L.A/; ŒA1 ; A2 D h./A1 ; A2 i; ŒL; Z D ad .L/.Z/ for L; L1 ; L2 2 l, A; A1 ; A2 2 a and Z 2 l . Then the Jacobi identity for Œ ; is equivalent to a certain cocycle condition for ˛ and . We will denote this condition by .˛; / 2 Z2Q .l; a/ and postpone its exact formulation to Section 2.3. Thus, if .˛; / 2 Z2Q .l; a/, then Œ ; is a Lie bracket and it is easy to check that h ; i is invariant with respect to this bracket. This gives the following result.


7

Proposition 2.1 ([35], Proposition 2.4). If .˛; / 2 Z2Q .l; a/, then d˛; .l; a/ WD .d; h ; i/ is a metric Lie algebra. Two special cases of this construction were known previously. For the case ˛ D D 0 our d˛; .l; a/ is the double extension d .a; l/ of the abelian metric Lie algebra a by l (in the sense of Medina and Revoy as explained above) and for a D 0 it coincides with the T -extension introduced by Bordemann [12]. Example 2.2. Take l D R. Let a be the standard Euclidean vector space R2m with (orthonormal) standard basis e1 ; : : : ; e2m . Take D .1 ; : : : ; m / 2 .l /m Š Rm . We define an orthogonal representation of l on a by .L/.e2i1 / D i .L/ e2i ;

.L/.e2i / D i .L/ e2i1

for L 2 l and i D 1; : : : ; m. We set a WD . ; a/. Then osc./ WD d0;0 .R; a / is a metric Lie algebra of signature .1; 2m C 1/. This Lie algebra is often called oscillator algebra. As explained above, osc./ can also be considered as double extension of a by R. In the following we will see that any metric Lie algebra without simple ideals is isomorphic to some d˛; .l; a/ for suitable data l, a, .˛; / 2 Z2Q .l; a/ and how this fact can be used to describe isomorphism classes of metric Lie algebras. 2.2 Metric Lie algebras and quadratic extensions. As already mentioned in the introduction we are especially interested in metric Lie algebras without simple ideals. In this section we will learn more about the structure of such metric Lie algebras. Later on we wish to equip metric Lie algebras with additional structures, e.g. with an involution when we want to study symmetric triples or with even more structure when we will be studying geometric structures on symmetric spaces. For this reason we develop a theory that is equivariant under a Lie algebra h and a Lie group K acting semisimply on h by automorphisms. We assume throughout the paper that K has only finitely many connected components. We suggest to take the trivial case .h; K/ D .0; feg/ on first reading. In particular, this means that you may omit all maps called ˆ in the following. An .h; K/-module .V; ˆV / consists of a finite-dimensional vector space V and a map ˆV W h [ K ! Hom.V / such that ˆV jh W h ! Hom.V / and ˆV jK W K ! GL.V / Hom.V / are representations of h and K satisfying ˆV .k X / D ˆV .k/ˆV .X /ˆV .k/1 for all k 2 K and X 2 h. There is a natural notion of an .h; K/-submodule. An .h; K/module is called semisimple if for any .h; K/-submodule there is a complementary .h; K/-submodule. Definition 2.3. 1. An .h; K/-equivariant Lie algebra .l; ˆ l / is a Lie algebra l that is equipped with the structure of a semisimple .h; K/-module such that im.ˆ l jh / Der.l/ and im.ˆ l jK / Aut.l/, where Der.l/ and Aut.l/ denote the Lie algebra of derivations and the group of automorphisms of l, respectively. 2. An .h; K/-equivariant metric Lie algebra is a triple .g; ˆ; h ; i/ such that

8


(i) .g; h ; i/ is a metric Lie algebra, (ii) .g; ˆ/ is an .h; K/-equivariant Lie algebra, (iii) ˆ.h/ Der a .g/ and ˆ.K/ Aut.g; h ; i/, where Der a .g/ denotes the Lie algebra of antisymmetric derivations on .g; h ; i/. The index (signature) of an .h; K/-equivariant metric Lie algebra .g; ˆ; h ; i/ is the index (signature) of h ; i. Sometimes we abbreviate .g; ˆ; h ; i/ as g. A homomorphism (resp., isomorphism) of .h; K/-equivariant Lie algebras F W .l1 ; ˆ1 / ! .l2 ; ˆ2 / is a homomorphism (resp., isomorphism) of Lie algebras F W l1 ! l2 that satisfies F ı ˆ1 .h/ D ˆ2 .h/ ı F for all h 2 h [ K. Isomorphisms of .h; K/-equivariant metric Lie algebras are in addition compatible with the given inner products. We have a natural notion of direct sums of .h; K/-equivariant (metric) Lie algebras. An .h; K/-equivariant (metric) Lie algebra is called decomposable if it is isomorphic to the direct sum of two non-trivial .h; K/-equivariant (metric) Lie algebras. Otherwise it is called indecomposable. In the following let .l; ˆl / always be an .h; K/-equivariant Lie algebra. Definition 2.4. 1. An .l; ˆl /-module .; a; â / consists of (i) a semisimple .h; K/-module .a; â /, (ii) a representation W l ! Hom.a; h ; i/ that satisfies .ˆl .k/L/ D â .k/ ı .L/ ı â .k/1 ;

.ˆl .X /L/ D Œâ .X /; .L/

for all k 2 K, X 2 h and L 2 l. 2. An orthogonal .l; ˆl /-module .; a; h ; ia ; â / consists of an .l; ˆl /-module .; a; â / and an inner product h ; ia on a such that (i) .a; h ; ia ; â / is an abelian .h; K/-equivariant metric Lie algebra, (ii) is an orthogonal representation, i.e. W l ! so.a; h ; i/. We often abbreviate .; a; â / and .; a; h ; ia ; â / as .; a/ or a. Let .li ; ˆli /, i D 1; 2, be two .h; K/-equivariant Lie algebras and let .i ; ai /, i D 1; 2, be orthogonal .li ; ˆli /-modules. Let S W l1 ! l2 be a homomorphism of .h; K/-equivariant Lie algebras and let U W a2 ! a1 be an .h; K/-equivariant isometric embedding. Suppose that U ı 2 .S.L// D 1 .L/ ı U holds for all L 2 l. Then we call .S; U / morphism of pairs. We will write this as .S; U / W ..l1 ; ˆl1 /; a1 / ! ..l2 ; ˆl2 /; a2 /, but remember that S and U map in different directions. We will say that an ideal of an .h; K/-equivariant Lie algebra .g; ˆ/ is ˆ-invariant if it is invariant under all maps belonging to im ˆ.


9

Definition 2.5. Let .; a; h ; ia ; â / be an orthogonal .l; ˆ l /-module. A quadratic extension of .l; ˆ l / by a is given by a quadruple .g; i; i; p/, where (i) g is an .h; K/-equivariant metric Lie algebra, (ii) i is an isotropic ˆ-invariant ideal of g, (iii) i and p are homomorphisms of .h; K/-equivariant Lie algebras constituting an exact sequence i

p

0 ! a ! g=i ! l ! 0 that is consistent with the representation of l on a and has the property that i is an isometry from a to i? =i. Example 2.6 (The standard model). First we consider a Lie algebra l without further structure, i.e. ˆ l D 0. Let a be an orthogonal l-module and take .˛; / 2 Z2Q .l; a/. Let d˛; .l; a/ D .d; h ; i/ be the metric Lie algebra constructed in Section 2.1. We identify d=l with a ˚ l and denote by i W a ! a ˚ l the injection and by p W a ˚ l ! l the projection. Then .d˛; .l; a/; l ; i; p/ is a quadratic extension of l by a. Now suppose that we have in addition an .h; K/-structure on l and a, i.e. let .l; ˆl / be an .h; K/-equivariant Lie algebra and let .; a; h ; ia ; â / be an orthogonal .l; ˆ l /module. Then we can define a map ˆ W h [ K ! Der.d/ [ Aut.d/ by ˆ.X/.Z C A C L/ D ˆ l .X / .Z/ C â .X /.A/ C ˆ l .X /.L/; ˆ.k/.Z C A C L/ D .ˆ l .k/ /1 .Z/ C â .k/.A/ C ˆ l .k/.L/: Then d˛; .l; ˆ l ; a/ WD .d; ˆ; h ; i/ is an .h; K/-equivariant metric Lie algebra if .˛; / satisfies a certain natural invariance condition with respect to ˆ l and â . We write .˛; / 2 Z2Q .l; ˆ l ; a/ for this condition whose exact formulation we will give in Section 2.3. Hence, if .˛; / 2 Z2Q .l; ˆ l ; a/, then .d˛; ; l ; i; p/ is a quadratic extension of .l; ˆ l / by a. It is called standard model since, as we will see, any quadratic extension of .l; ˆl / by a is in a certain sense equivalent to some .d˛; ; l ; i; p/ for a suitable cocycle .˛; / 2 Z2Q .l; ˆ l ; a/. What makes the theory of quadratic extensions so useful is the fact that any .h; K/equivariant metric Lie algebra without simple ideals admits such a structure. Essentially, this follows from Bérard-Bergery’s investigations of pseudo-Riemannian holonomy representations and symmetric spaces in [6], [7], [8]. He proved that for any metric Lie algebra .g; h ; i/ there exists an isotropic ideal i.g/ g such that i.g/? =i.g/ is abelian. We want to describe the construction of this ideal now. However, instead of following [6], [7] we will give a description that is more adapted to the structure theory that we wish to develop here. In particular, we will give an .h; K/-equivariant formulation. Let .g; ˆ; h ; i/ be an .h; K/-equivariant metric Lie algebra. There is a chain of ˆ-invariant ideals g D R0 .g/ R1 .g/ R2 .g/ Rl .g/ D 0

10


which is defined by the condition that Rk .g/ is the smallest ideal of g contained in Rk1 .g/ such that the g-module Rk1 .g/=Rk .g/ is semisimple. The ideal R.g/ WD R1 .g/ is called nilpotent radical of g. It has to be distinguished from the nilradical (i.e. the maximal nilpotent ideal) n and the (solvable) radical r. By Lie’s Theorem R.g/ D r \ g0 D Œr; g n

(1)

and R.g/ acts trivially on any semisimple g-module [13]. We define an ideal i.g/ g by l1 X Rk .g/ \ Rk .g/? i.g/ WD kD1

and call it the canonical isotropic ideal of g. Proposition 2.7 ([6], [7]; [35], Lemma 3.4). If .g; ˆ; h ; i/ is an .h; K/-equivariant metric Lie algebra, then i.g/ is a ˆ-invariant isotropic ideal and the g-module i.g/? =i.g/ is semisimple. If g does not contain simple ideals, then the Lie algebra i.g/? =i.g/ is abelian. In particular, g=i.g/? becomes an .h; K/-equivariant Lie algebra and i.g/? =i.g/ a semisimple orthogonal g=i.g/? -module. Moreover, p

i

0 ! i.g/? =i.g/ ! g=i.g/ ! g=i.g/? ! 0 is an exact sequence of .h; K/-equivariant Lie algebras. Corollary 2.8. For any .h; K/-equivariant metric Lie algebra .g; ˆ; h ; i/ without simple ideals the quadruple .g; i.g/; i; p/ is a quadratic extension of g=i.g/? by i.g/? =i.g/. This extension will be called the canonical quadratic extension associated with .g; ˆ; h ; i/. Example 2.9. The following example shows that for a given metric Lie algebra .g; h ; i/ there may exist other quadratic extensions .g; i; i; p/ than the canonical one. Let h.1/ D fŒX1 ; X2 D X3 g be the three-dimensional Heisenberg algebra and let 1 , 2 , 3 be the basis of h.1/ that is dual to X1 ; X2 ; X3 . Let us consider the metric Lie algebra g WD d0;0 .h.1/; 0/ D h.1/ Ì h.1/. Example 2.6 says that .g; h.1/ ; i; p/ is a quadratic extension of h.1/ by 0, where .i; p/ is defined by iD0

p

0 ! g=h.1/ Š h.1/ ! 0: However, this quadratic extension is not the canonical one. Indeed, we have R.g/ D g0 D spanfX3 ; 1 ; 2 g. In particular, R.g/ z.g/, hence R2 .g/ D 0. This implies i.g/ D R.g/? \ R.g/ D R.g/ D spanfX3 ; 1 ; 2 g: In particular, the canonical quadratic extension associated with g is a quadratic extension of g=i.g/? Š R3 6Š h.1/ by i.g/? =i.g/ D 0.


11

Hence, at first glance we have the same difficulty as for double extensions, namely, in general an .h; K/-equivariant metric Lie algebra can be obtained in different ways by quadratic extensions. However, now we can always distinguish one of these extensions, namely the canonical one. As a quadratic extension this extension is characterised by the property to be balanced in the following sense. Definition 2.10. A quadratic extension .g; i; i; p/ of an .h; K/-equivariant Lie algebra .l; ˆ/ by an orthogonal .l; ˆ l /-module a is called balanced if i D i.g/. Since our aim is to determine isomorphism classes of .h; K/-equivariant metric Lie algebras, Corollary 2.8 leads us to the problem to decide for which balanced quadratic extensions .g1 ; i1 ; i1 ; p1 / and .g2 ; i2 ; i2 ; p2 / the .h; K/-equivariant metric Lie algebras g1 and g2 are isomorphic. We will divide this problem into two steps. First we will introduce an equivalence relation for quadratic extensions that is stronger than isomorphy of the underlying .h; K/-equivariant metric Lie algebras. We will describe the corresponding equivalence classes. In the second step we have to decide which equivalence classes of quadratic extensions have isomorphic underlying .h; K/-equivariant metric Lie algebras. Definition 2.11. Two quadratic extensions .gj ; ij ; ij ; pj /, j D 1; 2, of .l; ˆ l / by a are called equivalent if there exists an isomorphism F W g1 ! g2 of .h; K/-equivariant metric Lie algebras that maps i1 to i2 and satisfies Fx ı i1 D i2 and p2 ı Fx D p1 , where Fx W g1 =i1 ! g2 =i2 is the map induced by F . Similar to the case of ordinary extensions of Lie algebras one can describe equivalence classes of quadratic extensions by cohomology classes. We will introduce a suitable cohomology theory in the next section. Actually, for a given .h; K/-equivariant Lie algebra .l; ˆ l / and an orthogonal .l; ˆ l /-module a we will define a cohomology 2 2 2 set HQ .l; ˆ l ; a/ and a subset HQ .l; ˆ l ; a/b HQ .l; ˆ l ; a/ such that the following holds. Theorem 2.2. There is a bijective map ‰ from the set of equivalence classes of quadratic 2 extensions of .l; ˆ l / by a to HQ .l; ˆ l ; a/. The image under ‰ of the subset of all 2 2 equivalence classes of balanced extensions equals HQ .l; ˆ l ; a/b HQ .l; ˆ l ; a/. 2 The set HQ .l; ˆ l ; a/ consists of equivalence classes Œ˛; of cocycles .˛; / 2 2 ZQ .l; ˆl ; a/ with respect to a certain equivalence relation. The inverse of ‰ then maps Œ˛; to the equivalence class of the standard model d˛; .l; ˆl ; a/ (Example 2.6). For

an explicit description of the map ‰ see Section 2.4. For a proof of this theorem in the non-equivariant case see [35], Theorem 2.7 and Theorem 3.12.

2.3 Quadratic cohomology. The aim of this section is the exact definition of the cohomology sets that appear in Theorem 2.2. Since quadratic extensions are not ordinary Lie algebra extensions we cannot expect to describe them by usual Lie algebra cohomology. We need a kind of non-linear cohomology. Such cohomology sets were first introduced by Grishkov [29] in a rather general setting. For the special case of

12


cohomology needed for quadratic extensions we gave a self-contained presentation in [35]. Neither [29] nor [35] deals with the equivariant situation. As we will see here, the .h; K/-action can be easily incorporated. Let us first recall the construction of the usual Lie algebra cohomology. Let W l ! gl.a/ be a representation of a Lie algebra l on a vector space a. Then we have V the standard Lie algebra cochain complex .C .l; a/; d /, where C p .l; a/ D Hom. p l; a/ and d W C p .l; a/ ! C pC1 .l; a/ is defined by d .L1 ; : : : ; LpC1 / D

pC1 X

.1/i1 .Li / .L1 ; : : : ; LO i ; : : : ; LpC1 /

iD1

C

X O i ; : : : ; LO j ; : : : ; LpC1 /: .1/iCj .ŒLi ; Lj ; L1 ; : : : ; L i<j

The corresponding cohomology groups are denoted by H p .l; a/. In the special case where a is the one-dimensional trivial representation of l we denote the standard cochain complex also by .C .l/; d / and the cohomology groups by H p .l/. Suppose that we have two Lie algebras li , i D 1; 2 and orthogonal li -modules ai and that .S; U / W .l1 ; a1 / ! .l2 ; a2 / is a morphism of pairs. Then we have pull back maps .S; U / W C p .l2 ; a2 / ! C p .l1 ; a1 / .S; U / ˛.L1 ; : : : ; Lp / WD U ı ˛.S.L1 /; : : : ; S.Lp //; .S; U / W C p .l2 / ! C p .l1 /; .S; U / .L1 ; : : : ; Lp / WD .S.L1 /; : : : ; S.Lp //: Now let .l; ˆl / be an .h; K/-equivariant Lie algebra and let a be an orthogonal .h; K/-module. Then we can consider .e ˆ l .X/ ; e â .X/ / W .l; a/ ! .l; a/; X 2 h; and

.ˆ l .k/; .â .k//1 / W .l; a/ ! .l; a/;

(2)

k 2 K;

(3)

as morphism of pairs (without .h; K/-structure). Let C p .l; a/.h;K/ C p .l; a/ denote the subspace of cochains that are invariant under these morphisms of pairs for all X 2 h and k 2 K. We define a product C p .l; a/.h;K/ C q .l; a/.h;K/ ! C pCq .l/.h;K/ by ^

h ;ia

h ^ i W C p .l; a/.h;K/ C q .l; a/.h;K/ ! C pCq .l; a ˝ a/.h;K/ ! C pCq .l/.h;K/ : Now we define the set of quadratic 1-cochains to be 1 .l; ˆ l ; a/ D C 1 .l; a/.h;K/ ˚ C 2 .l/.h;K/ : CQ


13

This set is a group with group operation defined by . 1 ; 1 / . 2 ; 2 / D . 1 C 2 ; 1 C 2 C 12 h 1 ^ 2 i/: Let us consider the set ˚ Z2Q .l; ˆ l ; a/ D .˛; / 2 C 2 .l; a/.h;K/ ˚ C 3 .l/.h;K/ j d˛ D 0; d D 12 h˛ ^ ˛i 1 .l; ˆ l ; a/ acts from whose elements are called quadratic 2-cocycles. Then the group CQ 2 the right on ZQ .l; ˆ l ; a/ by .˛; /. ; / D ˛ C d ; C d C h.˛ C 12 d / ^ i :

We define the quadratic cohomology set as the orbit space 2 1 HQ .l; ˆ l ; a/ WD Z2Q .l; ˆ l ; a/=CQ .l; ˆ l ; a/: 2 .l; ˆ l ; a/ is denoted by Œ˛; . The equivalence class of .˛; / 2 Z2Q .l; ˆ l ; a/ in HQ As usual, if .h; K/ is trivial, then we omit ˆ l in all the notation above. Now let us consider a morphism of pairs .S; U / W ..l1 ; ˆ1 /; a1 / ! ..l2 ; ˆ2 /; a2 /. As discussed above .S; U / acts on C p .l; a/. By .h; K/-equivariance .S; U / maps the subspace C p .l2 ; a2 /.h;K/ C p .l2 ; a2 / to C p .l1 ; a1 /.h;K/ C p .l1 ; a1 /. It is not hard to prove that this map induces a map 2 2 .S; U / W HQ .l2 ; ˆl2 ; a2 / ! HQ .l1 ; ˆl1 ; a1 /

(cf. [35] for a proof in the non-equivariant case). In particular, for a given .h; K/-equivariant Lie algebra .l; ˆ l / and an orthogonal 2 .l; ˆ l /-module a the morphisms of pairs in (2) and (3) induce maps on HQ .l; a/. 2 2 .h;K/ Let HQ .l; a/ HQ .l; a/ denote the subset of all cohomology classes that are invariant under these maps for all X 2 h and k 2 K. Proposition 2.12. Under our assumptions on .h; K/ the inclusion Z2Q .l; ˆ l ; a/ ,! Z2Q .l; a/ induces a bijection 2 2 .l; ˆ l ; a/ ! HQ .l; a/.h;K/ : HQ

You can find a proof of this proposition in the appendix. 2 2 Next we will define the subset HQ .l; ˆ l ; a/b HQ .l; ˆ l ; a/, which plays an important role in Theorem 2.2. It was first introduced in [35], where you can also find a proof of the fact that its elements correspond exactly to those extensions that are balanced. As usual, an .l; ˆ l /-module .; a/ will be called semisimple if every .l; ˆ l /-submodule has an .l; ˆ l /-invariant complement. This is the case if and only if is semisimple. In the following definition we need the notion of the socle S.l/ of a Lie algebra l, which is the maximal ideal of l on which l acts semisimply.

14


Definition 2.13. Let .l; ˆ l / be an .h; K/-equivariant Lie algebra, let .; a/ be a semisimple orthogonal .l; ˆ l /-module and take .˛; / 2 Z2Q .l; ˆ l ; a/. Since is semisimple we have a decomposition a D al ˚ .l/a and a corresponding decomposition ˛ D ˛0 C ˛1 . Let m be such that RmC1 .l/ D 0. Then .˛; / 2 Z2Q .l; ˆ l ; a/ is called balanced if it satisfies the following conditions .Ak / and .Bk / for 0 k m. .A0 / Let L0 2 z.l/ \ ker be such that there exist elements A0 2 a and Z0 2 l satisfying for all L 2 l (i) ˛.L; L0 / D .L/A0 , (ii) .L; L0 ; / D hA0 ; ˛.L; /ia C hZ0 ; ŒL; l i as an element of l , then L0 D 0. .B0 / The subspace ˛0 .ker Œ ; l / al is non-degenerate. .Ak / .k 1/ Let k S.l/ \ Rk .l/ be an l-ideal such that there exist elements ˆ1 2 Hom.k; a/ and ˆ2 2 Hom.k; Rk .l/ / satisfying for all L 2 l and K 2 k (i) ˛.L; K/ D .L/ˆ1 .K/ ˆ1 .ŒL; Kl /, (ii) .L; K; / D hˆ1 .K/; ˛.L; /ia C hˆ2 .K/; ŒL; l i C hˆ2 .ŒL; Kl /; i as an element of Rk .l/ , then k D 0. .Bk / .k 1/ Let bk a be the maximal submodule such that the system of equations h˛.L; K/; Bia D h.L/ˆ.K/ˆ.ŒL; Kl /; Bia ;

L 2 l; K 2 Rk .l/; B 2 bk ;

has a solution ˆ 2 Hom.Rk .l/; a/. Then bk is non-degenerate. One can prove that for a cocycle the property to be balanced depends only on its 2 cohomology class. Hence we may call a cohomology class Œ˛; 2 HQ .l; ˆ l ; a/ 2 balanced if .˛; / 2 ZQ .l; ˆ l ; a/ is balanced. For an .h; K/-equivariant Lie algebra .l; ˆ l / and an orthogonal .l; ˆ l /-module 2 2 .; a/ let HQ .l; ˆ l ; a/b HQ .l; ˆ l ; a/ be the set of all balanced cohomology classes 2 if is semisimple and put HQ .l; ˆ l ; a/b WD ; if is not semisimple. This finishes the definition of the cohomology sets used in Theorem 2.2. Example 2.14. Take l D h.1/ D fŒX; Y D Zg and let .; a/ be a semisimple orthogonal l-module. Then the following two maps are bijective: Zl WD f˛ 2 C 2 .l; a/ j ˛.X; Y / D 0; ˛.Z; l/ al g ! H 2 .l; a/; ˛ 7! Œ˛; 2 Zl;b WD f˛ 2 Zl j ˛ 6D 0; ˛.Z; l/ al is non-degenerateg ! HQ .l; a/b ;

˛ 7! Œ˛; 0: Proof. Let us consider the first map. It is well-defined and injective. We will prove that it is surjective. Take a 2 H 2 .l; a/. Since l is nilpotent and a is semisimple we have


15

H .l; a/ D H .l; al / [21]. Hence we can represent a by a cocycle ˛ that satisfies ˛.l; l/ al . We define 2 C 1 .l; a/ by .X / D .Y / D 0, .Z/ D ˛.X; Y /. Then ˛Q WD ˛ C d satisfies ˛.X; Q Y / D ˛.X; Y / .ŒX; Y / D 0 and ˛.Z; Q l/ al . Since Œ˛ D Œ˛ Q the assertion follows. Now let us turn to the second map. First we have to check that it is also well-defined. If ˛ 2 Zl , then h˛ ^ ˛i D 0. Hence .˛; 0/ 2 Z2Q .l; a/. We have to show that the cocycle .˛; 0/ is balanced if ˛.l; l/ is non-degenerate and ˛ 6D 0. We have to check the admissibility conditions .Ak /, .Bk / for k D 0; 1 (note that R2 .l/ D 0). Since is semisimple and Z 2 R.l/ we have .Z/ D 0, hence z.l/ \ ker D R Z. Because of al ˛.Z; l/ 6D 0 Conditions .A0 / and .A1 / are satisfied. Conditions .B0 / and .B1 / hold since ˛.Z; l/ al is non-degenerate by assumption. Hence the second map is well-defined. Obviously it is injective. Let us prove surjectivity. Suppose 2 a 2 HQ .l; a/b . By surjectivity of our first map we can represent a by a balanced cocycle .˛; / with ˛ 2 Zl . Clearly, ˛ D 0 would contradict Condition .A0 / (choose L0 D Z, A0 D 0, Z0 D 0). Hence 0 6D ˛ 2 Zl . Now it is easy to see that there exists a cochain 2 C 1 .l; a/ such that d D 0 and h˛ ^ i D . Consequently, a D Œ˛; D Œ˛; 0, which proves the assertion. Since we want to construct indecomposable metric Lie algebras by quadratic extensions we also need a notion of indecomposability for quadratic cohomology classes. Definition 2.15. A non-trivial decomposition of a pair ..l; ˆ l /; a/ consists of two nonzero morphisms of pairs .qi ; ji / W ..l; ˆ l /; a/ ! ..li ; ˆli /; ai /, i D 1; 2; such that .q1 ; j1 / ˚ .q2 ; j2 / W ..l; ˆ l /; a/ ! ..l1 ; ˆl1 /; a1 / ˚ ..l2 ; ˆl2 /; a2 / is an isomorphism. 2 .l; ˆ l ; a/ is called decomposable if it can be written A cohomology class ' 2 HQ as a sum ' D .q1 ; j1 / '1 C .q2 ; j2 / '2 2 for a non-trivial decomposition .qi ; ji / of ..l; ˆl /; a/ and certain 'i 2 HQ .li ; ˆli ; ai /, 2 i D 1; 2. Here addition is induced by addition in the vector space C .l; a/ ˚ C 3 .l/. A cohomology class which is not decomposable is called indecomposable.

Then we have the following relation between indecomposability of cohomology classes and indecomposability of metric Lie algebras. Proposition 2.16 ([35], Proposition 4.5). An .h; K/-equivariant metric Lie algebra .g; ˆ; h ; i/ is indecomposable if and only if the image under ‰ of the canonial quadratic extension associated with .g; ˆ; h ; i/ is an indecomposable cohomology class. 2.4 A classification scheme. According to Corollary 2.8, each .h; K/-equivariant metric Lie algebra without simple ideals comes with a distinguished structure of a quadratic extension which is balanced. By Theorem 2.2 balanced quadratic extensions are characterised by balanced quadratic cohomology classes. We obtain a (functorial)

16


assignment f.h; K/-equivariant metric Lie algebras .g; ˆ; h ; i/ without simple idealsg 2 H) fquadruples .l; ˆl ; a; Œ˛; 2 HQ .l; ˆ l ; a/b /g

(4)

In order to make (4) concrete let us write down the map ‰ appearing in Theorem 2.2 explicitly. Let .g; i; i; p/ be a quadratic extension of .l; ˆ l / by a. Let pQ W g ! l be the map induced by p. We choose an .h; K/-equivariant section s W l ! g of pQ with isotropic image (which exists by semisimplicity of ˆ) and define ˛ 2 C 2 .l; a/.h;K/ and 2 C 3 .l/.h;K/ by i.˛.L1 ; L2 // WD Œs.L1 /; s.L2 / s.ŒL1 ; L2 l / C i 2 g=i; .L1 ; L2 ; L3 / WD h Œs.L1 /; s.L2 /; s.L3 /i: 2 .l; ˆ l ; a/ is the desired cohomology Then .˛; / 2 Z2Q .l; ˆ l ; a/, and Œ˛; 2 HQ class. Note that Œ˛; does not depend on the choice of s while .˛; / does. It will turn out that the data on the right-hand side of (4) give a very useful description of the set of isomorphism classes of .h; K/-equivariant metric Lie algebras. In fact, the resulting classification scheme, Theorem 2.3 below, is the basis of most of the classification results for metric Lie algebras and symmetric spaces that will be presented in this article. ss Let .l; ˆ l / be an .h; K/-equivariant Lie algebra. We consider the category Ml;ˆ of l semisimple orthogonal .l; ˆ l /-modules, where the morphisms between two modules a1 ; a2 are given by morphism of pairs .S; U / W ..l; ˆ l /; a1 / ! ..l; ˆ l /; a2 /. We ss denote the automorphism group of an object a of Ml;ˆ by Gl;ˆ l ;a . The natural right l 2 2 action of Gl;ˆ l ;a on HQ .l; ˆ l ; a/ leaves HQ .l; ˆ l ; a/b invariant.

Theorem 2.3 (compare [35], Theorem 4.6). Let L be a complete set of representatives of isomorphism classes of .h; K/-equivariant Lie algebras. For each .l; ˆ l / 2 L we choose a complete set of representatives Al;ˆ l of isomorphism classes of objects ss . in Ml;ˆ l Then (4) descends to a bijective map from the set of isomorphism classes of .h; K/equivariant metric Lie algebras without simple ideals to the union of orbit spaces a a 2 HQ .l; ˆ l ; a/b =Gl;ˆ l ;a : .l;ˆ l /2L a2Al;ˆ l

2 The inverse of this map sends the orbit of Œ˛; 2 HQ .l; ˆ l ; a/b to the isomorphism class of d˛; .l; ˆl ; a/. An .h; K/-equivariant metric Lie algebra is indecomposable if and only if the cor2 responding Gl;ˆ l ;a -orbit in HQ .l; ˆ l ; a/b consists of indecomposable cohomology classes.

In particular, the theorem says that the .h; K/-equivariant metric Lie algebras d˛; .l; ˆl ; a/ exhaust all isomorphism classes of .h; K/-equivariant metric Lie algebras without simple ideals.


17

Its proof is word by word the same as the one for the non-equivariant case that is given in complete detail in [35]. The main ingredient is Theorem 2.2, which actually involves the construction of the standard model d˛; .l; ˆl ; a/. In addition, one has to show that two metric Lie algebras are isomorphic if and only if their associated balanced cohomology classes are mapped to each other by an isomorphism of pairs. This is a rather straightforward consequence of the functoriality of the canonical isotropic ideal i.g/ ([35], Proposition 4.2). The statement about indecomposability follows from Proposition 2.16. Theorem 2.3 provides a complete set of invariants for equivariant metric Lie algebras and therefore structures the set of all isomorphism classes of them in a certain way. At first glance, however, Theorem 2.3 might look rather useless for classification problems. Indeed, the classification scheme seems to involve the classification of all (equivariant) Lie algebras. While this reflects the real difficulty of the problem there are at least two circumstances that, nevertheless, allow for interesting applications of the theorem. First of all, the index of d˛; .l; ˆl ; a/ is at least dim l. Thus for classification of equivariant metric Lie algebras of small index only Lie algebras l of small dimensions are needed. In general, it is a good strategy to look for certain subclasses of all equivariant metric Lie algebras such that the corresponding ingredients in Theorem 2.3 become manageable. 2 .l; a/b D ; for all semi-simple orSecondly, a great many Lie algebras l satisfy HQ 2 thogonal l-modules a (in view of Proposition 2.12 this also implies HQ .l; ˆ l ; a/b D ; for any equivariant structure ˆl on l). We call these Lie algebras non-admissible and the remaining ones, which really occur in Theorem 2.3, admissible. E.g., the twodimensional non-abelian Lie algebra and Heisenberg Lie algebras of dimension 5 are non-admissible ([35], Proposition 5.2). It is easy to see that all reductive Lie algebras are admissible and that the class of admissible Lie algebras is closed under forming direct sums. Solvable admissible Lie algebras l with dim l0 2 are classified in [32] and [35], Section 5. Up to abelian summands, there are only finitely many of them. This result (together with its proof) shows that an a priori classification of Lie algebras (within a certain class) is not really needed for applications of Theorem 2.3. Up to now it is unknown how large the class of admissible Lie algebras really is. Any new structure result for admissible Lie algebras would give highly desirable new insight into the world of (equivariant) metric Lie algebras. Contrary to L, the determination of Al;ˆ l for given .l; ˆ l / is usually no problem (since only semisimple modules are needed). Also the computation of the cohomology 2 sets HQ .l; ˆ l ; a/b succeeds in many interesting cases (techniques of homological algebra are helpful here). To achieve a true classification, one would have to determine, 2 as a last step, the orbit space of the action of the group Gl;ˆ l ;a on the set HQ .l; ˆ l ; a/b . This may lead to unsolved classification problems again, e.g., to the description of V the orbit space 3 Rn = GL.n; R/ for large n (compare [34], Section 5). But often 2 HQ .l; ˆ l ; a/b is so small such that things can be carried out explicitly. 2.5 Classification results for metric Lie algebras. For trivial .h; K/, i.e. for .h; K/ D .0; feg/ Theorem 2.3 specialises to a classification scheme for metric Lie algebras. As mentioned above for certain classes of metric Lie algebras this can be used to obtain a

18


full classification (in the sense of a list). In particular we can use it to classify metric Lie algebras of small index. Let us consider the Lorentzian case, i.e. the case where the metric Lie algebra has index 1. We already know examples of Lorentzian metric Lie algebras, namely the oscillator algebras osc./ defined in Section 2.1. For a classification of indecomposable Lorentzian metric Lie algebras we have to restrict to l with dim l 1 and to Euclidean l-modules a in our classification scheme. In this way we reproduce the following well-known result, which was originally proved using double extensions. Theorem 2.4 (Medina [43]). Each indecomposable non-simple metric Lie algebra of signature .1; q/, q > 0, is isomorphic to an oscillator algebra osc./ for exactly one D .1 ; : : : ; m / 2 .R /m Š Rm , q D 2m C 1, with 1 D 1 2 m . For the study of Lorentzian metric Lie algebras the method of double extensions and the method of quadratic extensions are in some sense equivalent, since every indecomposable non-simple Lorentzian metric g Lie algebra has exactly one isotropic ideal, namely its centre z.g/. However, as already mentioned, for classification of metric Lie algebras of higher index quadratic extensions are more useful than double extensions, see for example the classification of metric Lie algebras of index 2 in [34] and the classification of metric Lie algebras of index 3 in [35]. Both of them are based on Theorem 2.3. As a further example that shows that Theorem 2.3 is a useful mean for concrete classification problems let us consider nilpotent metric Lie algebras of small dimension. Favre and Santharoubane classified such Lie algebras up to dimension 7 in [23]. Their proof is based on the double extension method. In [32], Theorem 2.3 has been used to give a classification of nilpotent metric Lie algebras of dimension at most 10. Most of the isomorphism classes are isolated ones, however, in dimension 10 also 1-parameter families occur. Here we will restrict ourselves to nilpotent metric Lie algebras of dimension at most 9. In the following theorem a stands for a pseudo-Euclidean vector space that we consider always as a trivial l-module. As usual, let e1 ; : : : ; epCq be the standard basis of Rp;q . Furthermore, 1 ; : : : ; l 2 l will denote the dual basis of a given basis X1 ; : : : ; Xl of l and i1 :::ij WD i1 ^ ^ ij . Theorem 2.5 ([32]). If .g; h ; i/ is an indecomposable non-abelian nilpotent metric Lie algebra of dimension at most 9, then it is isomorphic to d˛; .l; a/ for exactly one of the data in the following list. 1. l D g4;1 D fŒX1 ; X2 D X3 ; ŒX1 ; X3 D X4 g, a 2 fR0;1 ; R1;0 g, ˛ D 14 ˝ e1 , 2 f0; 234 ; 134 ; 134 g; 2. l D h.1/ ˚ R D fŒX1 ; X2 D X3 g ˚ R X4 , a 2 fR0;1 ; R1;0 g, .˛; / D . 13 ˝ e1 ; 234 /; 3. l D R4 , a 2 fR0;1 ; R1;0 g, .˛; / D . 13 ˝ e1 ; 234 /; 4. l D h.1/ D fŒX1 ; X2 D X3 g,


19

(a) a 2 fR0;1 ; R1;0 g, .˛; / D . 13 ˝ e1 ; 0/, (b) a 2 fR0;2 ; R2;0 ; R1;1 g, .˛; / D . 13 ˝ e1 C 23 ˝ e2 ; 0/; 5. l D R3 , (a) a D 0, .˛; / D .0; 123 /, (b) a 2 fR0;2 ; R2;0 ; R1;1 g, .˛; / D . 12 ˝ e1 C 13 ˝ e2 ; 0/, (c) a 2 fR0;3 ; R2;1 ; R1;2 ; R3;0 g, .˛; / D . 12 ˝e1 C 13 ˝e2 C 23 ˝e3 ; 0/; 6. l D R2 , a 2 fR0;1 ; R1;0 g, .˛; / D . 12 ˝ e1 ; 0/.

3 Symmetric spaces 3.1 Symmetric triples and quadratic extensions. In this section we are concerned with Z2 -equivariant objects, i.e., .h; K/-equivariant modules, Lie algebras, metric Lie algebras etc., where K D Z2 is the group consisting of two elements and h D 0. Any equivariant structure ˆ, ˆl : : : is determined by its value on the nontrivial element of Z2 , which is an involution. We will denote this involution by , l : : : : We will keep the notation of Section 2 but with all ˆ’s replaced by the corresponding ’s. Any Z2 -module V has a decomposition V D VC ˚ V into the .˙1/-eigenspaces of V . As explained in the introduction, one associates with a pseudo-Riemannian symmetric space the Lie algebra of its transvection group together with a natural non-degenerate symmetric bilinear form and an isometric involution on it. This leads to the notion of a symmetric triple. Definition 3.1. (a) A Z2 -equivariant Lie algebra .g; / satisfying Œg ; g D gC is called proper. (b) A symmetric triple is a proper Z2 -equivariant metric Lie algebra .g; ; h ; i/. (c) The index (signature) of a symmetric triple .g; ; h ; i/ is the index (signature) of the symmetric bilinear form h ; ijg . The Lie algebra of the transvection group of a pseudo-Riemannian symmetric space M of index p carries the structure of a symmetric triple of the same index in a natural way. We call it the symmetric triple of M . The notions of isomorphy and decomposability carry over from general Z2 -equivariant metric Lie algebras to symmetric triples. Then we have Proposition 3.2 (see e.g. [16]). The assignment which sends each pseudo-Riemannian symmetric space to its symmetric triple induces a bijective map between isometry classes of simply connected symmetric spaces and isomorphism classes of symmetric triples. A symmetric space is indecomposable if and only if its symmetric triple is so. All (not necessarily simply connected) symmetric spaces corresponding to a given symmetric triple .g; ; h ; i/ can be easily classified. Let us shortly discuss this classification. For the facts used in this discussion we refer to [40] and [30], Chapter VII, §§ 8, 9.

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Let GQ be the connected and simply connected Lie group with Lie algebra g. Then Q The group GQ Q is connected. integrates to an involutive automorphism Q W GQ ! G. Q \ GQ Q . Then Z0 is discrete. Q be the center of G, Q and set Z0 WD Z.G/ Let Z.G/ Q Q Q containing Z0 and set G WD G=Z. Choose a discrete -stable subgroup Z Z.G/ Then Q induces an automorphism W G ! G. Its fixed point group G has at most finitely connected components. The connected component G0 satisfies G0 \ Z.G/ D feg. We choose a group GC such that G0 GC G and GC \ Z.G/ D feg and define M WD G=GC . Let x0 D eGC be the base point of the homogeneous space M . Then g Š Tx0 M , and h ; ijg defines a GC -invariant scalar product on Tx0 M , which extends uniquely to a G-invariant pseudo-Riemannian metric on M . Moreover, Q induces an involutive isometry x0 of M having x0 as an isolated fixed point. Conjugating x0 with elements of G we get involutive isometries x for all x 2 M . Thus M has the structure of a pseudo-Riemannian symmetric space. Moreover, G is the transvection group of M . If Z and GC run through all possible choices as above, then the resulting spaces M exhaust the isometry classes of pseudo-Riemannian spaces having a symmetric triple isomorphic to .g; ; h ; i/. Two such spaces are isometric if and only if the defining Q Q ; h ; i/ consisting data .Z; GC / are conjugated by the automorphism group Aut.G; Q of all automorphisms of G that respect the involution as well as the pseudo-Riemannian metric on GQ induced by h ; i. The simply connected symmetric space associated with .g; ; h ; i/ arises if we choose Z D Z0 , GC D G0 . Thus the classification of symmetric spaces is reduced to the classification of symmetric triples, i.e., proper Z2 -equivariant metric Lie algebras. The theory developed in Section 2 associates with every Z2 -equivariant metric Lie algebra .g; ; h ; i/ (without simple ideals) via its canonical quadratic extension a quadruple .l; l ; a; Œ˛; /, where • .l; l / is a Z2 -equivariant Lie algebra, • a is a semisimple orthogonal .l; l /-module, and 2 .l; l ; a/b is a balanced quadratic cohomology class. • Œ˛; 2 HQ Then .g; ; h ; i/ is isomorphic to the Z2 -equivariant metric Lie algebra d˛; .l; l ; a/. Because we want to apply the classification scheme Theorem 2.3 to symmetric triples we have to express the properness condition for .g; ; h ; i/ Š d˛; .l; l ; a/ in Definition 3.1 in terms of .l; l ; a; Œ˛; /. Proposition 3.3 ([36], Section 5). Let .l; l ; a; Œ˛; / be a quadruple as above. By semisimplicity a D al ˚ .l/a, and we have a corresponding decomposition ˛ D ˛0 C ˛1 . Then d˛; .l; l ; a/ is a symmetric triple if and only if .T1 / the Z2 -equivariant Lie algebra .l; l / is proper, and .T2 / alC D ˛0 .ker Œ ; l /: Definition 3.4. Let .l; l / be a proper Z2 -equivariant Lie algebra, and let a be a semisimple orthogonal .l; l /-module. A quadratic extension .g; i; i; p/ of .l; l / by a is called admissible if it is balanced and g is proper, i.e., a symmetric triple. A cohomol2 .l; l ; a/ is called admissible, if it is balanced and satisfies .T2 /. ogy class Œ˛; 2 HQ


21

2 We denote the set of all admissible quadratic cohomology classes by HQ .l; l ; a/] and 2 its subset of all indecomposable admissible classes by HQ .l; l ; a/0 .

By Proposition 3.3, admissible cohomology classes correspond to admissible quadratic extensions. Let .l; l / be a proper Z2 -equivariant Lie algebra. As in Section 2.4 we consider ss of semisimple orthogonal .l; l /-modules and morphisms of pairs the category Ml; l ss ..l; l /; a1 / ! ..l; l /; a2 /. The automorphism group Gl;l ;a of an object a of Ml; l 2 2 acts on HQ .l; l ; a/0 and on HQ .l; l ; a/] from the right. Combining Proposition 3.3 with Theorem 2.3 we arrive at the following classification scheme for symmetric triples. We prefer to formulate it for indecomposable symmetric triples. One gets the corresponding statement for general symmetric triples 2 2 if one replaces HQ .l; l ; a/0 by HQ .l; l ; a/] . Theorem 3.1 ([36], Section 6). Let Lp be a complete set of representatives of isomorphism classes of proper Z2 -equivariant Lie algebras. For each .l; l / 2 Lp we choose ss . a complete set of representatives Al;l of isomorphism classes of objects in Ml; l Then there is a bijective map from the set of isomorphism classes of non-semisimple indecomposable symmetric triples to the union of orbit spaces a a 2 HQ .l; l ; a/0 =Gl;l ;a : (5) .l;l /2Lp a2Al;l 2 The inverse of this map sends the orbit of Œ˛; 2 HQ .l; l ; a/0 to the isomorphism class of d˛; .l; l ; a/.

The theorem says in particular that the set 2 fd˛; .l; l ; a/ j .l; l / 2 Lp ; a 2 Al;l ; Œ˛; 2 HQ .l; l ; a/0 g

exhausts all isomorphism classes of non-semisimple indecomposable symmetric triples. As discussed at the end of Section 2.4 one does not need the whole set Lp . Only those .l; l / such that the Lie algebra l is admissible really occur. We will explain in Section 3.3 how Theorem 3.1 can be used in order to give a full classification of all symmetric triples of index at most 2. Before doing this we want to discuss the implications of the theory of quadratic extensions for the geometry of symmetric spaces. 3.2 The geometry of quadratic extensions. Let M be a pseudo-Riemannian symmetric space without semisimple local factors. As discussed in the last section, its (local) geometry is completely determined by its symmetric triple and, moreover, this symmetric triple carries the additional structure of an admissible quadratic extension. It is this structure that leads to the classification scheme Theorem 3.1. But what does this structure mean for the geometry of M ? This is the question we are going to discuss now. In particular, it will turn out that any pseudo-Riemannian symmetric space

22


without semisimple local factors M comes with a distinguished fibration q W M ! N over an affine symmetric space N such that all fibres are flat. Let us first recall the notion of an affine symmetric space. A connected manifold with connection .M; r/ is called an affine symmetric space if for each x 2 M there is an involutive affine transformation x of .M; r/ such that x is an isolated fixed point of x . Note that x , if it exists, is uniquely determined by r. Forgetting about the metric and only remembering the Levi-Civita connection we can consider any pseudoRiemannian symmetric space as an affine symmetric space. There are, however, many affine symmetric spaces that do not admit any symmetric pseudo-Riemannian metric. In exactly the same way as in the pseudo-Riemannian case one constructs the group of transvections of .M; r/, which acts transitively on M . Its Lie algebra comes with an involution but without scalar product. We will call this Lie algebra with involution the symmetric pair of M . Definition 3.5. A symmetric pair is a proper Z2 -equivariant Lie algebra .g; / satisfying z.g/ g . It follows from the effectivity of the action of the transvection group that the symmetric pair of M is indeed a symmetric pair in the sense of this definition. Moreover, if .g; ; h ; i/ is a symmetric triple, then .g; / is a symmetric pair. Indeed, if X 2 z.g/, then hX; ŒY; Zi D hŒX; Y ; Zi D 0 holds for all Y; Z 2 g , i.e., X 2 Œg ; g ? D g . We have the following analogue of Proposition 3.2. Proposition 3.6. The assignment which sends each affine symmetric space to its symmetric pair induces a bijective map between affine diffeomorphism classes of simply connected affine symmetric spaces and isomorphism classes of symmetric pairs. Also the description of all affine symmetric spaces corresponding to a given symmetric pair proceeds in the same way as in the pseudo-Riemannian case. In practice, affine symmetric spaces often arise as follows: Let W G ! G be an involutive automorphism of a connected Lie group, and let GC G be a closed -stable subgroup having the same identity component as the fixed point group G . Then induces an involutive diffeomorphism x0 of M WD G=GC , and there is a unique G-invariant connection r on M making x0 affine. Then .M; r/ is an affine symmetric space. Note that G might be different from the transvection group of M . In fact, the transvection group is a subgroup of a quotient of G. If q W M1 ! M2 is an affine map between two affine symmetric spaces, then qıx D q.x/ ıq for all x 2 M1 . If q is surjective, then it follows that the fibres q 1 .m/, m 2 M2 , are totally geodesic (possibly disconnected) affine symmetric submanifolds of M1 and that q induces a surjective homomorphism between the transvection groups of M1 and M2 . A submanifold Y of a pseudo-Riemannian manifold .M; g/ is called coisotropic, if Ty Y ?gy Ty Y for all y 2 Y . The following notion is a geometric counterpart of the notion of a quadratic extension (in the context of symmetric triples). Definition 3.7. Let N be an affine symmetric space. A special affine fibration over N is a surjective affine map q W M ! N , where M is a pseudo-Riemannian symmetric space and the fibres of q are flat, coisotropic, and connected.


23

Let N be an affine symmetric space with corresponding symmetric pair .l; l /. Its cotangent bundle T N carries the structure of a pseudo-Riemannian symmetric space such that its symmetric triple is d0;0 .l; l ; 0/. Then the bundle projection p W T N ! N is the simplest example of a special affine fibration over N . Recall that d0;0 .l; l ; 0/ is a quadratic extension of .l; l /. We now want to construct special affine fibrations that correspond to quite general quadratic extensions in the same way as p W T N ! N corresponds to d0;0 .l; l ; 0/. Any proper Z2 -equivariant Lie algebra .l; l / gives rise to a symmetric pair .l0 ; l0 /, where l0 WD l=.z.l/ \ lC / and l0 is induced by l . Proposition 3.8. Let .l; l / be a proper Z2 -equivariant Lie algebra. Let .g; ; h ; i/ be a symmetric triple equipped with the structure .g; i; i; p/ of a quadratic extension of .l; l / by some orthogonal .l; l /-module. Let M be a pseudo-Riemannian symmetric space with symmetric triple .g; ; h ; i/. We assume in addition that at least one of the following two conditions is satisfied: (a) M is simply connected. (b) z.g/ i? . Then there is an affine symmetric space N , unique up to isomorphism, with associated symmetric pair .l0 ; l0 / and a unique special affine fibration q W M ! N such that dQe D p0 :

(6)

Here Q is the homomorphism of transvection groups induced by q and p0 is the comp position of natural maps g ! g=i ! l ! l0 . The symmetric space N can be written as a homogeneous space N D L=LC , where L; LC are certain Lie groups with Lie algebras lC l. If N D L=LC , then the transvection group of N equals L0 D L=.Z.L/ \ LC /. Note that Z.L/ \ LC acts trivially on N . Thus the group L (and the Lie algebra l) that corresponds to the data of the quadratic extension arises as a central extension of the geometrically visible transvection group L0 (of the Lie algebra l0 ). The proof of the proposition will be given in the appendix. The idea behind it is very simple. Let G be the transvection group of M , and let J G be the analytic subgroup corresponding to the ideal i? g. Then J acts on M , and we would like to set q to be the projection onto the orbit space N D J nM . That the orbits are flat and coisotropic is a simple consequence of the properties of i? . The main problem is to show that the orbit space is a manifold (the orbits have to be closed, in particular). It is this point, where we need Condition (a) or (b). Without these conditions, it is not difficult to construct examples with non-closed J -orbits. For them the resulting geometric structure will be a foliation only, not a fibration. We are mainly interested in admissible, hence balanced quadratic extensions. They always satisfy Condition (b). Indeed, using Equation (1) we find that i D i.g/ R.g/ g0 . Forming orthogonal complements yields z.g/ i? . Thus admissible quadratic extensions give rise to special affine fibrations.

24


Corollary 3.9. Let M be a pseudo-Riemannian symmetric space without semisimple local factors. Let .g; ; h ; i/ be the corresponding symmetric triple. Then the canonical quadratic extension associated with .g; ; h ; i/ .in the sense of Corollary 2.8/ defines a special affine fibration q W M ! N over an affine symmetric space N . We call this fibration the canonical fibration of M . Its base N is an important invariant of the pseudo-Riemannian space M . Since there are a lot of non-admissible Lie algebras (see the end of Section 2.4) not all affine symmetric spaces N can appear as the base of the canonical fibration of some pseudo-Riemannian symmetric space M . Let q W M ! N be a special affine fibration. Let us collect some of its basic properties. 1. q W M ! N is locally trivial, i.e., it is a fibre bundle with flat affine symmetric fibres. More precisely, there exist k; l 2 N0 , an open covering fUi g of N , and diffeomorphisms î W q 1 .Ui / ! Ui Rk .S 1 /l such that for all x 2 Ui the restriction of î to q 1 .x/ is an affine diffeomorphism from the fibre onto fxg Rk .S 1 /l . If l D 0 we call the fibration q very nice. Note that a very nice fibration is not a vector bundle, in general. There is no distinguished zero section. 2. M is simply connected if and only if N is simply connected and q is very nice. 3. The fibres of q are foliated by the null spaces of the restriction of the metric to the fibre. The leaves are totally geodesic subspaces of dimension n D dim N . If the leaves are closed we call q nice. Of course, very nice implies nice. 4. We look at the cotangent bundle T N as bundle of abelian groups. There is a natural action of the bundle T N on the bundle q W M ! N by translations on the fibres. Its orbits are precisely the null leaves described in 3. 5. If q is nice, then the space of null leaves (D the orbit space of the T N -action) is itself an affine symmetric space which is fibred over N . We obtain a factorisation q D s ı r, where r W M ! P and s W P ! N are affine maps which are fibre bundles with flat affine symmetric fibres. Moreover, the fibres of s come with a non-degenerate metric. They are quotients of pseudo-Euclidean spaces by a (often trivial) discrete group of translations. One should observe the analogy to the properties of quadratic extensions. We summarize the structure of a nice special affine fibration by the following diagram.

/o /o /o / M T N AA 33 AA r 33 AA 33 AA 33 q P p 33 33 }} } 33 } 3 ~}}}} s N

Using Lemma 6.3 it can be shown that the canonical fibration of every indecomposable symmetric space M is nice (in fact, the absence of flat local factors that are not global


25

factors is sufficient). Thus any such space M carries a canonical structure of the kind indicated by the diagram. The data .l; l ; a; Œ˛; / appearing in the classification scheme Theorem 3.1 should be regarded as a complete set of (local) invariants describing this structure. It would be an interesting project to work out the precise geometric meaning of each of these invariants. We conclude this section with a certain converse of Proposition 3.8 saying that all special affine fibrations come from quadratic extensions. Let us denote the special affine fibration constructed in Proposition 3.8 by q.M; i; i; p/. Proposition 3.10. Let q W M ! N be a special affine fibration. Then there exists a structure of a quadratic extension .g; i; i; p/ on the symmetric triple .g; ; h ; i/ of M such that q D q.M; i; i; p/. We remark that the quadratic extension .g; i; i; p/ is not completely determined by q. What is uniquely determined is i g . Indeed, if we identify g D Tx0 M , then .i /? g has to be the tangent space to the fibre of q. We then have to choose iC gC subject to the conditions (a) i D iC ˚ i g is an isotropic ideal, (b) i? =i is abelian. The proposition says that such a choice is always possible. Indeed, one can show that iC WD fX 2 Œg ; .i /? j ŒX; .i /? D 0g: always satisfies (a) and (b). Nevertheless, iC is not uniquely determined by i , (a), and (b), in general. 3.3 Symmetric spaces of index one and two. In this section we will comment on some classification results for symmetric triples of small index. First we want to reformulate the classification of indecomposable non-semisimple Lorentzian symmetric triples by Cahen and Wallach [17] in terms of quadratic extensions. Indeed, this result follows easily from Theorem 3.1. To see this we just have to check which elements of (5) correspond to Lorentzian symmetric triples. Clearly, d˛; .l; l ; a/ is Lorentzian if and only if aCdim l D 1, where a is the index of h ; ia restricted to a . Hence, either l D 0 and a D a D R1;0 or .l; l / D .R; id/ and h ; ia restricted to a is positive definite. The first case is trivial. Let us consider the second one, i.e. suppose .l; l / D .R; id/. Take a D Rp;p ˚ R0;2q , p; q 0. Let e1 ; : : : ; e2p be the standard basis of Rp;p 0 and let e10 ; : : : ; e2q be the standard basis of R0;2q . We define an involution a on a 0 0 ; : : : ; e2q g. by aC D spanfe1 ; : : : ; ep ; e10 ; : : : ; eq0 g and a D spanfepC1 ; : : : ; e2p ; eqC1 1 p p p 1 q q q For D . ; : : : ; / 2 .l / Š R and D . ; : : : ; / 2 .l / Š R we define a representation ; of l on a by ; .L/.ei / D i .L/eiCp ;

; .L/.eiCp / D i .L/ei ;

(7)

; .L/.ej0 / D j .L/ej0 Cq ;

; .L/.ej0 Cq / D j .L/ej0

(8)

26


for L 2 l, i D 1; : : : ; p and j D 1; : : : ; q. Then a; WD .; ; a/ is an orthogonal .l; l /-module and we can define d.p; q; ; / WD d0;0 .l; l ; a; /: It is not hard to prove that every semisimple orthogonal .R; id/-module for which the 1-eigenspace of the involution is positive definite is of the kind defined above. 2 .l; l ; a/ D f.0; 0/g and indecomposability Since l is one-dimensional we have HQ and admissibility conditions are easy to handle. Now Theorem 3.1 gives: Theorem 3.2 (Cahen–Wallach [17]). Every indecomposable non-semisimple Lorentzian symmetric triple is either one-dimensional or isomorphic to exactly one of the symmetric triples d.p; q; ; /, p; q 0, p C q > 0, .; / 2 Mp;q , where ˇ 1 p 1 q p q ˇ 0 < ; 0 < ; Mp;q WD .; / 2 R ˚ R ˇ 1 : D 1 if p > 0; 1 D 1 else Cahen and Wallach [17] constructed explicit models for all simply connected Lorentzian symmetric spaces. Let us describe the simply connected Lorentzian symmetric space M associated with the symmetric triple d WD d.p; q; ; /. Since d is isomorphic to the semidirect product of a Heisenberg algebra by R it is not hard to see that the simply connected group G with Lie algebra d is isomorphic to d D l ˚ a ˚ l with group multiplication x A; N L/ x D .Z C ZN C 1 e adLN .A/; AN ; e adLN .A/ C A; N L C L/: N .Z; A; L/.Z; 2

Here Œ ; and ad are the operations in d. The analytic subgroup GC of G with Lie algebra dC then equals aC . The projection G ! G=GC has the global section s W G=GC ! G;

.Z; AC C A ; L/ GC 7! .Z C 12 ŒAC ; A ; A ; L/;

where AC 2 aC and A 2 a . In particular, we can identify G=GC with d (as a set). Let .z; a1 ; : : : ; ap ; a10 ; : : : ; aq0 ; l/ denote the coordinates of a vector in d Š l ˚ a ˚ l Š R ˚ RpCq ˚ R. In these coordinates, the metric on G=GC determined by the symmetric triple d is given by 2dzd l C

p X iD1

dai2 C

q X j D1

2

daj0 C

p X iD1

2i ai2

q X

j2 aj02 d l 2 :

j D1

Now let us turn to symmetric triples of index two. First classification results for this case were already obtained by Cahen and Parker in [15] and [16]. In [15] symmetric spaces of index two with solvable transvection group were studied. However, the classification presented there turned out to be incomplete. In his diploma thesis [46] Th. Neukirchner elaborated the claimed results, found the gaps and gave a revised classification of indecomposable symmetric spaces of index two that have a solvable transvection group. We tried to reproduce this result using our classification scheme


27

and observed that also Neukirchner’s classification is not quite correct. Besides minor errors a series of spaces is missing and some of the normal forms contain too much parameters. In [36] we give the corrected classification result. We do not want to recall this result to its full extent here. However, let us give a rough classification of indecomposable non-semisimple symmetric triples of signature .2; n/, which shows how our classification scheme works. Following Theorem 3.1 we have first to determine all proper Z2 -equivariant Lie algebras .l; l / for which dim l 2 holds. For l besides R; R2 ; su.2/, sl.2; R/ and h.1/ the following Lie algebras appear: n.2/ D fŒX; Y D Z; ŒX; Z D Y g;

r3;1 D fŒX; Y D Z; ŒX; Z D Y g:

The involutions l are described in Table 1. The table also lists the associated simply connected affine symmetric spaces N D L=LC for all .l; l / (see Section 3.2 for the geometric meaning of N ). The Lie algebra n.2/ is isomorphic to so.2/ËR2 , .n.2/; l / is a symmetric pair and the associated simply-connected affine symmetric space equals the universal covering L.2/ of the space of (affine) lines in R2 . Analogously, r3;1 is isomorphic to so.1; 1/ Ë R1;1 , .r3;1 ; l / is also a symmetric pair and the associated simply-connected affine symmetric space equals the space L.1; 1/ of time-like (affine) lines in R1;1 . Note that we have two non-conjugate involutions on sl.2; R/. The two associated simply-connected symmetric spaces are the hyperbolic plane H 2 and the universal covering S 1;1 of the unit sphere S 1;1 WD fx 2 R1;2 j hx; xi1;2 D 1g in R1;2 . For all these .l; l / we have to determine a set Al; l as described in Theorem 3.1. More exactly, since we want to classify only indecomposable symmetric spaces of signature .2; n/ we may restrict ourselves to the subset Anl; l Al; l of all a 2 Al; l for 2 which HQ .l; l ; a/0 is not empty and for which .2; n/ D sign a C .dim l ; dim l /. For all .l; l / listed above Anl; l consists of finitely many families .1 ; a1 /; : : : ; .k ; ak / of .l; l /-representations, where for each family .j ; aj / the space aj is fixed and j depends in a certain sense on dj continuous parameters for some dj 2 N0 . We list the spaces a1 ; : : : ; ak explicitly in the table, where we use the following notation: p;q p;q ap;q ; id/, ap;q ; id/. For the families 1 ; : : : ; k we omit a detailed WD .R C WD .R description, we only give the number dj of parameters for each j . Next we have to 2 .l; l ; aj /=Gl; l ;aj for each element of the family .j ; aj /, j D 1; : : : ; k. compute HQ We do not give the result in the table, either. However, we give the number dH of 2 .l; l ; aj /=Gl; l ;aj depends for a generic element continuous parameters on which HQ in the family .j ; aj /. Here d D 0 or dH D 0 means that the corresponding set is discrete (in fact, it is finite). Note that the values of d and dH for N D R2 in the table are not correct for small n, namely for those n for which the value of d given in the table would be negative. We do not consider these special cases in the table. In order to not present only vague data here let us study one case in more detail. We consider the case, where l is the Heisenberg algebra h.1/ D fŒX; Y D Zg. In Exam2 ple 2.14 we computed HQ .l; a/b for l D h.1/ and any semisimple orthogonal h.1/2 modul a. We identified HQ .l; a/b with a certain subset Zl;b of C 2 .l; a/. Now let a be a semisimple orthogonal .l; l /-module for l given by lC D RZ; l D spanfX; Y g.

e

e

28


Table 1. Non-semisimple indecomposable symmetric triples of signature .2; n/ l; An l; l

N R1

R2

l D l D R 1. .1;p ; ap;np ˚ a1;n1 /; p D 0; : : : ; n 1 C p;np 2. .2;p ; aC ˚ a1;n1 /; p D 1; : : : ; n p;np 1;n1 3. .3;p ; aC ˚ a /; p D 1; : : : ; n 1

d

dH

n1 n1 n1

0 0 0

l D l D R2 1.

.1;p ; ap;n2p ˚ a0;n2 /; p D 0; : : : ; n 2 C (occurs only for n 5)

2n 8

0

2.

.2;p ; ap;n1p ˚ a0;n2 /; p D 0; : : : ; n 2 C

2n 8

1

2n 8

1

2n 10

2

2n 12

3

.1;p ; ap;n2p ˚ a0;n2 /; p D 0; : : : ; n 2 C

n2

0

p D 0; : : : ; n 3

n3

0

p D 0; : : : ; n 3

n3

0

p D 0; : : : ; n 4

n4

1

.1;p ; ap;n2p ˚ a0;n2 /; p D 0; : : : ; n 2 C

n2

0

p D 0; : : : ; n 3

n3

0

p D 0; : : : ; n 3

n3

0

p D 0; : : : ; n 4

n4

1

0

1

0

1

0

1

3.

.3;p ; ap;n1p C

˚

a0;n2 /;

p D 1; : : : ; n 1

l D h.1/ D fŒX; Y D Zg; lC D R Z; l D spanfX; Y g 1.

e

L.2/

2.

2. 3. 4.

2. 3.

e H

2

˚

a0;n2 /;

p D 0; : : : ; n 4

.2;p ; ap;n3p C .3;p ; ap;n2p C .4;p ; ap;n3p C

˚ a0;n2 /; 0;n2 ˚ a /; ˚ a0;n2 /;

l D r3;1 ; lC D R Z; l D spanfX; Y g 1.

S 1;1

.2;p ; ap;n4p C

l D n.2/; lC D R Z; l D spanfX; Y g 1.

L.1; 1/

.1;p ; ap;n3p ˚ a0;n2 /; p D 0; : : : ; n 3 C

4.

.2;p ; ap;n3p C .3;p ; ap;n2p C .4;p ; ap;n3p C

˚ a0;n2 /; 0;n2 ˚ a /; ˚ a0;n2 /;

l D sl.2; R/ D fŒH; X D 2Y; ŒH; Y D 2X; ŒX; Y D 2H g; lC D R H; l D spanfX; Y g .; an2;n2 ˚ a0;n2 / C l D sl.2; R/; lC D R X; l D spanfH; Y g pqr n2Cpq;0 .klm ; aC ˚ a0;n2 /; k 2 N p ; l 2 N q ; m 2 N r ; k1 k2 kp ; l1 : : : ; lq ; m1 mr jkj C jlj C 2jmj C q D n 2

S2

l D su.2/ D fŒH; X D 2Y; ŒH; Y D 2X; ŒX; Y D 2H g; lC D R H; l D spanfX; Y g pqr 0;n2Cpq .klm ; aC ˚ a0;n2 /; p; q; r; k; l; m as in the previous entry


29

Then Zl;b is invariant under the morphism of pairs . l ; a /. Using Proposition 2.12 we 2 .l; l ; a/b corresponds bijectively to f˛ 2 Zl;b j ˛.Z; l/ al g. Furthersee that HQ 2 more, since l is indecomposable, HQ .l; l ; a/0 corresponds bijectively to f˛ 2 Zl;b j l ˛.Z; l/ D a g. Summarising we see that 2 Zl;0 WD f˛ 2 C 2 .l; a/ j ˛.X; Y / D 0; ˛.Z; l/ D al 6D 0g ! HQ .l; l ; a/0

˛ 7! Œ˛; 0

(9)

is a bijection. Let us now determine a suitable set Anl; l . Let .; a/ be an orthogo2 nal .l; l /-module such that HQ .l; l ; a/0 6D ;. In particular, is semisimple, which implies .Z/ D 0 since R.l/ D RZ. Hence, can be considered as a semisimple representation of the abelian Lie algebra l , which is determined by its weights. Moreover, l 0;2 1 p p we know from (9) that al D a0;1 or a D a . For D . ; : : : ; / 2 .l n 0/ and p;q 1 q q 0;pCq

D . ; : : : ; / 2 .l n 0/ we define a representation ; of l on aC ˚ a 0;pCq with Rp;p ˚ R0;2q . Now we define by (7) and (8), where we identify ap;q C ˚ a 0;pCq ˚ a0;1 a1; WD .; ˚ 0 ; ap;q /; C ˚ a

0;pCq a2; WD .; ˚ 0 ; ap;q ˚ a0;2 /; C ˚ a

where 0 denotes the trivial representation of dimension one and two, respectively. It is easy to see that .; a/ is equivalent to one of these representations. Next we ss have to decide which of these representations are isomorphic as objects of Ml; . The l automorphism group of .l; l / equals ˇ A 0 ˇ Aut.l; l / D ˇ A 2 GL.2; R/; det A D u ; 0

u

where the automorphisms are written with respect to the basis X; Y; Z. Let Sp denote the symmetric group of degree p. We define an action of .Sp Ë .Z2 /p / GL.2; R/ on .l n 0/p by .1 ; : : : ; p / " A D ."1 A .1/ ; : : : ; "p A .p/ / for .1 ; : : : ; p / 2 .l n 0/p , 2 Sp , ."1 ; : : : ; "p / 2 .Z2 /p and A 2 GL.2; R/. We define the sets ƒp WD .l n 0/p =.Sp Ë .Z2 /p /;

ƒp;q D ƒp ƒq :

ss by Š we obtain a1; 6Š a2;0 0 and Denoting the isomorphy relation in Ml; l

ai; Š ai;0 0 , p D p 0 ; q D q 0 and Œ; D Œ0 ; 0 2 ƒp;q = GL.2; R/ for i D 1; 2. Hence, the elements of the following 2n 5 families of .l; l /-modules constitute a suitable set Anl; l : .1;p ; ap;n3p ˚ a0;n2 / WD fa1; j Œ; 2 ƒp;n3p = GL.2; R/g; C Q pQ .2;pQ ; ap;n4 ˚ a0;n2 / WD fa2; j Œ; 2 ƒp;n4 Q pQ = GL.2; R/g; C

where p D 0; : : : ; n 3, pQ D 0; : : : ; n 4. In particular, for each family 1;p , p D 0; : : : ; n 3, we have d D 2n 10 if n 5 and d D 0 for n D 3; 4. Similarly,

30


for each 2;pQ , pQ D 0; : : : ; n 4, we have d D 2n 12 if n 6 and d D 0 for n D 4; 5. 2 .l; l ; a/=Gl;l ;a for a 2 Anl; l . Take a D a1; . If Finally, let us determine HQ n 5, then in the generic case ..X /; .X // 2 Rn3 and ..Y /; .Y // 2 Rn3 are 2 linearly independent and we get HQ .l; l ; a/=Gl;l ;a Š Zl;0 =Z2 . Hence, dH D 2. In the non-generic case Gl;l ;a becomes larger and we get dH D 0. For n D 3; 4 we have dH D 0. Now take a D a2; . If n 6, then in the generic case ..X/; .X// 2 Rn4 and ..Y /; .Y // 2 Rn4 are linearly independent and we 2 have HQ .l; l ; a/=Gl;l ;a Š Zl;0 =O.2/. Hence, dH D 3. In the non-generic case we get dH D 1. For n D 5 we have dH D 1 and for n D 4 we get dH D 0.

4 Special geometric structures on symmetric spaces 4.1 Examples of geometric structures. We are now going to discuss pseudo-Riemannian symmetric spaces that are equipped with certain geometric structures coming from complex and quaternionic geometry. Let .M; g/ be a pseudo-Riemannian manifold. The Levi-Civita connection induces a connection on the bundle so.TM / End.TM / of endomorphisms that are skewsymmetric w.r.t. g. A Kähler structure on .M; g/ is a parallel section I of so.TM / satisfying I 2 D idTM . In particular, I is an integrable almost complex structure, and thus .M; I / is a complex manifold. A pseudo-Hermitian symmetric space is a tuple .M; g; I /, where .M; g/ is a pseudo-Riemannian symmetric space and I is a Kähler structure on .M; g/. A hyper-Kähler structure on .M; g/ is a pair of Kähler structures .I; J / satisfying IJ D JI . A quaternionic Kähler structure on M arises if we weaken the parallelity conditions on I , J : it consists of a three-dimensional parallel subbundle E so.TM / that can be locally spanned by almost complex structures I , J , and K WD IJ D JI . We have the corresponding notions of a hyper-Kähler symmetric space .M; g; I; J / and a quaternionic Kähler symmetric space .M; g; E/. In particular, hyper-Kähler symmetric spaces form a subclass of all quaternionic Kähler symmetric spaces. In the pseudo-Riemannian world all these structures have their “para”-versions. If we replace the condition I 2 D idTM for a Kähler structure by I 2 D idTM we are lead to the notion of a para-Kähler structure. A para-Kähler structure on .M; g/ is equivalent to a parallel splitting TM D TMC ˚TM into totally isotropic subbundles. Thus paraKähler structures can exist for metrics of neutral signature .m; m/, only. A pair .I; J /, where I is a Kähler structure and J is a para-Kähler structure such that IJ D JI , is called a hypersymplectic structure (sometimes also para-hyper-Kähler structure). Note that then K WD IJ D JI is a second para-Kähler structure on M . A parallel subbundle E so.TM / locally spanned by (not necessarily parallel) sections I; J; K of this kind is called a para-quaternionic Kähler structure. There are the corresponding notions of para-Hermitian, hypersymplectic, and para-quaternionic Kähler symmetric spaces.


31

Recall the notion of an .h; K/-module .V; ˆV / from Section 2.2. As usual, .V; ˆV / is called irreducible if it has no proper submodules. Let .h; K/ be the set of equivalence classes of irreducible .h; K/-modules. Let .V; ˆV / be arbitrary and fix 2 .h; K/. The -isotypic component V ./ V is the sum of all irreducible submodules of V belonging to the equivalence class . If .V; ˆV / is semisimple, then M V D V ./:

1

1

b 1K/-graded vector space. In particular, we can consider V as a .h; 1K/. An .h; K/-module .V; ˆ / is called …-graded, if it Definition 4.1. Let … .h; 2.h;K/

is semisimple and V ./ D 0 for all 62 ….

V

Looking at the underlying .h; K/-module structure we can speak of …-graded .h; K/-equivariant (metric) Lie algebras, …-graded .l; ˆl /-modules, etc. In order to save words we call a …-graded .h; K/-equivariant (metric) Lie algebra .l; ˆl / simply a (metric) Lie algebra with …-grading. Thus, the term …-grading stands for the whole equivariant structure ˆl , not only for the decomposition into isotypic components. Let us explain the examples of …-gradings relevant for the geometric structures discussed above. In all these cases we can take h D 0. Another interesting …-grading for .h; K/ D .R; Z2 / will appear in Section 5.1. • K D U.1/, … D f1; g. Here 1 stands for the one-dimensional trivial representation, and for the standard representation of U.1/ on C Š R2 . A …-grading of this kind is called complex grading. • K D R , … D f1; ; g. Here stands for the standard representation of the multiplicative group R on R1 , and denotes the dual of . The corresponding …-grading is called para-complex grading. • K D Sp.1/, … D f1; g. Here stands for the standard representation of Sp.1/ on H Š R4 . The corresponding …-grading is called quaternionic grading. • K D SL.2; R/, … D f1; g. Here stands for the two-dimensional standard representation of SL.2; R/. The corresponding …-grading is called paraquaternionic grading. In all cases above we have a natural embedding of Z2 Š O.1/ ,! K such that .w/ D id for the non-trivial element w 2 Z2 . Therefore objects V with such a …-grading are special Z2 -equivariant objects. Thus they come with a splitting V D VC ˚ V . We call such a …-grading of a Lie algebra g proper if Œg ; g D gC . In particular, metric Lie algebras with a proper …-grading of this kind are symmetric triples, which are equipped with an additional structure. Definition 4.2. A pseudo-Hermitian (para-Hermitian, hyper-Kähler, hypersymplectic) symmetric triple .g; ˆ; h ; i/ is a metric Lie algebra .g; h ; i/ with proper complex (para-complex, quaternionic, para-quaternionic) grading ˆ.

32


Then we have the following variant of Proposition 3.2. Proposition 4.3. There is a bijective map between isometry classes of simply connected “geostruc” symmetric spaces and isomorphism classes of “geostruc” symmetric triples, where “geostruc” stands for pseudo-Hermitian, para-Hermitian, hyper-Kähler, or hypersymplectic. Let us explain the correspondence for hyper-Kähler symmetric spaces. The other cases are similar. Let .M; g; I; J / be a hyper-Kähler symmetric space with base point x0 , and let .g; ; h ; i/ be the associated symmetric triple. Then I; J; K WD IJ span a Lie algebra k Š sp.1/ which acts orthogonally on g Š Tx0 M . This action commutes with the one of gC . We extend the k-action to g by the trivial action on gC . For X; Y 2 g , Z 2 gC , and Q 2 k we compute hŒQX; Y C ŒX; QY ; Zi D hQX; ŒY; Zi C hX; ŒQY; Zi D hQX; ŒY; Zi C hX; QŒY; Zi D 0: It follows that k acts by derivations on g. Integrating the resulting homomorphism from sp.1/ into antisymmetric derivations of g we obtain the desired homomorphism ˆ W Sp.1/ ! Aut.g/. Vice versa, if .g; ˆ; h ; i/ is a hyper-Kähler symmetric triple and .M; g/ is the simply connected symmetric space with symmetric triple .g; ˆ.1/;h ; i/, then I D ˆ.i/jg and J D ˆ.j /jg are gC -invariant anticommuting complex structures on g Š Tx0 M respecting the metric. They induce a G-invariant hyper-Kähler structure on M . It is now easy to specify the classification scheme Theorem 2.3 for .h; K/-equivariant metric Lie algebras to pseudo-Hermitian (para-Hermitian, …) symmetric triples (compare also Theorem 3.1). We do not want to write down the complete results. We only remark that any indecomposable non-semisimple “geostruc” symmetric triple is isomorphic to some d˛; .l; ˆl ; a/, where (O1) .l; ˆl / is a Lie algebra with proper …-grading, … chosen according to the geometric structure, (O2) a is a …-graded orthogonal .l; ˆl /-module, and 2 .l; ˆl ; a/b is indecomposable and satisfies Condition (T2 ) in Propo(O3) Œ˛; 2 HQ sition 3.3. 4.2 Pseudo-Hermitian symmetric spaces. We have seen in the previous section that in order to classify indecomposable pseudo-Hermitian symmetric spaces one would have to classify the objects (O1)–(O3) for ˆl being a complex grading. This task is of a similar complexity as the classification of all symmetric spaces discussed in Section 3. However, there are structural restrictions coming from U.1/-equivariance and making some aspects of the theory simpler then for general symmetric spaces. In addition, we will see that only very few symmetric spaces of index 2 (as listed in Section 3.3, Table 1) admit a Kähler structure.


33

We treat pseudo-Hermitian symmetric spaces together with para-Hermitian ones because of the similarity of their behaviour. Indeed, Kähler and para-Kähler structures could be viewed as different real forms of only one complexified structure as the common complexification of U.1/ and R is C . Proposition 4.4. Let .l; ˆl / be a Lie algebra with proper complex or para-complex grading. Then the radical r l is nilpotent and acts trivially on every semisimple .l; ˆl /-module. In particular, every solvable pseudo-Hermitian or para-Hermitian symmetric triple is nilpotent. Proof. Differentiating ˆl we can consider .l; ˆl / as a k-equivariant Lie algebra, where k is the Lie algebra of U.1/ or R , respectively. Properness of .l; ˆl / implies that lk l0 . Now we apply Lemma 6.1. Proposition 4.4 has also the following consequence: For fixed .l; ˆl /, the set of isomorphism classes of (para-)complex graded semisimple orthogonal .l; ˆl /-modules is discrete. Indeed, such an orthogonal .l; ˆl /-module is essentially determined by the action of the (semisimple) Levi factor of l on it. The smallest possible nonzero index of a pseudo-Hermitian symmetric triple is two. For this case the objects (O1)–(O3) can be classified completely. Thanks to Proposition 4.4 this is considerably simpler than to classify all symmetric triples of index 2 (compare Section 3.3). Theorem 4.1 (compare [36], Section 7.3). If .g; ˆ; h ; i/ is an indecomposable pseudoHermitian symmetric triple of signature .2; 2q/ that is neither semisimple nor abelian, then .g; ˆ; h ; i/ is isomorphic to d˛; .l; ˆl ; a/ for exactly one of the data in the following list: 1. q D 1 W l D R2 Š C, ˆl D is the standard action of U.1/, (a) a D R with the standard scalar product, ; â trivial, ˛.z1 ; z2 / D Im zN1 z2 , z1 ; z2 2 C, D 0; (b) all data as in (a) except for the opposite sign of the scalar product; 2. q D 2 W l D h.1/, ˆl D ˚ 1 via l Š C ˚ z.l/ as a vector space, a D C with the real standard scalar product, trivial, â D , ˛.z; x/ D z x, ˛.z1 ; z2 / D 0, z; z1 ; z2 2 C, x 2 z.l/ Š R, D 0; 3. q D 1 C p; p 0 W l D su.2/; ˆl D Ad ıi1 , where i1 W U.1/ ! U.2/ is the standard embedding into the left upper corner, a D .a1 /pr ˚ .a2 /r , 0 r p, where a1 D C 2 with positive definite standard scalar product, 1 is the standard representation of su.2/, â1 D i1 , and a2 D su.2/ with B, B the Killing form, 2 D ad, â2 D Ad ı i1 ; ˛ D 0, .X1 ; X2 ; X3 / D cB.ŒX1 ; X2 ; X3 /; c 2 R.

34


p 4. q D 1 C p; p 0 W l D sl.2; R/; ˆl .z/ D Ad.i2 . z //, z 2 U.1/, where i2 W U.1/ Š SO.2/ ! SL.2; R/ is the natural embedding, a D .a1 /pr ˚ .a2 /r , 0 r p, where a1 D R2 ˝ C with the scalar product given by !R2 ˝!C , !R2 ; !C being the standard symplectic forms of the factors, 1 is p the complexified standard representation of sl.2; R/, â1 .z/ D .i2 ˝ /. z/, and

p a2 D sl.2; R/ with the Killing form B, 2 D ad, â2 .z/ D Ad.i2 . z//; ˛ D 0, .X1 ; X2 ; X3 / D cB.ŒX1 ; X2 ; X3 /; c 2 R.

The bases N D L=LC of the canonical fibrations of the simply connected pseudoHermitian symmetric spaces corresponding to 1.–4. are C; C; S 2 Š CP 1 , and H 2 , respectively. The complex gradings of l correspond to the natural complex structures of these spaces. Note that the data defined in 4. do not depend on the choice of the square root of z. The classification of complex graded semisimple orthogonal .l; ˆl /-modules in cases 3. and 4. can be found already in [16], Chapter V, Proposition 3.3. We remark that the statement of Theorem 4.1 differs slightly from the corresponding statement in [36]. In [36] we determined all symmetric triples of signature .2; 2q/ that admit the structure of a pseudo-Hermitian symmetric triple, whereas here we have determined isomorphism classes of pseudo-Hermitian symmetric triples. Comparing both results we find that all of these symmetric triples admit exactly one complex grading (up to isomorphism). In a similar manner one can classify para-Hermitian symmetric triples of small index. All para-Hermitian symmetric spaces of signature .1; 1/ are locally isomorphic to either the flat space R1;1 or to the one-sheeted hyperboloid S 1;1 R1;2 , which is semisimple. The complexifications of the pseudo-Hermitian symmetric triples of signature .2; 2q/ listed in Theorem 4.1 admit real forms that are para-Hermitian symmetric triples of signature .q C 1; q C 1/. Concerning index 2, we obtain: Proposition 4.5. There are exactly two isolated isomorphism classes and one 1-parameter family of isomorphism classes of indecomposable para-Hermitian symmetric triples of index at most 2 that are neither semisimple nor abelian, namely d˛; .l; ˆl ; a/ for: 1. l D R2 Š R1 ˚ .R1 / , ˆl D ˚ , a D R with the standard scalar product, ; â trivial, ˛ induced by the dual pairing, D 0; 2. all data as in 1. except for the opposite sign of the scalar product of a; 3. l D sl.2; R/; ˆl D Ad ı i3 , where i3 W R ! GL.2; R/ is the standard embedding into the left upper corner, a D f0g, ˛ D 0, .X1 ; X2 ; X3 / D cB.ŒX1 ; X2 ; X3 /; c 2 R, B being the Killing form. Note that the family in 3. corresponds to a 1-parameter family of para-Hermitian metrics on the symmetric space T S 1;1 . As in the cases of metric Lie algebras and general symmetric triples there seems to be no hope for a complete classification of pseudo-Hermitian and para-Hermitian


35

triples without index restrictions. The reason is that the Lie algebra structure of nilpotent pseudo-Hermitian and para-Hermitian symmetric triples can be arbitrarily complicated. E.g., a basic invariant of a nilpotent Lie algebra g is its nilindex, which is by definition the smallest non-negative integer k such that gkC1 D f0g. Nilpotent Lie algebras of nilindex k are sometimes also called k-step nilpotent Lie algebras. The following series of examples shows that there are nilpotent pseudo-Hermitian symmetric triples with an arbitrary large nilindex. This is in sharp contrast to the theory of hyper-Kähler symmetric triples discussed in the next section, see Theorem 4.4. Example 4.6. For each m 2 N0 we define a pseudo-Hermitian symmetric triple .g.m/; ˆ; h ; i/ as follows: As a vector space with complex grading we set g.m/ D C mC1 ˚ Rm ;

ˆ D mC1 ˚ 1m :

Let Ei , i D 1; : : : ; m C 1, Zk , k D 1; : : : ; m, be the standard basis vectors of C mC1 and Rm , respectively. Set Fj WD iEj . Then fEi ; Fj ; Zk g is a basis of the real vector space g.m/. The nonzero Lie brackets between the basis vectors are defined as ŒEi ; Fj D ZiCj 1 ;

ŒZk ; Ei D FiCk ;

ŒZk ; Fj D EkCj :

Here Zl D Eq D Fq D 0 for l > m, q > m C 1. Finally, the scalar product is given by C mC1 ? Rm and hEi ; Fj i D 0;

hEi ; Ej i D hFi ; Fj i D ıiCj;mC2 ;

hZk ; Zl i D ıkCl;mC1 :

It is easy to check that .g.m/; ˆ; h ; i/ is indeed a pseudo-Hermitian symmetric triple. It is indecomposable and nilpotent of nilindex 2m C 1. Note that g.1/ and g.2/ appear in Theorem 4.1 under 1.(a) and 2., respectively. Observe that the holonomy algebras g.m/C D Rm are abelian. However, there exist nilpotent pseudo-Hermitian symmetric triples having even holonomy algebras of arbitrary large nilindex. 4.3 Quaternionic Kähler and hyper-Kähler symmetric spaces. Let .M; g; E/ be a pseudo-Riemannian manifold of dimension 4n > 4 with quaternionic or paraquaternionic Kähler structure E. Then .M; g/ is Einstein. We have to distinguish between two cases: If the scalar curvature of .M; g/ is non-zero, then .M; g/ is indecomposable and E has no nontrivial parallel section. Otherwise E can be spanned by parallel sections, i.e., .M; g/ carries a hyper-Kähler or hypersymplectic structure, respectively. For these facts we refer to [3] and [11]. The latter reference deals only with the Riemannian case, but the arguments work in the indefinite case as well. For symmetric spaces we have: Proposition 4.7 (Alekseevsky–Cortés [3]). Let .M; g; E/ be a .para-/quaternionic Kähler symmetric space of non-zero scalar curvature. Then the transvection group G of .M; g/ is simple. Indeed, the Ricci curvature of .M; g/ is essentially given by the Killing form of the Lie algebra g of G. The Einstein property now implies that the Killing form is

36


non-degenerate. It follows that G is semisimple. Simplicity of G follows from the indecomposability of .M; g/. Complete lists of these spaces can be found in [3] for the quaternionic and in [19] for the para-quaternionic case. The above discussion and Proposition 4.3 reduces the classification of (para-) quaternionic symmetric spaces to the classification of hyper-Kähler (hypersymplectic) symmetric triples. First of all we have the following counterpart of Proposition 4.4 and Proposition 4.7. Proposition 4.8 ([37], Proposition 2.1). Let .l; ˆl / be a Lie algebra with proper quaternionic or para-quaternionic grading. Then l is nilpotent. In particular, every hyperKähler or hypersymplectic symmetric triple is nilpotent. Proof. We can consider .l; ˆl / as an .k; Z2 /-equivariant Lie algebra, where k is sp.1/ or sl.2; R/, respectively. Properness implies that this .k; Z2 /-equivariant Lie algebra satisfies the assumptions of Lemma 6.2, which says that l has to be nilpotent. That hyper-Kähler and hypersymplectic symmetric triples are solvable has been already observed in [2]. Recently, we obtained an important sharpening of Proposition 4.8. We shall discuss it at the end of the present section, see Theorem 4.4. First we want to give an overview on the results on hyper-Kähler symmetric triples obtained by Alekseevsky, Cortés, and the authors in [2],[18], and [37]. Following our classification scheme we have to study the objects (O1)–(O3) for ˆl being a quaternionic grading. Let us begin, however, with an alternative approach to hyper-Kähler symmetric triples due to Alekseevsky–Cortés [2] that provides additional information. Let .E; !/ be a complex symplectic vector space. Any S 2 S 4 E defines a complex linear subspace hS sp.E; !/ Š S 2 E by hS D spanfSv;w 2 S 2 E j v; w 2 Eg; where Sv;w is the contraction of S with v and w via the symplectic form !. If S 2 .S 4 E/hS ;

(10)

then hS sp.E; !/ is a Lie subalgebra and, moreover, there is a natural Lie bracket on gS WD hS ˚ .H ˝C E/ such that hS gS is a subalgebra and hS acts on H ˝C E by the natural action on the second factor. The remaining part of the commutator maps H ˝C E H ˝C E to hS as follows: Œp ˝ v; q ˝ w D !H .p; q/Sv;w ;

p; q 2 H; v; w 2 E;

where !H is the alternating complex bilinear 2-form on H such that !H .1; j / D 1. Equation (10) is a system of quadratic equations for S 2 S 4 E. It admits families of particularly simple solutions, namely all S 2 S 4 EC S 4 E, where EC E is a Lagrangian subspace. Let us call solutions of this kind tame. If S is tame, then the Lie algebra hS is abelian. N Then J induces a real Let J be a quaternionic structure on E such that J ! D !. structure on each of the spaces S 4 E, S 2 E Š sp.E; !/, and H ˝C E. We denote all


37

these structures by the same symbol . If S 2 .S 4 E/ satisfies (10), then the real Lie algebra gJ;S WD .gS / D .hS / ˚ .H ˝C E/ carries a canonical structure of a hyper-Kähler symmetric triple. The Sp.1/-action on gJ;S is given by ˝ 1 on .H ˝C E/ and the trivial action on .hS / . This construction produces all hyper-Kähler symmetric triples. Proposition 4.9 (Alekseevsky–Cortés [2]). Let .g; ˆ; h ; i/ be a hyper-Kähler symmetric triple, then there exist data .E; !; J; S/ as above, S 2 S 4 E being a -invariant solution of (10), such that .g; ˆ; h ; i/ Š gJ;S as a hyper-Kähler symmetric triple. The tuple .E; !; J; S/ is uniquely determined by .g; ˆ; h ; i/ up to complex linear isomorphisms. Thus the classification of hyper-Kähler symmetric triples is equivalent to the classification of all -invariant solutions of (10). Recall that there is a family of easy-to-find solutions called tame. Unfortunately, the claim in [2] (repeated in [3] and [1]) that all solutions of (10) are tame is false. Recall that tame -invariant solutions S produce hyper-Kähler symmetric triples with abelian holonomy algebra gC D .hS / . Indeed, in the beginning of 2005 we found examples of hyper-Kähler symmetric triples with non-abelian holonomy (see [37] and Example 4.12 below) and noticed that the contradiction relies on a sign mistake in [2]. It appears that it is impossible to find all solutions of (10) in a straightforward way. In the recent paper [18] Cortés reconsiders the situation and is able to prove the following: Proposition 4.10 (Cortés [18], Theorem 10). Let S be a solution of (10). If hS is abelian, then S is tame. Thus, Propositions 4.9 and 4.10 provide a classification of all hyper-Kähler symmetric triples with abelian holonomy. In order to compare it with the other results of this paper we want to give the precise formulation of this classification in terms of our standard models d˛; .l; ˆl ; a/. In the following Examples 4.11 and 4.12 we consider quaternionic vector spaces V in two different ways: as complex vector spaces equipped with a quaternionic structure J and as Sp.1/-modules. Symmetric powers are symmetric powers of complex vector spaces. Then J induces a real structure on the complex vector spaces S 2k E. The l-action on all appearing .l; ˆl /-modules a is trivial.

Example 4.11. We fix n 2 N and S 2 .S 4 Hn / . Let l D l D Hn be abelian. The polynomial S defines a symmetric bilinear form bS on the real vector space .S 2 Hn / . We set aS D .aS /C WD .S 2 Hn / =rad.bS / and equip aS with the scalar product induced by bS . We define ˛S 2 C 2 .Hn ; aS /Sp.1/ by ˛S .v; w/ D vJ.w/ wJ.v/ mod rad.bS / 2 aS ;

v; w 2 Hn :

Then .˛S ; 0/ 2 Z2Q .Hn ; n ; aS /. Moreover, ˛S satisfies Condition (T2 ) in Proposition 3.3, and d˛S ;0 .Hn ; n ; aS / is a hyper-Kähler symmetric triple. It has abelian

38


holonomy aS and signature .4n; 4n/. One can check that d˛S ;0 .Hn ; n ; aS / is iso morphic to gJ;S for .E; !/ D Hn ˚ Hn , ! induced by the dual pairing. Note that Hn E is a Lagrangian subspace, and thus S is a tame solution of (10).

We call S 2 .S 4 Hn / indecomposable if S ¤ 0 and S … S 4 V ˚ S 4 W for all non-trivial decompositions Hn D V ˚ W into two quaternionic subspaces. We denote the set of all indecomposable S by .S 4 Hn / 0 . There is a natural right action of the group GL.n; H/ on .S 4 Hn / 0 . Moreover, if S is indecomposable, then the 2 cohomology class Œ˛S ; 0 2 HQ .Hn ; n ; aS / is balanced and indecomposable. Now we have the following consequence of Propositions 4.9 and 4.10. Theorem 4.2 (Alekseevsky–Cortés). The assignment

.S 4 Hn / 0 3 S 7! d˛S ;0 .Hn ; n ; aS /

yields a bijection between the union of the orbit spaces .S 4 Hn / 0 = GL.n; H/, n 2 N, and the set of isomorphism classes of indecomposable non-abelian hyper-Kähler symmetric triples with abelian holonomy. Note that this rather satisfactory classification result is not a classification in the sense of a list since for large n the orbit spaces .S 4 Hn / 0 = GL.n; H/ are not explicitly known. Example 4.12. Fix p; n 2 N0 , p n. We define a Lie algebra .ln ; ˆn / with proper quaternionic grading as follows: ln WD Hn , lnC WD .S 2 Hn / , lnC D z.ln /, Œv; wl WD vJ.w/ wJ.v/ 2 lnC ; v; w 2 Hn : We set an D .an / WD S 3 Hn . The quaternionic structure of Hn induces one on S 3 Hn . Thus an is a quaternionic vector space. The standard Sp.1/-invariant complex Hermitian form on Hn of signature .2p; 2.n p// induces a Hermitian form on an . We equip an with the real part of this Hermitian form and denote the resulting orthogonal .ln ; ˆn /-module by an;p . We define ˛n 2 C 2 .ln ; an /Sp.1/ by ˛n .v; L/ D vL 2 S 3 Hn ;

v 2 Hn ; L 2 .S 2 Hn / ;

˛n .ln ; ln / D ˛n .lnC ; lnC / D 0:

The above mentioned Hermitian form on Hn induces a natural identification of lnC D .S 2 Hn / with the Lie algebra sp.p; n p/ and a scalar product h ; ip on lnC . We denote the resulting Lie bracket on lnC by Œ ; p . Eventually, we define a 3-form p 2 C 3 .lnC / C 3 .ln /Sp.1/ by p .L1 ; L2 ; L3 / WD hŒL1 ; L2 p ; L3 ip ; Li 2 lnC : Then .˛n ; p / 2 Z2Q .ln ; ˆn ; an;p /. Moreover, the associated cohomology class is balanced, indecomposable, and satisfies (T2 ). Thus d˛n ;p .ln ; ˆn ; an;p / is an indecomposable hyper-Kähler symmetric triple. Because of the form of p its holonomy algebra .lnC / ˚ lnC is non-abelian.


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These hyper-Kähler symmetric triples are natural generalisations of Example 1 in [37], which is isomorphic to d˛1 ;0 .l1 ; ˆ1 ; a1;0 / and has signature .4; 12/. The paper [37] contains further examples with non-abelian holonomy. The smallest possible index of a non-abelian hyper-Kähler symmetric triple is 4. In the following classification result we use the notation of Examples 4.11, 4.12. Theorem 4.3 ([37], Theorem 7.4). Let .g; ˆ; h ; i/ be a non-abelian indecomposable hyper-Kähler symmetric triple of signature .4; 4q/. Then q D 1 or q D 3, and .g; ˆ; h ; i/ is isomorphic to exactly one of the following triples: q D 1: d˛S ;0 .H; ; aS /, 2 Œ2; 2, S .z C wj / WD z 4 C z 2 w 2 C w 4 , z; w 2 C; q D 3: d˛1 ;0 .l1 ; ˆ1 ; a1;0 /. The proof of the theorem in [37] is based on Theorem 2.3, i.e., it classifies directly the relevant objects (O1)–(O3). It does not rely on Theorem 4.2. There is a completely parallel theory for hypersymplectic symmetric triples. E.g., in the Alekseevsky–Cortés construction one has simply to replace the real structure induced by J by a real structure coming from real structures on both factors H and E. In other words, one can work from the beginning with R2 ˝R E0 instead of H ˝C E, where E0 is a real symplectic vector space. Hypersymplectic triples with abelian holonomy can be classified as in Theorem 4.2, see [18], [19], [1]. In addition, there is a variant of Example 4.12 producing hypersymplectic symmetric triples with non-abelian holonomy (one has to replace Hn by R2 ˝ R2n , the parameter p disappears). Note that all examples of hyper-Kähler symmetric triples presented so far have nilindex at most 5. As the following theorem and its corollary show, this is not an accident. Theorem 4.4 ([47]). Let .l; ˆl / be a Lie algebra with proper quaternionic or paraquaternionic grading. Then the nilindex of l is at most 6. Corollary 4.13. Let .g; ˆ; h ; i/ be a non-abelian hyper-Kähler or hypersymplectic symmetric triple. As usual, let l D g=i.g/? , where i.g/ g is the canonical isotropic ideal. Then there are only two possibilities for the nilindices of g, gC , and l as listed in the following table: holonomy

g

gC

l

1.

abelian

3

1

1 :

2.

non-abelian

5

2

2

Proof. Let us denote the nilindex of a nilpotent Lie algebra h by n.h/. It is easy to see that n.g/ is odd and that 2n.gC / < n.g/ for any nilpotent symmetric pair g D gC ˚g . Thus in our case n.g/ D 3 or n.g/ D 5 by Theorem 4.4. The possibility that at the same time n.g/ D 5 and n.gC / D 1 is excluded by Theorem 4.2. The corollary now follows from the inequality 2n.l/ n.g/ 2n.l/ C 1, which holds for any nilpotent metric Lie algebra and can be derived from the definition of i.g/.

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Corollary 4.14. The base N of the canonical fibration .see Section 3.2/ of a hyperKähler or hypersymplectic symmetric space M is flat. Proof. By Corollary 4.13 the Lie algebra l is at most two-step nilpotent. It follows that N D L=LC is flat. Corollary 4.13 shows that the structure of hyper-Kähler and hypersymplectic symmetric triples is strongly restricted. Therefore the goal of a full classification of these triples might be not unrealistic. The (lengthy) proof of Theorem 4.4, which we do not want to explain here, gives further inside into the structure of these triples: it computes certain universal Lie algebras Ln with proper (para-) quaternionic grading such that any Lie algebra with proper (para-)quaternionic grading in n generators is a quotient of Ln . We conclude this section with the following open question. Is the dimension of every hyper-Kähler symmetric space without flat local factors divisible by 8? Of course, this is true for all known examples. Note that indecomposable hypersymplectic symmetric spaces exist in all dimensions that are multiples of 4.

5 Further applications 5.1 Extrinsic symmetric spaces. Let us consider a non-degenerate connected submanifold M Rp;q . For x 2 M let sx be the reflection of Rp;q at the normal space Tx? M of M at x, i.e., sx is an affine isometry and sx jTx M D id, sx jTx? M D id. Here and throughout the section we consider Tx M and Tx? M as affine subspaces of Rp;q . Then M is called extrinsic symmetric if sx .M / D M holds for each point x 2 M . Extrinsic symmetric spaces in Rp;q are exactly those complete submanifolds whose second fundamental form is parallel. Extrinsic symmetric spaces in the Euclidean space are well understood. A classification in this case follows from Ferus’ results discussed below and the classification of symmetric R-spaces due to Kobayashi and Nagano. The case of a pseudo-Euclidean ambient space seems to be more involved. In Section 3.1 we discussed the correspondence between pseudo-Riemannian symmetric spaces and symmetric triples. Here we will see that there is a similar correspondence for (a certain class of) extrinsic symmetric spaces. While symmetric triples are special (namely proper) Z2 -equivariant metric Lie algebras the algebraic objects that we will use here are certain .R; Z2 /-equivariant metric Lie algebras. The group Z2 D f1; 1g acts on R by multiplication. Hence for an .R; Z2 /-equivariant metric Lie algebra .g; ˆ; h ; i/ we can regard ˆ as a pair .D; / that consists of a derivation D 2 Der.g/ and an involution 2 Aut.g/ such that D D D. Assume that for such an .R; Z2 /-equivariant metric Lie algebra D 3 D D holds. Then the eigenvalues of D are in fi; i; 0g. We put gC WD ker D;

g WD spanfX 2 g j D 2 .X / D X g

and define an involution D on g by D W g ! g; D jgC D id; D jg D id: Obviously


41

D and commute, hence gC and g are invariant under D . We introduce the notation C gC C WD gC \ g ;

g C WD gC \ g ;

C gC WD g \ g ;

g WD g \ g :

(11)

Definition 5.1. An extrinsic symmetric triple is an .R; Z2 /-equivariant metric Lie algebra .g; ˆ; h ; i/, ˆ D .D; /, for which (i) the derivation D is inner and satisfies D 3 D D, (ii) the Z2 -equivariant metric Lie algebras .g; ; h ; i/ and .gC ; D jgC ; h ; ijgC / are proper (i.e. symmetric triples). Two extrinsic symmetric triples are called isomorphic if they are isomorphic as .R; Z2 /-equivariant metric Lie algebras. Remark 5.2. We consider the subset

2

… D f.R; ˆ1 /; .R; ˆ2 /; .R2 ; ˆ3 /g .R; Z2 /; where ˆ1 .r/ D ˆ2 .r/ D 0; ˆ1 .z/ D 1, ˆ2 .z/ D z, and 0 r 1 0 ˆ3 .r/ D ; ˆ3 .z/ D r

0

0

z

for r 2 R and z 2 Z2 . In other words, … contains exactly those representations of .R; Z2 / that integrate to one of the following representation of O.2/: the onedimensional trivial representation, the one-dimensional representation via the determinant or the two-dimensional standard representation. Then an .R; Z2 /-equivariant (metric) Lie algebra .g; ˆ; h ; i/, ˆ D .D; /, satisfying D 3 D D is the same as a …-graded (metric) Lie algebra. Let .g; ˆ; h ; i/, ˆ D .D; /, be an extrinsic symmetric triple and choose 2 g such that D D ad./. We consider the subgroup GC WD h exp.adX /jg j X 2 gC i of O.g / and define Mg; WD GC g : Proposition 5.3 ([33]). 1. For any extrinsic symmetric triple .g; ˆ; h ; i/ and for each 2 g with D D ad./ the submanifold Mg; g is an extrinsic symmetric space. The abstract symmetric space Mg; with the induced metric is associated with the symmetric triple .gC ; D jgC ; h ; ijgC /. 2. Let .gi ; î ; h ; ii /, i D 1; 2, be extrinsic symmetric triples and let Mgi ; i be constructed as above. Then there exists an affine isometry f W .g1 / ! .g2 / mapping Mg1 ; 1 to Mg2 ; 2 if and only if .g1 ; ˆ1 ; h ; i1 / and .g2 ; ˆ2 ; h ; i2 / are isomorphic. Now let us turn to a construction that may be considered as a converse of Proposition 5.3, 1. Given an extrinsic symmetric space M Rp;q satisfying certain additional assumptions this construction yields an extrinsic symmetric triple .g; ˆ; h ; i/ and an element 2 g such that M D Mg; . Let us first discuss these additional assumptions.

42


Definition 5.4. A submanifold M Rp;q is called full if it is not contained in any proper affine subspace of Rp;q . It is called Tnormal if the intersection of the normal spaces of all points of M is not empty, i.e. x2M Tx? M 6D ;. To be full is not really a restriction for submanifolds M of the Euclidean space Rn since we may always consider the smallest affine subspace that contains M . Contrary to that, there are many submanifolds of the pseudo-Euclidean space Rp;q that are contained in an affine subspace that is degenerate with respect to the inner product but not in a proper non-degenerate one. This is also the case if one restricts oneself to extrinsic symmetric spaces. As far as normality is concerned, Ferus proved in [24] that extrinsic symmetric spaces in the Euclidean space decompose into a product of an affine subspace and a normal extrinsic symmetric space. This seems to be not true for a pseudo-Euclidean ambient space. Let us now describe the construction. It was developed by Ferus [25], [26] who used it to prove that every full and normal extrinsic symmetric space in Rn is a standard imbedded symmetric R-space. In our language this means that any such extrinsic symmetric space arises from an extrinsic symmetric triple as described in Proposition 5.3, 1. Here we will present the more elementary description of this construction given by Eschenburg and Heintze [22]. We will also include the necessary modifications for the pseudo-Riemannian situation discussed in [33], [38]. Let M Rp;q be a full and normal extrinsic symmetric space. Fix a point x0 2 M . Since M is normal assume that 0 2 Rp;q is contained in the intersection of T we may ? normal spaces x2M Tx M . This implies that sx 2 O.p; q/ for all x 2 M . We define the transvection group of M Rp;q by K WD hsx ı sy j x; y 2 M i O.p; q/: Obviously, K is isomorphic to the transvection group of the abstract symmetric space M . Let k so.p; q/ be the Lie algebra of K. Since it is isomorphic to the Lie algebra of the transvection group of the abstract symmetric space M it is the underlying Lie algebra of a symmetric triple .k; k ; h ; ik /. Besides k we can associate with M Rp;q the following metric Lie algebra g. As a vector space g equals g D k ˚ V;

V D Rp;q :

Using the standard scalar product h ; ip;q on V we define a scalar product on g by h ; i D h ; ik ˚ h ; ip;q . Furthermore, we define a bracket Œ ; W g g ! g by (i) Œ ; restricted to k k equals the Lie bracket of k, (ii) ŒA; v D A.v/ for A 2 k and v 2 V , (iii) Œ ; restricted to V V is given by the condition Œ ; V V W V V ! k and by hA; Œx; yi D hŒA; x; yi for all A 2 k and x; y 2 V . Using the fullness of M Rp;q one can prove that this bracket satisfies the Jacobi identity. Moreover, we have an involution on g given by gC D k, g D V and an inner derivation D WD ad.x0 /. We obtain:


43

Theorem 5.1 (Ferus [25], [26], see also [22], [33], [38]). Let M Rp;q be a full and normal extrinsic symmetric space. Let g; h ; i and ˆ D .D; / be as constructed above. Then .g; ˆ; h ; i/ is an extrinsic symmetric triple and M D Mg;x0 . In [38] Kim discusses this construction also for the case of a non-normal extrinsic symmetric space. Remark 5.5. Let .g; ˆ; h ; i/, ˆ D .D; /, be an extrinsic symmetric triple and suppose D D ad./, 2 g . Then Mg; is normal by construction. It is full if and only if C Œg (12) C ; g D g (see (11) for the notation used here). We call an extrinsic symmetric triple satisfying (12) full. Having explained the relation between extrinsic symmetric spaces and extrinsic symmetric triples we now want to apply our classification scheme to extrinsic symmetric triples. Recall that we can understand an extrinsic symmetric triple as a …-graded metric Lie algebra satisfying additional conditions, where … is given as in Remark 5.2. We can proceed as in Section 4.1 and we see that any extrinsic symmetric triple is isomorphic to some d˛; .l; ˆl ; a/, where now (O1) .l; ˆl / is a Lie algebra with …-grading such that the Z2 -equivariant Lie algebras .l; l / and .lC ; Dl jlC / are proper, (O2) a is a …-graded semisimple orthogonal .l; ˆl /-module, 2 .l; ˆ l ; a/b is indecomposable and satisfies besides .T2 / two further (O3) Œ˛; 2 HQ conditions .T2C / and .AC 0 / (see [33]), which ensure that d˛; .l; ˆl ; a/ satisfies the properness condition (ii) in Definition 5.1, (O4) the derivation D D Dl ˚ Da ˚ Dl is inner. Condition (O4) is equivalent to 9l 2 l 9a 2 a C 0 .l; a/ 9z 2 l C 1 .l/: Dl D adl .l/;

Da D .l/;

da D i.l/˛;

dz D ha ^ ˛i C i.l/:

Combining (O1)–(O4) with Theorem 2.3 we obtain a classification scheme for extrinsic symmetric triples. Example 5.6 (Full and normal extrinsic Cahen–Wallach spaces). We want to answer the question which indecomposable non-semisimple non-flat Lorentzian symmetric spaces can be embedded into a pseudo-Euclidean space as full and normal extrinsic symmetric spaces. Recall from Theorem 3.2 that every indecomposable non-semisimple non-flat Lorentzian symmetric space is associated with a symmetric triple of the kind d.p; q; ; /, p; q 0, p C q > 0, .; / 2 Mp;q . Theorem 5.2 (cf. [33]). The symmetric triple d.p; q; ; /, p; q 0, p C q > 0, .; / 2 Mp;q admits an associated symmetric space M 1;nC1 , n D p C q, that can be embedded as a full and normal extrinsic symmetric space if and only if

44


1. q D 0 and D .1; : : : ; 1/ or 2. p D 0 and D .1; : : : ; 1/. In both cases the ambient space is R2;nC2 and one obtains a one-parameter family of mutually non-isomorphic extrinsic symmetric spaces. To verify the above theorem we have to determine all extrinsic symmetric triples .g; ˆ; h ; i/, ˆ D .D; /, that satisfy (12) and .gC ; D jgC ; h ; igC / Š d.p; q; ; / with .; / 2 Mp;q (see Section 3.3). In particular, since gC is indecomposable and non-reductive the Lie algebra g is indecomposable and non-semisimple. This implies g Š d WD d˛; .l; ˆl ; a/ for a suitable .R; Z2 /-equivariant Lie algebra .l; ˆl /, an orthogonal .l; ˆl /-module a and some .˛; / 2 Z2Q .l; ˆ l ; a/. In particular, we have gC Š dC D d˛jlC ;jlC .lC ; ˆ l jlC ; aC /. Hence dim l C D dim l D 1 and

C C lC C D ŒlC ; lC D 0. Equation (12) implies ŒlC ; l D l . Since l 6D 0 we also obtain C dim l D 1. Now it is not hard to see that l is isomorphic to one of the Lie algebras

1. sl.2; R/ D fŒH; X D 2Y; ŒH; Y D 2X; ŒX; Y D 2H g, 2. su.2/ D fŒH; X D 2Y; ŒH; Y D 2X; ŒX; Y D 2H g, and that in both cases ˆ l D .Dl ; l / is given by lC D R H , l D spanfX; Y g and Dl D .1=2/ adX . Since H 2 .l; a/ D 0 for l 2 fsl.2; R/; su.2/g we may assume ˛ D 0. Since d is indecomposable we get al D 0. Because of D 3 D D the eigenvalues of .X / are in f0; 2i; 2i g. Consequently, a is the direct sum of submodules that are all equivalent to the adjoint representation of l. Finally, we obtain the following result, which proves Theorem 5.2 and gives all embeddings explicitly. Proposition 5.7 (cf. [33]). If M 1;nC1 ,! Rr;s is a full and normal extrinsic symmetric space of Lorentz signature and if M 1;nC1 is solvable and indecomposable, then .r; s/ D .2; n C 2/ and we are in one of the following cases: 1. M 1;nC1 is associated to the symmetric triple d.n; 0; ; 0/, D .1; : : : ; 1/ 2 Rn and M 1;nC1 R2;nC2 is extrinsic isometric to Md; for d D d˛; .l; ˆl ; a/ with L l D sl.2; R/; ˆ l as above ; a D niD1 .ad; l; Bl ; â /; â D .Dl ; l /; ˛ D 0;

D cBl .; Œ ; /;

c 2 R;

and D .1=2/ X. Here Bl is the Killing form of l. 2. M 1;nC1 is associated to the symmetric triple d.0; n; 0; /, D .1; : : : ; 1/ 2 Rn and M 1;nC1 R2;nC2 is extrinsic isometric to Md; for d D d˛; .l; ˆl ; a/ with L l D su.2/; ˆ l as above ; a D niD1 .ad; l; Bl ; â /; â D .Dl ; l /; ˛ D 0;

D cBl .; Œ ; /;

c 2 R;

and D .1=2/ X. Again Bl denotes the Killing form of l. We remark that there are decomposable solvable non-flat Lorentzian symmetric spaces that can be embedded as full and normal extrinsic symmetric spaces such that


45

the embedding is indecomposable, i.e., the associated extrinsic symmetric triple is indecomposable. In [33] we classify all extrinsic symmetric triples without simple ideals that are associated with a Lorentzian extrinsic symmetric space. Of course, that classification contains all extrinsic symmetric triples that we obtained above. However, it contains also extrinsic symmetric triples whose associated extrinsic symmetric spaces are not full. Remark 5.8. Let .g; ˆ; h ; i/, ˆ D .D; /, be an extrinsic symmetric triple. Exponentiating D defines a complex grading ˆ0 on .g; h ; i/. Equation (12) implies that this grading is proper if .g; ˆ; h ; i/ is full. Thus, forgetting about , a full extrinsic symmetric triple can be considered as a pseudo-Hermitian symmetric triple such that the complex grading is given by inner automorphisms. If .gC ; D jgC ; h ; igC / has signature .p; q/, then the pseudo-Hermitian symmetric triple .g; ˆ0 ; h ; i/ has signature .2p; 2q/. Moreover, Proposition 4.4 implies that g has a non-trivial Levi factor and that the radical of g is nilpotent. In view of these facts, the reader should compare Proposition 5.7 with Theorem 4.1. 5.2 Manin triples. Manin triples are algebraic objects that are associated with Poisson–Lie groups. A Poisson–Lie group is a Lie group equipped with a Poisson bracket that satisfies a compatibility condition with the group multiplication. The infinitesimal object associated with such a Poisson–Lie group G is the Lie algebra g of G together with a 1-cocycle W g ! g ^ g that satisfies co-Jacobi identity, i.e., W g ^ g ! g is a Lie bracket on g . Such a pair .g; / is called Lie bialgebra. Given such a Lie bialgebra .g; / it is easy to see that there exists a unique Lie algebra structure on the vector space g ˚ g such that the inner product on g ˚ g defined by the dual pairing is invariant. What we obtain is an example of a so-called Manin triple. Definition 5.9. A Manin triple .g; h1 ; h2 / consists of a metric Lie algebra g and two complementary isotropic subalgebras h1 and h2 . Here we only consider finite-dimensional Lie bialgebras and Manin triples. In this case the above described assignment associating a Manin triple to a given Lie bialgebra is a one-to-one correspondence. For a detailed introduction to this subject we refer to [41]. Manin triples .g; h1 ; h2 / for which g is a complex reductive Lie algebra were classified by Delorme [20]. Having learned about the difficulties in classifying non-reductive metric Lie algebras it is not surprising that there is little known about non-reductive Manin triples. However, there are some results in low dimensions. Figueroa-O’Farrill [27] studied Manin triples in the context of conformal field theory and the N D 2 Sugawara constructions. By rather heavy calculations he achieved a classification of complex six-dimensional Manin triples. A more conceptual proof of his result, which also yields a classification in the real case is due to Gomez [28]. Unfortunately, the method of quadratic extensions seems to be not adequate for the description of Manin triples. The difficulty is to handle both isotropic subalgebras at the same time. However, we can describe Manin pairs.

46


Definition 5.10. A Manin pair .g; h/ consists of a metric Lie algebra of signature .n; n/ and an n-dimensional isotropic subalgebra h g. Applying the method of quadratic extensions to Manin pairs we obtain the following result. Proposition 5.11. Let l be a Lie algebra and l0 l a subalgebra. Let a be an orthogonal l-module of signature .m; m/ and a0 a an m-dimensional isotropic l0 invariant subspace. Let .˛; / 2 Z2Q .l; a/ satisfy ˛.l0 ; l0 / a0 and .l0 ; l0 ; l0 / D 0. Then d WD d˛; .l; a/ has signature .m C dim l; m C dim l/ and .d; Ann.l0 / ˚ a0 ˚ l0 / is a Manin pair, where Ann.l0 / l denotes the annihilator of l0 . Conversely, every Manin pair .g; h/ for which g does not contain simple ideals is isomorphic to some pair .d; Ann.l0 /˚a0 ˚l0 / constructed in this way, where, moreover, d is balanced. Proof. The first part of the proposition is easy to check. The second part relies on the functorial assignment (4) and the fact that the section s W l ! g defining .˛; / 2 Z2Q .l; a/ can be chosen in the following way. Decompose h as a vector space into h D h1 ˚ h2 ˚ h3 , where h1 D h \ i.g/ and h2 is a complement of h1 in h \ i.g/? . Then choose s W l ! g as described in Section 2.4 such that it satisfies in addition h3 s.l/ and s.l/ ? h2 . Combining this with Theorem 2.3 we obtain a classification scheme for those Manin pairs .g; h/ for which g does not have simple ideals. Let us remark that now the absence of simple ideals is a real restriction contrary to the case of metric Lie algebras without distinguished subalgebra. In small dimensions, Proposition 5.11 is a helpful tool not only for classification of Manin pairs but also for classification of Manin triples, since a complementary isotropic subalgebra of h g (if it exists) can be determined by hand. E.g., using this method it is easy to recover the classification in dimension six without extensive calculations.

6 Appendix: Some lemmas and proofs 6.1 Implications of .h; K /-equivariance. We first consider the case of the trivial group K D feg. Lemma 6.1. Let .l; ˆl / be an h-equivariant Lie algebra such that lh l0 . Let r l be the radical of the Lie algebra l. Then r is nilpotent and acts trivially on every semisimple .l; ˆl /-module. Proof. Recall the notion of the nilpotent radical R.g/ of a Lie algebra g from Section 2.2. The ideal R.g/ is nilpotent and acts trivially on any semisimple g-module. We consider the Lie algebra h Ë l, where the action of h on l is given by ˆl . A semisimple .l; ˆl /-module can be considered as a semisimple h Ë l-module. It is therefore sufficient to show that r R.h Ë l/.


47

Formula (1) implies that ˆl .h/r C r \ l0 R.h Ë l/. Since h acts semisimply on l and r l is h-invariant we have r D rh ˚ ˆl .h/r. By assumption rh r \ l0 . We conclude that r R.h Ë l/. The lemma follows. We now look at the case K D Z2 . Any .h; Z2 /-equivariant Lie algebra has a decomposition l D lC ˚ l w.r.t. the Z2 -action. Lemma 6.2. Let h be a semisimple Lie algebra. Let .l; ˆl / be an .h; Z2 /-equivariant Lie algebra such that .a/ lC D lh , and .b/ lC D Œl ; l . Then l is nilpotent. Proof. Let r be the radical of l. Then the semisimple Lie algebra s WD l=r inherits an .h; Z2 /-equivariant structure ˆs . Since any derivation of s is inner we may identify ˆs .h/ with a semisimple subalgebra k s acting on s by the adjoint representation. ˆs .h/ respects the decomposition s D sC ˚ s . Thus k sC . By Condition (a) we have sC D sk . Thus k is abelian, hence zero. Applying (a) again we obtain s D sC , thus s D f0g. Now Condition (b) yields s D 0. Thus l is solvable. Now we consider l as an h-equivariant solvable Lie algebra and apply Lemma 6.1. 6.2 Proof of Proposition 2.12. Any semisimple .h; K/-module V has a canonical decomposition V D V .h;K/ ˚V 1 , where V 1 is the sum of all irreducible submodules of V carrying a non-trivial .h; K/-action. We denote the corresponding components of any v 2 V by v 0 ; v 1 . Tensor products of semisimple .h; K/-modules are again semisimple. In particular, the natural .h; K/-actions on C p .l; a/, C q .l/ are semisimple. 2 2 .l; ˆ l ; a/ ! HQ .l; a/.h;K/ . Let Let us first prove injectivity of the natural map HQ .˛i ; i / 2 Z2Q .l; ˆ l ; a/ D Z2Q .l; a/.h;K/ , i D 1; 2, be two cocycles that represent the 2 same element in HQ .l; a/, i.e., there exists . ; / 2 C 1 .l; a/ ˚ C 2 .l/ such that .˛2 ; 2 / D .˛1 C d ; 1 C d C h.˛1 C 12 d / ^ i/: The invariance of ˛i implies that d D .d /0 D d 0 . It follows that .d ^ /0 D .d 0 ^ /0 D d 0 ^ 0 : This and the invariance of i and ˛1 implies 2 D 1 C d 0 C h.˛1 C 12 d 0 / ^ 0 i: 1 Thus .˛2 ; 2 / D .˛1 ; 1 /. 0 ; 0 /. Since . 0 ; 0 / 2 CQ .l; ˆ l ; a/ the cocycles .˛i ; i /, 2 i D 1; 2, represent the same class in HQ .l; ˆ l ; a/. This shows injectivity. Surjectivity is a little bit more involved. Let .˛; / 2 Z2Q .l; a/ such that Œ˛; 2 2 HQ .l; a/.h;K/ . This means that for all X 2 h, k 2 K there exist elements . X ; X /, . k ; k / 2 C 1 .l; a/ ˚ C 2 .l/ such that

k˛ D ˛ C d k ; X˛ D d X ;

k D C dk C h.˛ C 12 d k / ^ k i ; X D dX C h˛ ^ X i :

(13) (14)

48


We have to show that Œ˛; can be represented by an .h; K/-invariant cocycle. We first consider the case that K is connected. Then .h; K/-invariance is the same as k Ë h-invariance, where k is the Lie algebra of K which acts by derivations on h. Since the action of k Ë h on l and a is semisimple it factors over a reductive quotient of k Ë h. Thus without loss of generality we may assume that K is trivial and h D s ˚ t, where s is semisimple and t is abelian. Let C be the Casimir operator of s. For any semisimple h-module V there exists an element XV 2 t such that the operator DV WD XV C C acts invertibly on any h-submodule of V 1 . In particular, im DV D V 1 , ker DV D V h . Now we fix D WD DV for the semisimple h-module V D C 2 .l; a/ ˚ C 1 .l; a/ ˚ C 3 .l/ ˚ C 2 .l/. P The operator D is of the form D D X C Yi Zi for some X; Yi ; Zi 2 h. Now (14) implies that there exists an element D 2 C 1 .l; a/ such that D˛ D d D . Thus D˛ 2 B 2 .l; a/ \ V 1 , where B 2 .l; a/ C 2 .l; a/ denotes the h-submodule of coboundaries. Therefore there exists a coboundary d such that D˛ D Dd . We now consider the quadratic cocycle .˛ 0 ; 0 / WD .˛; /. ; 0/. Then D˛ 0 D 0, thus ˛ 0 is h-invariant. The cocycle .˛ 0 ; 0 / satisfies (14) again. We claim that 0 0 D 0 D dD C h˛ 0 ^ D i

(15)

0 0 0 for some quadratic cochain . D ; D / satisfying d D D 0. It is obvious that (15) is valid for D replaced by X 2 h. We have to show that it also holds for D replaced by a second order monomial Y Z for Y; Z 2 h. Thus we can assume (15) for Z and apply Y . Note that Y ˛ 0 D 0 by h-invariance. Setting Y0 Z WD Y Z0 and Y0 Z WD Y Z0 we obtain Y Z 0 D d Y Z0 C Y h˛ 0 ^ Z0 i D dY0 Z C h˛ 0 ^ Y0 Z i:

This justifies (15). We denote by B 3 .l/ C 3 .l/ and Z 1 .l; a/ C 1 .l; a/ the corresponding submodules of coboundaries and cocycles. Then (15) says D 0 2 .B 3 .l/ C h˛ 0 ^ Z 1 .l; a/i/ \ V 1 DW W 1 : Thus we can solve

D 0 D D.d 0 C h˛ 0 ^ 0 i/

in W 1 , in particular with d 0 D 0. Now we set .˛ 00 ; 00 / WD .˛ 0 ; 0 /. 0 ; 0 /. Note that ˛ 00 D ˛ 0 , thus D˛ 00 D 0. We have D 00 D D 0 Dd 0 Dh˛ 0 ^ 0 i D 0. Thus .˛ 00 ; 00 / is h-invariant and Œ˛; D Œ˛ 00 ; 00 . This implies surjectivity of the natural 2 2 map HQ .l; ˆ l ; a/ ! HQ .l; a/.h;K/ in the case of connected K. 2 If K is disconnected and Œ˛; 2 HQ .l; a/.h;K/ we can assume by the above that .˛; / 2 Z2Q .l; a/.h;K0 / , where K0 K is the identity component. We now work in the vector space V D ŒC 2 .l; a/ ˚ C 1 .l; a/ ˚ C 3 .l/ ˚ C 2 .l/.h;K0 / , which carries an action of the finite group G D K=K0 . By injectivity of the canonical map 2 2 .l; ˆl j.h;K0 / ; a/ ! HQ .l; a/.h;K0 / , the Equations (13) are valid with k replaced HQ by g 2 G and with g ; g invariant under .h; K0 /. We now proceed similar as in 1 P the connected case. We will use the projection operator P D jGj g2G g. We set


49

1 P 0 0 0 WD jGj g2G g and .˛ ; / WD .˛; /. ; 0/. Then we have ˛ D P ˛, i.e., ˛ is 0 0 G-invariant. Equation (13) for .˛ ; / provides for g 2 G certain .h; K0 /-invariant elements . g0 ; g0 / with d g0 D 0. We now define

0 WD

1 X 0 g ; jGj g2G

0 WD

1 X 0 g jGj g2G

and set .˛ 00 ; 00 / WD .˛ 0 ; 0 /. 0 ; 0 /. Then ˛ 00 D ˛ 0 is G-invariant. Using d g0 D 0 and (13) we eventually obtain 00 D 0 C d 0 C h˛ 0 ^ 0 i D P 0 : Thus .˛ 00 ; 00 / is G-invariant. In other words, .˛ 00 ; 00 / 2 Z2Q .l; a/.h;K/ . Since 2 Œ˛; D Œ˛ 00 ; 00 2 HQ .l; a/ this finishes the proof of surjectivity of the canonical 2 2 map HQ .l; ˆ l ; a/ ! HQ .l; a/.h;K/ . 6.3 Proof of Proposition 3.8. We first assume Condition (b). Let G be the transvection group of M , and let J G be the analytic subgroup corresponding to the ideal i? g. By the definition of a quadratic extension (Definition 2.5) the Lie algebra i? =i Š a is abelian, i.e., Œi? ; i? i. Moreover, hŒi? ; i; gi D hi? ; Œi; gi hi? ; ii D f0g: It follows that i? is 2-step nilpotent: Œi? ; Œi? ; i? D 0:

(16)

The following lemma together with Condition (b) implies that J G is closed. Lemma 6.3. Let G be a connected Lie group with Lie algebra g. Let j g be a nilpotent ideal containing the center z.g/. Then the analytic subgroup J G corresponding to j is closed. Proof. The assertion of the lemma is well known, if j g is the maximal nilpotent ideal (see e.g. [14], Chapter III §9). In this case J is called the nilradical of G. We look at the adjoint group G1 WD Adg .G/ Š G=Z.G/ acting on the Lie algebra g. Let N1 be the nilradical of G1 . Then N1 G1 is closed. By Engel’s Theorem, there is a basis of g such that all elements of N1 are represented by upper triangular matrices with 1’s on the diagonal with respect to this basis. It follows that the exponential is a diffeomorphism from the Lie algebra n1 of N1 to N1 . Now Adg .J / N1 . Via the exponential we see that Adg .J / Š adg .j/ is closed in N1 Š n1 . Hence Adg .J / is closed in G1 . It follows that J Z.G/ is closed in G. Since z.g/ j the group J is the identity component of J Z.G/. Therefore, J is closed as well. We set L WD G=J . We identify the Lie algebra of L via p with l. Let QL W G ! L be the natural projection. Let GC G be the stabilizer of the base point x0 , and

50


let be the corresponding involution of G. Then GC G . Since J is -stable induces an involution L W L ! L. Then LC WD QL .GC / is contained in LL and has Lie algebra lC . Since LL has the same Lie algebra we see that LC L is closed. Then N WD L=LC is an affine symmetric space and the map q W M Š G=GC ! N induced by QL is affine and surjective. The transvection group of N is equal to L0 D L=.Z.L/ \ LC /. The natural map Q between the transvection groups of M and N is therefore the composition of QL with the projection of L to L0 . It is now evident that Condition (6) is satisfied. The fibres of q are connected since they are precisely the orbits of the connected group J . Let us show that they are coisotropic and flat. By homogoneity it is sufficient to look at tangent space Tx0 M at the base point which can be identified with g . Then .i? / corresponds to the tangent space of the orbit. Since i .i? / the orbits are coisotropic. For X; Y; Z 2 g Š Tx0 M the curvature tensor is given by R.X; Y /Z D ŒŒX; Y ; Z. Flatness of the orbits now follows from (16). The uniqueness of N and q is a simple consequence of Condition (6) and the required connectedness of the fibres. If Condition (b) is not satisfied, but M is simply connected, we make the following modifications of the above proof. First of all it is more convenient to work with the universal covering group of the transvection group of M . We now denote this universal cover by G. The involution of g induces one of G which we again denote by . G acts on M . Let GC G be the stabilizer of the base point x0 2 M . The Lie algebra of GC is gC . Since M and G are simply connected GC is connected. It follows that GC G . (This is the crucial point where the simple connectedness of M enters the proof. If M were not simply connected we would have G ¨ GC .) Let pQ W g ! l be the homomorphism induced by p. Let L be the connected and simply connected group with Lie algebra l. The surjection pQ integrates to a surjection QL W G ! L. We set J WD ker QL which is closed in G. The group J has Lie algebra i? . Moreover, J is connected since L Š G=J and G are simply connected. With these changes understood the proof runs as in the first case. Lemma 6.3 is no longer needed.

References [1] Alekseevsky, D. V., Blažić, N., Cortés, V., Vukmirović, S., A class of Osserman spaces. J. Geom. Phys. 53 (3) (2005), 345–353. 37, 39 [2] Alekseevsky, D. V., Cortés, V., Classification of indefinite hyper-Kähler symmetric spaces. Asian J. Math. 5 (2001), 663 – 684. 36, 37 [3] Alekseevsky, D. V., Cortés, V., Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type. In Lie groups and invariant theory, Amer. Math. Soc. Transl. Ser. (2) 213, Amer. Math. Soc., Providence, RI, 2005, 33–62. 35, 36, 37 [4] Astrahancev, V. V., Decomposability of metrizable Lie algebras. Funct. Anal. Appl. 12 (1979), 210–212. 3 [5] Baum, H., Kath, I., Doubly extended Lie groups – curvature, holonomy, and parallel spinors. Differential Geom. Appl. 19 (2003), 253–280. 6


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Holonomy groups of Lorentzian manifolds: classification, examples, and applications Anton Galaev and Thomas Leistner

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Holonomy groups . . . . . . . . . . . . . . . . . . . . 2.1 Holonomy groups of linear connections . . . . . 2.2 Holonomy groups of semi-Riemannian manifolds 2.3 Lorentzian holonomy groups . . . . . . . . . . .

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3 Algebraic aspects of non-irreducible, indecomposable Lorentzian holonomy 3.1 Basic algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Four types of indecomposable subalgebras of p . . . . . . . . . . . . 3.3 Berger algebras and weak-Berger algebras . . . . . . . . . . . . . . . 3.4 Decomposition of the space of curvature endomorphisms . . . . . . . 3.5 Consequences for Lorentzian holonomy . . . . . . . . . . . . . . . .

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4 The classification of weak-Berger algebras . . 4.1 Real and complex weak-Berger algebras 4.2 Weak-Berger algebras of unitary type . 4.3 Weak-Berger algebras of real type . . .

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Lorentzian manifolds with indecomposable, non-irreducible holonomy 5.1 Local description and coordinates . . . . . . . . . . . . . . . . 5.2 Metrics that realise all types of Lorentzian holonomy . . . . . . 5.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 pp-waves and generalisations . . . . . . . . . . . . . . . . . . . 5.5 Holonomy of space-times . . . . . . . . . . . . . . . . . . . . .

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6 Applications and outlook . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Parallel spinors on Lorentzian manifolds . . . . . . . . . . . . . . 6.2 Holonomy of indecomposable, non-irreducible Einstein manifolds 6.3 Open problems and outlook on higher signatures . . . . . . . . .

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1 Introduction The holonomy group is a very useful concept for studying the structure of semiRiemannian manifolds. It links geometric and algebraic properties and allows to apply the tools of algebra to geometric questions. In particular, it enables us to describe parallel sections in geometric vector bundles associated to the manifold, such as the

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tangent bundle, tensor bundles, or the spin bundle, as holonomy-invariant objects and by algebraic means. Many fruitful developments in differential geometry were initiated or driven by the study and the knowledge of holonomy groups, such as the study of so-called special geometries in Riemannian geometry. These developments were based on the classification of Riemannian holonomy groups, which was achieved by the de Rham decomposition theorem [28] and the Berger list of irreducible semi-Riemannian holonomy groups [13] (see Section 2.2 of the present article). For manifolds with indefinite metric, in particular for Lorentzian manifolds, this question was widely open and untackled – apart from a classification in 4 dimensions by J. F. Schell [57] and R. Shaw [62] (presented in Section 5.5) – until a groundbreaking paper by L. Bérard-Bergery and A. Ikemakhen [11] based on Ikemakhen’s PhD thesis [43]. The results recently obtained by the authors in [48], [50], [51] and [38] (see also [52], [55] and [34]) complement the results of Bérard-Bergery and Ikemakhen to a full classification of Lorentzian holonomy groups. There are many applications of holonomy theory for Lorentzian manifolds such as the study of equations motivated by physics in relation to the possible holonomy groups. On the one hand these are the Einstein equations, on the other hand certain spinor field equations in supergravitiy theories (see e.g. [33]). Similarly to the Riemannian situation, the generalisation of the de Rham decomposition theorem to arbitrary signature by H. Wu [67] allows to reduce the general classification problem for pseudo-Riemannian holonomy groups to a problem concerned with indecomposably acting groups. For Lorentzian manifolds this means that the connected holonomy group is a product of Riemannian holonomy groups with a Lorentzian holonomy group which is either equal to SO0 .1; n/ or acts indecomposably, but non-irreducibly. Hence, the task was to classify holonomy groups acting in this way (see Section 2.3). In order to explain the classification result we have to recall that the holonomy group of an indecomposable, non-irreducible Lorentzian manifold of dimension .nC2/ admits an invariant, light-like line. Hence, it is contained in the parabolic subgroup P D .RC SO.n//ËRn of the conformal group SO0 .1; nC1/. For an indecomposable subgroup of P the main ingredient is the projection onto SO.n/, which is called the orthogonal part. These facts will be explained in Section 3.1. The classification of Lorentzian holonomy groups in P now consists of two theorems. The first describes the algebraic structures of indecomposable, non-irreducible subalgebras in so.1; nC1/ and is due to L. Bérard-Bergery and A. Ikemakhen [11]. Theorem 1.1 (Bérard-Bergery, Ikemakhen [11]). Let h be a subalgebra of p D .R ˚ so.n// Ë Rn , acting indecomposably on RnC2 , and let g WD pr so.n/ .h/ D z ˚ g0 be the Levi-decomposition of its orthogonal part. Then h belongs to one of the following types. 1. If h contains Rn , then there are three types: Type 1: h contains R. Then h D .R ˚ g/ Ë Rn .

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Type 2: pr R .h/ D 0, i.e. h D g Ë Rn . Type 3: Neither type 1 nor type 2. In this case there exists an epimorphism ' W z ! R, such that h D .l ˚ g0 / Ë Rn ; where l WD graph ' D f.'.T /; T / j T 2 zg R ˚ z. Or written in matrix form: 80 9 1 ˇ vt 0 < '.A/ ˇ = ˇ ACB v A ˇ A 2 z; B 2 g0 ; v 2 Rn : hD @ 0 : ˇ ; 0 0 '.A/ 2. In the case where h does not contain Rn we have Type 4: There exists a non-trivial decomposition Rn D Rk ˚ Rl , 0 < k; l < n and an epimorphism W z ! Rl , such that g so.k/ and h D .g0 ˚ l/ Ë Rk p where l WD f.'.T /; T / j T 2 zg D graph Rl ˚ z. Or, written in matrix form, 80 9 1 0 .A/t vt 0 ˆ ˇ > ˆ > ˆ > : ; 0 0 0 0 In [34] we gave a more geometric proof of this theorem which is presented in Section 3.2. The second theorem gives a classification of the orthogonal part and was proved in [48], [50], [51], [55], [38]. Theorem 1.2. Let H be a connected subgroup of SO0 .1; n C 1/ which acts indecomposably and non-irreducibly. Then H is a Lorentzian holonomy group if and only if its orthogonal part is a Riemannian holonomy group. Naturally, the proof of this theorem consists of two main steps. The first is to show that the orthogonal part of H has to be a Riemannian holonomy group. This involves the notion of weak-Berger algebras (see Section 3.3 and the following sections) and their classification, which is explained in Section 4. This step uses similar methods as the classification of irreducible holonomy groups of torsion free connections and was done in [48], [50], [51], [55]. The second step consists of showing that each of the arising groups can actually be realised as holonomy group. This is easy for the types 1 and 2 in Theorem 1.1 (see for example [49], [52]) but more involved for the coupled types 3 and 4 and was achieved recently in [38]. This method is explained in Section 5.2. In further parts of Section 5 we will explain some holonomy related structures of indecomposable Lorentzian manifolds, such as the existence of certain distributions and coordinates on these manifolds. In Section 5.3 we present a large variety of examples, including metrics for which the orthogonal part of the holonomy is given by a symmetric pair, by so.3/ irreducibly included in so.5/, by the exceptional Lie algebra g2 , and by

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spin.7/. Furthermore, we will review our result that a Lorentzian manifold has Abelian holonomy if it is a pp-wave and present some holonomy-related generalisations of ppwaves made in [54]. As an application of these results, in Section 6.1 we will present the classification of holonomy groups of indecomposable Lorentzian manifolds which admit a parallel spinor. For these manifolds one can show that the holonomy group has to be of type 2. Together with the de Rham–Wu decomposition on gets the following consequence. Theorem 1.3. Let .M; h/ be a simply connected, complete Lorentzian spin manifold which admits a parallel spinor. Then .M; h/ is isometric to a product .M 0 ; h0 / .N1 ; g1 / .Nk ; gg /, where the .Ni ; gi / are flat or irreducible Riemannian manifolds with a parallel spinor and .M 0 ; h0 / is either .R; dt / or it is an indecomposable, non-irreducible Lorentzian manifold of dimension n C 2 > 2 with holonomy G Ë Rn where G is the holonomy group of a Riemannian manifold with parallel spinor. In particular, the holonomy group of .M; h/ is the following product y .G Ë Rn / G; y being holonomy groups of Riemannian manifolds for some n 0, and G and G admitting a parallel spinor, i.e. both being a product of the possible factors f1g, SU.p/, Sp.q/, G2 , or Spin.7/ .with G trivial if n < 2/. As another, new application we show that the holonomy of indecomposable, nonirreducible Lorentzian Einstein manifolds cannot be of the coupled types 3 or 4. Theorem 1.4. Let .M; h/ be an indecomposable non-irreducible Lorentzian Einstein manifold. Then the holonomy of .M; h/ is of uncoupled type 1 or 2. If the Einstein constant of .M; h/ is non-zero, then the holonomy of .M; h/ is of type 1. This implies on the one hand that a Lorentzian Einstein manifold .M; h/ that admits a light-like parallel vector field is Ricci flat. On the other hand it implies that the holonomy group of an indecomposable, non-irreducible Lorentzian Einstein manifold is given by .R G/ Ë Rn or G Ë Rn ; with G a Riemannian holonomy group, and in the second case the manifold is Ricci flat and G is a product of SO.k/, SU.p/, Sp.q/, G2 , Spin.7/, or the holonomy of a Riemannian symmetric space. This review concludes with some short remarks about results in higher signatures such as .2; n/ and .n; n/.

2 Holonomy groups 2.1 Holonomy groups of linear connections. Let M be a smooth m-dimensional manifold equipped with a linear connection r, i.e. a connection on the tangent bundle TM . Such a setting .M; r/ is sometimes called affine manifold because it allows to


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define geodesics, which are a generalisation of straight lines. If X 2 Tp M is a tangent vector at the point p 2 M , r allows us to parallel translate this vector along any given curve W Œ0; 1 ! M starting at p, i.e. .0/ D p. The parallel displacement, denoted by X.t/ is a vector field along satisfying the equation r.t/ P X.t / D 0 for all t in the domain of the curve. This is a linear ordinary differential equation, and thus, for any curve the map P.t/ W T.0/ M ! T.t/ M X 7! P.t/ .X / WD X.t / is a vector space isomorphism which is called parallel displacement. Hence, r enables us to link the tangent spaces in different points, which is the reason why it bears the name connection. Then the holonomy group of r at p is the group defined by parallel displacements along loops about this point, Holp .M; r/ WD fP.1/ j .0/ D .1/ D pg: This group is a Lie group which is connected if the manifold is simply connected. Its connected component is called connected holonomy group, it is denoted by Holp0 .M; r/, and it is the group generated by parallel displacements along homotopically trivial loops. Its Lie algebra holp .M; r/ is called holonomy algebra. Obviously, both are given together with their representation on the tangent space Tp M which is usually identified with the Rm . In this sense Holp .M; r/ GL.m; R/. Holonomy groups at different points in the manifold are conjugated by an element in GL.m; R/, which is obtained by the parallel displacement along a curve joining these different points. It is worthwhile to note that the holonomy group is closed if it acts irreducibly (for a proof of this fact see [65] or [29]). This is not true in general, there are examples of non-closed holonomy groups. The main result on which the calculation of holonomy groups is based is the Ambrose–Singer holonomy theorem. It states that for a connected affine manifold the holonomy algebra is given by the following ˚ 1 holp .M; r/ D span P.t/ ı R.P.t/ X; P.t/ Y / ı P .t / j .0/ D p; X; Y 2 Tp M ; for R being the curvature of r, R.X; Y / D ŒrX ; rY rŒX;Y . For connections that have no torsion, i.e. rX Y rY X D ŒX; Y , the curvature satisfies the first Bianchiidentity R.X; Y /Z C R.Y; Z/X C R.Z; X /Y D 0; which imposes very strong algebraic conditions on the holonomy algebra. Later on we will explain this more thoroughly. These conditions make it possible to classify holonomy groups of torsion-free linear connections which act irreducibly. This was

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done recently by S. Merkulov and L. Schwachhöfer [56], [59], [60]. We refer to this list of irreducible holonomy groups of torsion-free connections as the Schwachhöfer– Merkulov list. It extends the well-known Berger list of irreducible holonomy groups of semi-Riemannian manifolds (c.f. next section). However, the assumption of irreducibility is essential for these classification results because their proof uses the theory of irreducible representations of Lie algebras. Because of the following result of J. Hano and H. Ozeki [42] a classification problem for holonomy groups only arises if one poses further conditions on the linear connection, such as conditions on the torsion: Any closed subgroup of GL.m; R/ can be obtained as a holonomy group of a linear connection, but possibly a connection with torsion. They also gave examples of holonomy groups which were not closed. We will return to this question later. Concluding this introductory section we want to point out a general principle in holonomy theory which says that any subspace which is invariant under the holonomy group corresponds to a distribution (i.e. a sub-bundle of the tangent bundle) which is invariant under parallel transport. Obviously, this distribution is obtained by parallel transporting the invariant subspace, and this procedure is independent of the chosen path because of the holonomy invariance of the subspace. This distribution is called parallel, which means that sections of it are mapped onto sections of it under rX for any X 2 TM . 2.2 Holonomy groups of semi-Riemannian manifolds. If .M; g/ is a semi-Riemannian manifold of dimension m D r C s and signature .r; s/ then there a uniquely defined linear torsion-free connection r D r g which parallelises the metric, called Levi-Civita connection. The holonomy group of a semi-Riemannian manifold then is the holonomy group of this connection, Holp .M; g/ WD Holp .M; r/. As the Levi-Civita connection is metric, the parallel displacement preserves the metric. This implies on the one hand that the holonomy group is a subgroup of O.Tp M; g/ and – by fixing a basis in Tp M – can be understood as a subgroup of O.r; s/, which is only defined up to conjugation in O.r; s/. On the other hand it ensures that for a subspace V Tp M which is invariant under the holonomy group the orthogonal complement V ? is invariant as well. Hence, for a Riemannian metric the holonomy group acts completely reducibly, i.e. the tangent space decomposes into subspaces on which it acts trivially or irreducibly, but for indefinite metrics the situation is more subtle. We say that the holonomy group acts indecomposably if the metric is degenerate on any invariant proper subspace. In this case we also say that the manifold is indecomposable. Of course, for Riemannian manifolds, this is the same as irreducibility. A remarkable property is that the holonomy group of a product of Riemannian manifolds (i.e. equipped with the product metric) is the product of the holonomy groups of these manifolds (with the corresponding representation on the direct sum). Even more remarkable is the fact that a converse of this statement is true in the following sense: Any semi-Riemannian manifold whose tangent space at a point admits a decomposition into non-degenerate, holonomy-invariant subspaces is locally isometric to a


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product of semi-Riemannian manifolds corresponding to the invariant subspaces, and moreover, the holonomy group is a product of groups acting on the corresponding invariant subspaces. These groups are the holonomy groups of the manifolds in the local product decomposition if the original manifold is complete (see [15, Theorem 10.38 and Remark 10.42]). This was proven by A. Borel and A. Lichnerowicz [17], and the property that a decomposition of the representation space entails a decomposition of the acting group is sometimes called Borel–Lichnerowicz property. A global version of this statement was proven under the assumption that the manifold is simply-connected and complete by G. de Rham [28, for Riemannian manifolds] and H. Wu [67, in arbitrary signature]. Summarising we have the following result: Theorem 2.1 (G. de Rham [28] and H. Wu [67]). Any simply-connected, complete semiRiemannian manifold .M; g/ is isometric to a product of simply connected, complete semi-Riemannian manifolds one of which can be flat and the others have an indecomposably acting holonomy group and the holonomy group of .M; g/ is the product of these indecomposably acting holonomy groups. The other groundbreaking result in holonomy theory is the list of irreducible holonomy groups of non locally-symmetric semi-Riemannian manifolds, which was obtained by M. Berger [13] and completed by several other authors [63], [2], [21], [22]. Theorem 2.2 (M. Berger [13]). Let .M; g/ be a simply connected semi-Riemannian manifold of dimension m D r C s and signature .r; s/, which is not locally-symmetric. If the holonomy group of .M; g/ acts irreducibly, then it is either SO0 .r; s/ or one of the following .modulo conjugation in O.r; s//: U.p; q/ or SU.p; q/ SO.2p; 2q/; m 4; Sp.p; q/ or Sp.p; q/ Sp.1/ SO.4p; 4q/; m 8; SO.r; C/ SO.r; r/;

m 4;

Sp.p; R/ SL.2; R/ SO.2p; 2p/; m 8; Sp.p; C/ SL.2; C/ SO.4p; 4p/; m 16; G2 SO.7/; G2.2/

SO.4; 3/;

G2C SO.7; 7/; Spin.7/ SO.8/; Spin.4; 3/ SO.4; 4/; Spin.7/C SO.8; 8/: In case of symmetric spaces with irreducible holonomy group, the holonomy group is equal to the isotropy group of the symmetric space, which were classified by E. Cartan [27, for Riemannian signature] and [14, for arbitrary signature].

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For Riemannian manifolds one can combine this result with the de Rham decomposition and obtains a comprehensive holonomy classification. Theorem 2.3. Any simply-connected, complete Riemannian manifold .M; g/ is isometric to a product of simply-connected, complete Riemannian manifolds one of which may be flat and the others are either locally symmetric or have one of the following groups as holonomy, SO.n/, U.n/, SU.n/, Sp.n/, Sp.n/ Sp.1/, G2 , or Spin.7/. The holonomy group of .M; g/ is a product of these groups. For indefinite metrics there is the possibility that one of the factors in Theorem 2.1 is indecomposable, but non-irreducible, i.e. has a degenerate invariant subspace. An attempt to classify holonomy groups for indefinite metric has to provide a classification of these indecomposable, non-irreducible holonomy groups. 2.3 Lorentzian holonomy groups. It is remarkable that the Berger list in Theorem 2.2 implies that the only irreducible holonomy group of Lorentzian manifolds is the full SO0 .1; n/. Stating it this way conceals that this result has nothing to do with holonomy but is due to the algebraic fact that the only connected irreducible subgroup of O.1; n/ is SO0 .1; n/ which was proven by A. J. Di Scala and C. Olmos [30] (see also [19], [10], [29] for other proofs). Hence, if one is interested in Lorentzian manifolds with special holonomy, i.e. with proper subgroups of SO0 .1; n/ as holonomy, one has to look at manifolds admitting a holonomy-invariant subspace. The decomposition due to Theorem 2.1 gives the following result for Lorentzian manifolds. Corollary 2.4. Any simply-connected, complete Lorentzian manifold is isometric to the following product of simply-connected complete semi-Riemannian manifolds, .M; h/ .M1 ; g1 / .Mk ; gk /; where the .Mi ; gi / are either flat or irreducible Riemannian manifolds and .M; h/ is either .R; dt 2 / or an indecomposable Lorentzian manifold, the holonomy of which is either SO0 .1; n/ or contained in the stabiliser of a light-like line. This decomposition obviously follows from the de Rham–Wu decomposition of Theorem 2.1. If the Lorentzian factor is not flat, it is indecomposable in which case it can be irreducible, and hence with holonomy SO0 .1; n/, or not. The latter means that it has an degenerate invariant subspace V . The intersection of this subspace with its orthogonal complement V ? yields a light-like line which is holonomy invariant. Hence, the holonomy group of .M; h/ is contained in the stabiliser of this light-like line. We should point out that, due to the general principle mentioned above, this holonomy-invariant light-like line corresponds to a distribution „ of light-like lines which are invariant under parallel transport. Locally, this distribution is spanned by a recurrent light-like vector field. A vector field X is called recurrent if there is a one-form such that rX D ˝ X: If d D 0, e.g. if the length of X is not zero, X can be re-scaled to


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a parallel vector field. If the light-like vector field is not only recurrent but parallel, the holonomy group is contained in the stabiliser of a light-like vector. A parallel distribution „ of light-like lines equips a Lorentzian manifold .M; g/ with further, holonomy related structure. As „ is light-like, it is contained in its orthogonal complement „? . This enables us to define the vector bundle S whose fibres are the quotients „p? =„p , and equip it with a metric g S induced by the Lorentzian metric g. Since both distributions are parallel, the Levi-Civita connection of g equips S also with a metric connection r S . Definition 2.5. If .M; g; „/ is a Lorentzian manifold with a parallel distribution „ of light-like lines, the vector bundle .S; g S ; r S / is called screen bundle of .M; g; „/. The holonomy group of the vector bundle connection r S is called screen holonomy group. We introduced the screen holonomy in [52] and studied it further in [54]. We will return to it in the next sections. We can summarise the situation as follows: In order to classify Lorentzian holonomy groups, we have to classify Lorentzian holonomy groups which act indecomposably, but non-irreducibly. If Holp .M; g/ is such a connected holonomy group, it is contained in the stabiliser in SO0 .Tp M / of a light-like line in Tp M . This stabiliser can be understood as a parabolic group in the conformal group SO0 .1; n C 1/ if n C 2 is the dimension of M . In the following sections we will describe further the Lie algebra of this group and its indecomposable subalgebras and, more importantly, derive some holonomy related conditions.

3 Algebraic aspects of non-irreducible, indecomposable Lorentzian holonomy 3.1 Basic algebraic properties. We consider the Minkowski space R1;nC1 of dimension n C 2 and fix a basis .X; E1 ; : : : ; En ; Z/ in which the scalar product has the form 1 0 0 0t 1 @0 In 0A ; (1) 1 0t 0 where In is the n-dimensional identity matrix. The connected stabiliser of R X is a parabolic group P inside the conformal group SO0 .1; n C 1/. Its Lie algebra p can be written as follows: 80 9 1 ˇ < a vt 0 = ˇ ˇ p D @0 A v A ˇ a 2 R; v 2 Rn ; A 2 so.n/ : (2) : ; ˇ t 0 0 a This Lie algebra is a semi-direct sum in an obvious way, p D .R ˚ so.n// Ë Rn , the commutator relations are given as follows: Œ.a; A; v/ ; .b; B; w/ D .0; ŒA; Bso.n/ ; .A C a Id/ w .B C b Id/ v/:

(3)

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In this sense we will refer to R, Rn and so.n/ as subalgebras of p. R is an abelian subalgebra of p, commuting with so.n/, and Rn an abelian ideal in p. so.n/ is the semisimple part and co.n/ D R ˚ so.n/ the reductive part of p. Now one can assign to a subalgebra h p the projections pr R .h/, pr Rn .h/ and pr so.n/ .h/. The subalgebra g WD pr so.n/ .h/ associated to a h is called the orthogonal part of h. Then the following properties are obvious. Lemma 3.1. If h is an indecomposably acting subalgebra of p, then 1. 2. 3. 4.

pr Rn .h/ D Rn , h has a trivial sub-representation if and only if pr R h D 0, h is abelian if and only if h D Rn , g WD pr so.n/ .h/ is compact and hence reductive.

A Lie algebra is compact if there exists a positive definite invariant symmetric bilinear form on it. Subalgebras of compact algebras are compact, in particular, subalgebras of so.n/ are compact. g is called reductive if its Levi decomposition is g D z˚g0 where z is the centre of g and g0 WD Œg; g is the derived Lie algebra, which is semisimple. In compact Lie algebras we have that g0 D z? , hence compact Lie algebras are reductive. 3.2 Four types of indecomposable subalgebras of p. An important result about indecomposable subalgebras of p is their distinction into four types due to the relation between their projections which was obtained by L. Bérard-Bergery and A. Ikemakhen [11]. This is stated as Theorem 1.1 of the Introduction. This distinction obviously gives four types for the corresponding connected, indecomposable groups in the parabolic P We should remark that these types are independent of conjugation within O.1; n C 1/. The proof of Theorem 1.1 given by L. Bérard Bergery and A. Ikemakhen was purely algebraic. We describe now a more geometric proof of this theorem given in [34], which works directly for the groups and provides a geometric interpretation for the different types. Let P be the parabolic subgroup in the conformal group SO0 .1; nC1/ corresponding to the Lie algebra p defined above, and let RC , SO.n/, and Rn be the connected subgroups in P corresponding to the subalgebras R, so.n/, and Rn in p. Then P is a semi-direct product, P D .RC SO.n// Ë Rn : Now we equip the Rn with the Euclidean scalar product. Denote by Sim.n/ the connected component of the Lie group of similarity transformations of Rn . Then RC , SO.n/, and Rn in Sim.n/ are the connected identity components of the Lie groups of homothetic transformations, rotations and translations, respectively. We obtain for Sim.n/ the same decomposition as for P , i.e. we have a Lie group isomorphism W P ! Sim.n/: The isomorphism can be defined geometrically. For this consider the vector model of the real hyperbolic space H nC1 R1;nC1 and its boundary @H nC1 P R1;nC1 that consists of isotropic lines of R1;nC1 and is isomorphic to the n-dimensional


63

sphere. Any element f 2 P induces a transformation .f / of the Euclidean space @H nC1 nfR Xg ' Rn . In fact, .f / is a similarity transformation of Rn . This defines the isomorphism . Now, in [34] we have proven that a connected Lie subgroup H P is indecomposable if and only if the subgroup .H / Sim.n/ acts transitively on Rn . Then, using a description for connected transitive subgroups of Sim.n/ given in [3] and [4], one can show the following theorem. Theorem 3.2. A connected Lie subgroup H Sim.n/ is transitive if and only if H belongs to one of the following types for which G SO.n/ is a connected Lie subgroup: Type 1: H D .RC G/ Ë Rn ; Type 2: H D G Ë Rn ; n Type 3: H D .Rˆ C G/ËR , where ˆ W RC ! SO.n/ is a non-trivial homomorphism and Rˆ C D fa ˆ.a/ j a 2 RC g RC SO.n/ is a group of screw dilations of Rn that commutes with G; Type 4: H D .G .Rnm /‰ / Ë Rm ; where 0 < m < n, Rn D Rm ˚ Rnm is an orthogonal decomposition, ‰ W Rnm ! SO.m/ is a homomorphism with ker d‰ D f0g, and .Rnm /‰ D f‰.u/ u j u 2 Rnm g SO.m/ Rnm is a group of screw isometries of Rn that commutes with G. The indecomposable Lie algebras of the corresponding Lie subgroups of P have the same type as in Theorem 1.1. 3.3 Berger algebras and weak-Berger algebras. Here we will recall the notion of weak-Berger algebras which we have introduced in [48] where also their basic properties presented in this section are proven. Let K be the real or complex numbers. For a subalgebra g gl.n; K/ we define the following space by the Bianchi-identity, K.g/ WD fR 2 ƒ2 .Kn / ˝ g j R.x; y/z C R.y; z/x C R.z; x/y D 0g; gK WD spanfR.x; y/ j x; y 2 Kn ; R 2 K.g/g; and for g orthogonal, i.e. g so.n; C/ or g so.r; s/ we set B.g/ WD fQ 2 .Kn / ˝ g/ j hQ.x/y; zi C hQ.y/z; xi C hQ.z/x; yi D 0g; gB WD spanfQ.x/ j x 2 Kn ; Q 2 B.g/g; where h ; i is the corresponding scalar product. Both, K.g/ and B.g/ are spaces of curvature endomorphisms. For distinction, one may call B.g/ the space of weak curvature endomorphisms. Both are g-modules. gK and gB are ideals in g. For g orthogonal, a curvature endomorphism R 2 K.g/ enjoys more symmetries than only the Bianchi-identity, namely hR.x; y/u; vi D hR.x; y/v; ui;

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and thus hR.x; y/u; vi D hR.u; v/x; yi: Definition 3.3. Let g gl.n; K/ be a subalgebra. g is called Berger algebra if gK D g. g so.n; C/ or g 2 so.r; s/ with gB D g, is called weak-Berger algebra. Equivalent to the (weak-)Berger property is the fact that there is no proper ideal h in g such that K.h/ D K.g/ (respectively B.h/ D B.g/). The following lemma relates both curvature endomorphism modules to each other. Its proof is straightforward. Lemma 3.4. The vector space R.g/ spanned by fR.x; / 2 B.g/ j R 2 K.g/; x 2 Kn g is a g-sub-module of B.g/. This implies gK gB , so we get a justification of the nomenclature. Proposition 3.5. Every orthogonal Berger algebra is a weak-Berger algebra. For a weak-Berger algebra the Bianchi-identity which defines B.g/ yields a decomposition property similar to the Borel–Lichnerowicz property mentioned in Section 2.1. Theorem 3.6. Let g so.n/ be a weak-Berger algebra. To the decomposition of Rn into invariant subspaces Rn D E0 ˚ E1 ˚ ˚ Ek , where E0 is a trivial submodule and the Ei are irreducible for i D 1; : : : ; k, corresponds a decomposition of g into ideals g D g1 ˚ ˚ gk such that gi acts irreducibly on Ei and trivially on Ej . Each of the gi so.dimEi / is a weak Berger algebra and it holds that B.g/ D B.g1 / ˚ ˚ B.gk /. If Rn D E0 ˚ E1 ˚ ˚ Ek be the complete decomposition of Rn under g then the gi ’s are given by gi WD spanfQ.xi / j Q 2 B.g/; xi 2 Ei g which are therefore weak-Berger algebras by definition and span g. Of course, they are ideals and the Bianchi-identity which defines weak-Berger algebras then ensures that gi acts trivial on Ej and that gi \ gj D f0g for i 6D j . This verifies the statement. We should point out that the same statement holds for Berger algebras for a complete decomposition of Rn into g-invariant subspaces. By the Ambrose–Singer holonomy theorem holonomy algebras of torsion free connections – in particular of a Levi-Civita-connection – are Berger algebras. The list of all irreducible Berger algebras is known [13], [14], [56]. The so.n/-projection of an indecomposable, non-irreducible Lorentzian manifold a priori is no holonomy algebra, and therefore not necessarily a Berger algebra. But we will show that it is a weak-Berger algebra. Before we recall this result we will give a description of the space of curvature endomorphism of an indecomposable h p. 3.4 Decomposition of the space of curvature endomorphisms. If we now consider a subalgebra h p so.1; n C 1/ we can exploit the symmetries of the curvature endomorphisms K.h/ together with the description of the four types in Theorem 1.1 in order to obtain a further description of K.g/ which involves the space B.g/. This was done in [37], where we proved the following theorem, for which we fix a basis .X; E1 ; : : : ; En ; Z/ of the Minkowski space R1;nC1 as in Section 3.1.


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Theorem 3.7. Let h be a subalgebra of the parabolic algebra p in so.1; n C 1/ and g D z ˚ g0 its orthogonal part. Then it holds: (1) Any R 2 K.h/ is uniquely given by 2 R; L 2 .Rn / ; Q 2 B.g/; R0 2 K.g/; and T 2 End.Rn / with T D T in the following way: R.X; Z/ D .; 0; L .1//; R.U; V / D .0; R0 .U; V /; 12 Q .U ^ V // R.U; Z/ D .L.U /; Q.U /; T .U //; R.X; U / D 0; where U; V 2 span.E1 ; : : : ; En /. (2) If h is indecomposable of type 2, any R 2 K.h/ is given as in .1/ with D 0 and L D 0. (3) If h is indecomposable of type 3 defined by the epimorphism ' W z ! R, any R 2 K.h/ is given as in .1/ with D 0, L D 'z ı Q and R0 2 K.g0 ˚ ker '/, where 'z is the extension of ' to g set to zero on g0 . (4) If h is of type 4 defined by the epimorphism W z ! Rnk , any R 2 K.h/ is given as in .1/ with D 0, L D 0, pr Rnk ıT D z ıQ and R0 2 K.g0 ˚ker /, where z is the extension of to g set to zero on g0 . Here U ^ V denotes the identification of ƒ2 with so.n/ and denotes the adjoint with respect to the scalar product h ; i in Rn and the Killing form in so.n/. In particular, T is a symmetric matrix and Q W g ! Rn is given by Q .U ^ V / D Pn 2 iD1 hQ.Ei /U; V iEi . Using Theorem 3.7, Lemma 3.4, and Proposition 3.5 one obtains the following consequence. Corollary 3.8. An indecomposable subalgebra h p is a Berger algebra if and only if its orthogonal part g so.n/ is a weak-Berger algebra. The ‘only if’-direction of this corollary was proved in [48] independently of the description of the space of curvature endomorphisms by restricting the Bianchi-identity to the space span.E1 ; : : : ; En /. 3.5 Consequences for Lorentzian holonomy. Now we want to draw the consequences for an indecomposable, non-irreducible Lorentzian holonomy group from the previous sections. To this end let p 2 M be a point in an indecomposable, non-irreducible Lorentzian manifold .M; h/ and let „p Tp M be the fibre of the parallel light-like distribution of lines, i.e. „p is the holonomy-invariant light like line. Now we fix a basis .X; E1 ; : : : ; En ; Z/ in Tp M such that the metric hp is of the form (1) and X 2 „p . Then the holonomy algebra holp .M; h/ is contained in the parabolic algebra p defined above. In this setting we can apply the results of the previous sections. In particular,

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the (connected) holonomy group H WD Holp .M; h/ belongs to one of the four types corresponding to the characterisation of the Lie algebra h in Theorem 1.1. The Lie group corresponding to the orthogonal part g is denoted by G SO.n/. If h is of uncoupled type 1 or 2, then we have either H D .RC G/ Ë Rn ;

or

H D G Ë Rn ;

(4)

respectively. If h is of one of the coupled types 3 or 4 it is defined by an epimorphism ' W z ! R or W z ! Rl where z is the centre of g due to Theorem 1.1. For type 3 we have that (5) H D L G 0 Ë Rn ; where G 0 is the Lie group corresponding to the derived Lie algebra g0 of g and L is the Lie group corresponding to the graph of '. For type 4 we get that H D L G 0 Ë Rnl ; where L is the Lie group corresponding to the graph of The first proposition is an obvious consequence.

(6) .

Proposition 3.9. A Lorentzian manifold with indecomposable, non-irreducible holonomy group H admits a parallel light-like vector field if and only if pr RC .H / D 0, i.e. if and only if its Lie algebra is of type 2 or 4. The second consequence relates the screen holonomy and the orthogonal part. It was proven in [52], [54]. Proposition 3.10. The orthogonal part of an indecomposable, non-irreducible Lorentzian holonomy group is equal to the screen holonomy. This result enables us to describe algebraic properties of the orthogonal part of the holonomy by invariant structures of the screen bundle. E.g., if the orthogonal part is contained in the unitary group, then there is a parallel complex structure on the screen bundle. The main result is the one we obtain from Corollary 3.8 by applying the Ambrose– Singer holonomy theorem. Theorem 3.11. The orthogonal part g of an indecomposable, non-irreducible Lorentzian holonomy algebra is a weak-Berger algebra. In particular, g decomposes into irreducibly acting weak-Berger algebras as in Theorem 3.6. This theorem has several important consequences. Not only gives it an algebraic criterion for the orthogonal part from which a classification attempt can start, it also provides a proof of the Borel–Lichnerowicz decomposition property for the orthogonal part proved by L. Bérard-Bergery and A. Ikemakhen [11, Theorem II], which is given in our Theorem 3.6. This ensures that we are at a similar point as in the Riemannian situation,

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but reaching it in a different way. This is shown schematically in the following diagram. PP geometric level:

gD

PP P

pr so.n/ hol Ambrose–Singer + Bianchi-identity

g D hol

PP PP

g D g1 ˚ ˚ g r ; with gi D holi

de Rham-

Theorem 3.11

?

algebraic level:

?

? -

Corollary 3.6 g weak-Berger

PP

PP gi irreducible gi irreduciblePP Berger PP weak-Berger P

Theorem 3.11 also has implication for algebras of coupled type 4. They were defined by an epimorphism W z ! Rp for 0 < p < n where z is the centre of the orthogonal part g. If Rn decomposes as Rn D Rn0 ˚ Rn1 ˚ ˚ Rns ; where g acts irreducibly on the Rni and trivial on Rn0 , inducing the decomposition of g D g1 ˚ ˚gs as in Theorem 3.6, then, first of all, we have that 0 < p n0 < n1. Moreover, as the gi ’s act irreducibly, their centre has to be at most one-dimensional. Since is surjective this implies that 0 < p s. In particular, Type 4 only occurs for n 3, i.e. dim M 5. Before we explain the classification of weak-Berger algebras we want to conclude this section by implications of Theorem 3.11 for the closedness of indecomposable, non-irreducible holonomy groups, which were obtained in [11]. Let H be an indecomposable, non-irreducible Lorentzian holonomy group and, as above, let G be its orthogonal part, and let h and g be the corresponding Lie algebras. For the uncoupled types 1 and 2 it depends only on G if H is closed. But due to Theorem 3.11, G is a product of irreducibly and orthogonally acting Lie groups, which are closed. Therefore G, and thus H is closed in this case. If H is of one of the coupled types 3 or 4 it is defined by an epimorphism ' W z ! R or W z ! Rl . For these types, H is closed if and only if L as in formulae (5) and (6) is closed. But L is closed if and only if its intersection with the torus Z, which is the centre of G, is closed. We can summarise this in the following result obtained in [11]. Corollary 3.12. If the Lie algebra of an indecomposable, non-irreducible Lorentzian holonomy group H is of type 1 and 2, then H is closed. If it is of type 3 or 4, defined by an epimorphism ', then H is closed if and only if the Lie group generated by the subalgebra ker.'/ is a compact subgroup of the torus. This corollary implies that holonomy groups of Lorentzian manifolds of dimension less or equal to 5 are closed.

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4 The classification of weak-Berger algebras 4.1 Real and complex weak-Berger algebras. As we will use representation theory of complex semisimple Lie algebras, we have to describe the transition of a real weakBerger algebra to its complexification. The spaces K.g/ and B.g/ for g so.r C s/ are defined by the following exact sequences: / K.g/

0 0

/ ƒ2 .Rn / ˝ g

/ B.g/

/ .Rn / ˝ g

/ ƒ3 .Rn / ˝ Rn ;

/ ƒ3 .Rn / ;

where the map is the skew-symmetrisation and the dualisation by the scalar product and the skew-symmetrisation. If we consider a real Lie algebra g acting orthogonally on Rn , then the scalar product extends by complexification to a complex-linear scalar product which is invariant under gC , i.e. gC so.r C s; C/. The complexification of the above exact sequences gives K.g/C D K.gC /; C

C

B.g/ D B.g /

(7) (8)

and leads to the following statement. Proposition 4.1. A subalgebra g so.r; s/ is a .weak-/ Berger algebra if and only if gC so.r C s; C/ is a .weak-/ Berger algebra. Thus complexification preserves the weak-Berger as well as the Berger property. But irreducibility is a property which is not preserved under complexification. In order to deal with this problem we have to recall briefly the following distinctions (for details of the following see [48] or [55]). Let g so.n/ be a real orthogonal Lie algebra which acts irreducibly on Rn . Then one can consider the complexification of this representation, i.e. the representation of the real Lie algebra g on C n given by g so.n/ so.n; C/. This representations can still be irreducible, in which case we say that the representation of g, or simply that g is of real type, or it can be reducible and we say it is of unitary type. In the second case n D 2k has to be even and C n decomposes into two g-invariant subspaces, C n D C k ˚ C k for which we obtain that g u.k/ is unitary and irreducible. Note that this implies that g 6 so.k; C/. This distinction was made by E. Cartan [26] (see also [46] and [39]) for arbitrary irreducible real representations, where these are called “representations of first type” and of “second type”. Now we complexify also the Lie algebra g because we want to use the tools of the theory of irreducible representations of complex Lie algebras. Of course, it holds that g so.n/ is irreducible if and only if gC so.n; C/ is irreducible. Hence, for an irreducible g so.n/ we end up with two cases: If g is of real type, then gC so.n; C/ is irreducible, or if g is of unitary type, i.e. n D 2k, then gC gl.k; C/ with gC 6 so.k; C/.


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4.2 Weak-Berger algebras of unitary type. In this section we will present the main steps in the proof that any unitary weak-Berger algebra is a Berger algebra. In order to avoid confusion we have to introduce some subtleties in the notation. Let g0 so.2n/ be of unitary type, i.e. g0 u.n/. As seen above gC 0 so.2n; C/ does not act irreducibly, but C 2n decomposes into gC -invariant subspaces C n and C n . We denote by g the complex Lie algebra gC acting irreducibly on C n , in contrast to the nonirreducible action of gC on C 2n . W.r.t. this decomposition of C 2n , the scalar product is given by the matrix 0 In : In 0 This means that hu; vi D 0 for u; v 2 C n and the g0 -invariant Hermitian form on N These facts imply for Q 2 B.gC C n is given by .u; v/ D hu; vi. 0 / that .Q.u/v; w/ D hQ.u/v; wi x D hQ.v/w; x ui hQ.w/u; x vi D .Q.v/u; w/; „ ƒ‚ … D0

i.e. Q.u/v D Q.v/u. Hence, the Bianchi-identity implies that Q is contained in the first prolongation g.1/ D fQ 2 Hom.C n ; g/ j Q.u/v D Q.v/ug of g gl.n; C/. Thus, there is a mapping .1/ B.gC 0 / ! g ; Q 7! QjC n C n :

(9)

For this situation, in [48] we proved the following statement. Proposition 4.2. Let g0 u.n/ so.2n/ of unitary type, g gl.n/ defined as above. .1/ Then (9) defines an isomorphism between B.gC 0 / and g . An analogous result can be proven for the space K.g/ (see [48]). We obtain two corollaries. Corollary 4.3. Let h0 ¤ g0 u.n/ so.2n/ of unitary type, h and g defined as above. If h.1/ D g.1/ , then g0 cannot be a weak-Berger algebra. Corollary 4.4. Let g0 u.n/ so.2n/ of unitary type, g gl defined as above. n .1/ Then gC 0 is a weak-Berger algebra if and only if g D spanfQ.u/ j u 2 C ; Q 2 g g Analogous results can be obtained for Berger algebras leading – with the same reasoning as below – to a classification of irreducible Berger algebras of non-real type [48]. As a result of the previous and this section we have to investigate complex irreducible representations of complex Lie algebras with non-vanishing first prolongation. There

70


are only a few complex Lie algebras irreducibly contained in gl.n; C/ which have non vanishing first prolongation. The classification is due to E. Cartan [25] and S. Kobayashi and T. Nagano [47]. We collect them in two tables, following [60]. Table 1. Complex Lie algebras with g.1/ 6D 0 and g.1/ 6D V .

V

g.1/

Cn; n 2 Cn, n 1 C 2n , n 2 C 2n , n 2

.V ˝ ˇ2 V /0 V ˝ ˇ2 V ˇ3 V ˇ3 V

g sl.n; C/ gl.n; C/ sp.n; C/ C ˚ sp.n; C/

1. 2. 3. 4.

Table 2. Complex Lie algebras with first prolongation g.1/ D V .

V

g n

1. 2. 3. 4.

co.n; C/ gl.n; C/ gl.n; C/ sl.gl.n; C/ ˚ gl.m; C//

C , n3 ˇ2 C n , n 2 ^2 C n , n 5 n C ˝ C m , m; n 2

5. 6.

C ˚ spin.10; C/ C ˚ e6

16 C 10 ' C C 27

For a details of these representations see [1]. Regarding Table 1, its first three entries are complexifications of the Riemannian holonomy algebras su.n/, u.n/ acting on R2n and sp.n/ acting on R4n , which are Berger algebras. The fourth has the compact real form so.2/ ˚ sp.n/ acting irreducibly on R4n . Since the representation of sp.n/ on R4n is of unitary type we are in the situation of Corollary 4.3, because .C Id ˚sp.n; C//.1/ sp.n; C/.1/ . Hence, so.2/˚sp.2n/ is not a weak-Berger algebra. Considering at the compact real forms of the Lie algebras and the reellification of the representations in Table 2, one sees that they correspond to the holonomy representations of Riemannian symmetric spaces which are Kählerian (for a detailed proof see the appendix of [48]), i.e. to the symmetric space of type BD I, C I, D III, A III, E III and E VII. We can summarise this result as follows. Theorem 4.5. If g u.n/ so.2n/ is an irreducible weak-Berger algebra of unitary type, then it is a Riemannian holonomy algebra, in particular a Berger algebra. 4.3 Weak-Berger algebras of real type. In this situation we start with an irreducible Lie algebra g0 so.n/ such that g WD gC so.n; C/ is irreducible as well. The first thing to notice is that by the Schur lemma g has no center, and thus g is not only


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reductive but semisimple. Considering the four different types of indecomposable, nonirreducible holonomy algebras from Theorem 1.1, this fact already yields the following observation. Corollary 4.6. Let g so.n/ be the so.n/-projection of an indecomposable, nonirreducible Lorentzian holonomy algebra, which is supposed to be of coupled type 3 or 4. Then at least one of the irreducibly acting ideals of g so.n/ is of non-real type. As we found that g is semisimple, we may use all the tools provided by roots and weights. In the following we will sketch how to transform the weak-Berger property into conditions on roots and weights of the corresponding representation (for the proofs see [50], [51], [55]). Let t be the Cartan subalgebra of g, t be the roots of g, and set 0 WD [ f0g. g decomposes into its root spaces g˛ WD fA 2 g j ŒT; A D ˛.T / A for all T 2 tg 6D f0g: M gD g˛ ; where g0 D t. ˛20

Let t be the weights of g so.n; C/. Then C n decomposes into weight spaces V WD fv 2 V j T .v/ D .T / v for all T 2 tg 6D f0g, M V D V : 2

As g so.n; C/, the weight spaces of are related as follows. Proposition 4.7. Let g so.n; C/ be a complex, semisimple Lie algebra with weight space decomposition. Then V ?V if and only if 6D . In particular, if is a weight, then too. If g is a weak-Berger algebra, then B.g/ is a non-zero g–module. If we denote by … its weights, it decomposes into weight spaces M B.g/ D B : 2…

Now we define a subset of t , ˇ ˇ 2 ; 2 … and there is an u 2 V WD C ˇ t : and a Q 2 B such that Q.u/ 6D 0 It is not difficult to see that 0 . But for weak-Berger algebras also the other inclusion is true. In fact, Q .u / 2 gC implies that M gB D spanfQ .u / j C 2 g gˇ : ˇ 2

We obtain:

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Proposition 4.8. If g so.n; C/ is an irreducible, semisimple Lie algebra which is weak-Berger, then D 0 . For a root ˛ 2 we denote by ˛ the following subset of : ˛ WD f 2 j C ˛ 2 g : Then ˛ C ˛ are the weights of g˛ V . Then the weak-Berger property can be expressed as follows. Proposition 4.9. Let g be a semisimple Lie algebra with roots and 0 D [f0g. Let g so.n; C/ irreducible, weak-Berger with weights . Then the following properties are satisfied: (PI) There is a 2 and a hyperplane U t such that f C ˇ j ˇ 2 0 g [ U [ f C ˇ j ˇ 2 0 g :

(10)

(PII) For every ˛ 2 there is a ˛ 2 such that ˛ f ˛ ˛ C ˇ j ˇ 2 0 g [ f ˛ C ˇ j ˇ 2 0 g :

(11)

Of course, it is desirable to find weights and ˛ which are extremal in order to handle criteria (PI) and (PII). To this end we notice that C n as an irreducible g-module is generated as vector space by elements of the form A1 : : : Ak u with Ai 2 g and k 2 N (see for example [61]) which yields the following fact. Lemma 4.10. Let g so.n; C/ be an irreducible, complex semisimple Lie algebra with B.g/ 6D 0. Then for any extremal weight vector u 2 Vƒ there is a weight element Q 2 B.g/ such that Q.u/ 6D 0. Finally, we obtain: Proposition 4.11. Let g so.n; C/ be an irreducibly acting, semisimple weak-Berger algebra with roots and 0 D [ f0g and weights . Then there is a partial order of .i.e. a set of simple roots/ with the following property. If ƒ is the highest weight of g so.n; C/ with respect to this partial order, then (Q I) there is a ı 2 C [ f0g and a hyperplane U t such that fƒ ı C ˇ j ˇ 2 0 g [ U [ fƒ C ı C ˇ j ˇ 2 0 g :

(12)

If ı cannot be chosen to be zero, then (Q II) there is an ˛ 2 such that ˛ fƒ ˛ C ˇ j ˇ 2 0 g [ fƒ C ˇ j ˇ 2 0 g :

(13)

Example 4.12. Representations of sl.2; C/. To illustrate how these criteria will work we apply them to irreducible representations of sl.2; C/, for which we obtain the following statement: Let V be an irreducible, complex, orthogonal sl.2; C/-module of highest weight ƒ. If it is weak-Berger then ƒ 2 f2; 4g.


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Proof. Let sl.2; C/ so.N; C/ be an irreducible representation of highest weight ƒ. I.e. ƒ.H / D l 2 N for sl.2; C/ D span.H; X; Y / where X has the root ˛. Since the representation is orthogonal, l must be even (see for example [64]) and 0 is a weight. The hypersurface U is the point 0. Now property (10) ensures that l 2 f2; 4; 6g. If

D ƒ we obtain l 2 f2; 4g. If 6D ƒ we can apply (QII): We have that ˛ D nfƒg and ˛ D n fƒg. Then (QII) implies l 2 f2; 4g. The conditions given in Proposition 4.11 still are not the ones which will be exploited in order to find the roots and weights of weak-Berger algebras. To this end we shall describe a result of [59], and [60], where irreducible Berger algebras are classified. For a Berger algebra holds that for every ˛ 2 0 there is a weight element R 2 K.g/ and weight vectors u1 2 V1 and u2 2 V2 such that 0 6D R.u1 ; u2 / 2 g˛ . Choosing u1 ; u2 such that 0 6D R.u1 ; u2 / 2 t, by the Bianchi identity one gets for any 2 and v 2 V that .R.u1 ; u2 //v D R.v; u2 /u1 C R.u1 ; v/u2 : This implies 2 .R.u1 ; u2 //? t or V gV1 ˚ gV2 , and hence (RI) There are weights 1 ; 2 2 such that f 1 C ˇ j ˇ 2 0 g [ U [ f 2 C ˇ j ˇ 2 0 g: If one chooses u1 ; u2 such that 0 6D R.u1 ; u2 / D A˛ 2 g˛ with ˛ 2 , for 2 we get that A˛ V gV1 ˚ gV2 . Hence, the weights of A˛ V are contained in f 1 C ˇ j ˇ 2 0 g [ f 2 C ˇ j ˇ 2 0 g: (RII) For every ˛ 2 there are weights 1 ; 2 2 such that ˛ f 1 ˛ C ˇ j ˇ 2 0 g [ f 2 ˛ C ˇ j ˇ 2 0 g: Of course (PI) is a special case of (RI) with 1 D 2 . (PII) is not a special case of (RII) since ˛ C ˛ is not a weight, a priori. Definition 4.13. Let g gl.n; C/ be an irreducibly acting complex Lie algebra, 0 be the roots and zero of the semisimple part of g, the weights of g and ˛ defined as above. 1. A triple . 1 ; 2 ; ˛/ 2 is called spanning triple if ˛ f 1 ˛ C ˇ j ˇ 2 0 g [ f 2 ˛ C ˇ j ˇ 2 0 g : 2. A spanning triple . 1 ; 2 ; ˛/ is called extremal if 1 and 2 are extremal. 3. A triple . 1 ; 2 ; U / with 1 ; 2 extremal weights and U an affine hyperplane in t is called planar spanning triple if every extremal weight different from 1 and

2 is contained in U and f 1 C ˇ j ˇ 2 0 g [ U [ f 2 C ˇ j ˇ 2 0 g. In [59] the following conclusion is deduced from (RI) and (RII).

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Proposition 4.14 ([59, Proposition 3.13]). Let g gl.n; C/ be an irreducible complex Berger algebra. Then, for every root ˛ 2 there is a spanning triple. Furthermore there is an extremal spanning triple or a planar spanning triple. Returning to weak-Berger algebras we reformulate Proposition 4.11: Proposition 4.15. Let g so.n; C/ be an irreducible complex weak-Berger algebra. Then there is an extremal weight ƒ such that one of the following properties is satisfied. (SI) There is a planar spanning triple of the form .ƒ; ƒ; U /. (SII) There is an ˛ 2 with ˛ fƒ ˛ C ˇ j ˇ 2 0 g [ fƒ C ˇ j ˇ 2 0 g: There is a fundamental system such that the extremal weight in (SI) and (SII) is the highest weight. Obviously, we are in a slightly different situation as in the Berger case since ƒC˛ is not necessarily a weight and in case it is a weight, it is not extremal in general. Now the argument continues as follows: First of all one applies the result of Proposition 4.15 to all simple complex irreducibly acting Lie algebras. This is done step by step in [50] under the following special conditions: 1. 2. 3. 4.

The highest weight of the representation is a root. The representation satisfies (SI), i.e. admits a planar spanning triple .ƒ; ƒ; U /. The representation satisfies (SII) and has weight zero. The representation satisfies (SII) and does not have weight zero.

It turns out that any simple, irreducible complex weak-Berger algebra is a Berger algebra, in fact it is a Riemannian holonomy algebra. Then semisimple, non-simple weak-Berger algebras are considered. Here one uses the fact that to a decomposition of g into ideals g D g1 ˚ g2 corresponds a decomposition of the irreducible module V into factors V D V1 ˝ V2 which are irreducible g1 - resp. g2 -modules. Moreover, the Cartan subalgebra t of g is the sum of the Cartan subalgebras of g1 and g2 , and if are the roots of g and i the roots of gi then D 1 [ 2 . For the weights it holds D 1 C 2 . On the base of these facts in [51] a similar analysis as for the simple Lie algebras can be carried out in order to find the possible simple summands of g. It will lead to the result that any semisimple, irreducible complex weak-Berger algebra is a Berger algebra, and in fact a Riemannian holonomy algebra. We can summarise: Theorem 4.16. Let g so.n/ be an irreducible weak-Berger algebra of real type. Then g is a Riemannian holonomy algebra, and in particular a Berger algebra. Of course, Theorems 4.5 and 4.16 will give the ‘only if’ implication of Theorem 1.2. In order to get the other direction of Theorem 1.2 we need to prove that those indecomposable, non-irreducible subalgebras of so.1; n C 1/ described in Theorem 1.1 which have Riemannian holonomy as orthogonal part can be realised as holonomy algebra of a Lorentzian manifold. This will be one of the aims of the next section.


75

5 Lorentzian manifolds with indecomposable, non-irreducible holonomy 5.1 Local description and coordinates. In this section we will recall some results about the local shape of indecomposable Lorentzian manifolds. Based on foliations of the manifold which are defined by the holonomy invariant subspaces, we will give different coordinates and the local form of the metric. These coordinates are due to [20], [66] and [18]. Let .M; h/ be a Lorentzian manifold with indecomposable, non-irreducible holonomy algebra h. We recall that its tangent bundle admits a filtration „ „? TM;

(14)

where „ is a parallel distribution of light-like lines and „? its orthogonal complement. Let S be the screen bundle as defined in Section 2.3. We have seen that its fibre Sp D „p? =„p decomposes into a subspaces which are invariant under the orthogonal component g of the holonomy algebra h, Sp

D E0 ˚ E1 ˚ ˚ Es ;

such that g acts trivial on E0 and irreducibly on Ei for 1 i s. Now, let the spaces ‡pi be the pre-image under the canonical projection „p? ! Sp of those Ei . They have common intersection „p . and are holonomy invariant. Therefore they are the fibres of parallel distributions ‡ 0 ; : : : ; ‡ s on M with „ D ‡0 \ \ ‡s:

(15)

In this context hold the following [18] for the curvature R of .M; h/ Lemma 5.1. If X 2 „; Y i 2 ‡ i and Y 2 „? , then: 1. R.Y i ; Y j / D R.Y 0 ; Y / D R.X; Y / D 0 for i 6D j . 2. For U; V 2 TM and i 6D j it holds R.Y i ; U; Y j ; V / D R.Y i ; V; Y j ; U /. All the foliations „ ‡ i „? are parallel, hence, they are involutive and therefore integrable. I.e. for every point p 2 M , there are integral manifolds Xp , Ypi and Xp? of „ and „? passing through it. Each leaf of Y i and X ? again is foliated in leaves of X. The leaves of X are a light-like geodesic lines, the leaves of Y i are lightlike totally geodesic submanifolds, and the ones of X ? are light-like totally geodesic hypersurfaces. Lemma 5.1 then can be used to prove the existence of several coordinate systems. The first type of coordinates only respects the foliation X X ? and is sometimes called Walker coordinates. Proposition 5.2. Let .M; h/ be a Lorentzian manifold of dimension n C 2.

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1. .M; h/ admits recurrent, light-like vector field if and only if there exist coordinates .x; .yi /niD1 ; z/ such that h D 2 dxdz C

n X

2

ui dyi dz C f dz C

iD1

n X

gij dyi dyj

(16)

i;j1

@g

i D 0, f 2 C 1 .M /. These coordinates are called Walker with @xij D @u @x coordinates ([66], [44]). The recurrent vector field is parallel if and only if @f D 0. Then the coordinates are called Brinkmann coordinates ([20], [32]). @x 2. .M; h/ is a Lorentzian manifold with parallel light-like vector field if and only if there exist coordinates .x; .yi /niD1 ; z/ such that

h D 2 dxdz C

n X

gij dyi dyj

with

i;j D1

@gij D 0: @x

(17)

These coordinates are due to R. Schimming ([58]). @ corresponds to the recurrent/parallel lightIn these coordinates the vector field @x @f like vector field. We should remark that if f is sufficient general (e.g. @y 6D 0 for all i i D 1; : : : ; n), then .M; h/ is indecomposable. If one considers small n-dimensional submanifolds in U through the point p D ' 1 .x; y1 ; : : : ; yn ; z/ defined by

W.x;z/ WD f' 1 .x; y1 ; : : : yn ; z/ j .y1 ; : : : yn / 2 Rn \ '.U /g; then one can understand the gij as coefficients of a family of Riemannian metrics gz and the ui as coefficients of a family of 1-forms z on W.x;z/ depending on a parameter z. A direct calculation gives the following relation between the holonomy of these Riemannian metrics and the orthogonal component of the full holonomy Hol.x;y;z/ .W.x;z/ ; gz / pr so.n/ Hol.x;y;z/ .M; h/ ; which was proven in [44]. But one can go further and prove the existence of coordinates adapted to the foliations X Y i X ? . This was done by C. Boubel in [18], where he found also an additional condition under which these coordinates are unique. For the sake of completeness we will present these coordinates here. Theorem 5.3. Let .M; h/ be an indecomposable Lorentzian manifold of dimension n C 2 > 2 with a recurrent light-like vector field. Then there exist coordinates U; ' D .x; y10 ; : : : ; yn00 ; : : : ; y1s ; : : : ; yns s ; z/ around the point p 2 M , such that h D 2 dxdz C dz C

ni s X X

i gkl dyki dyli ;

iD0 k;lD1

which are adapted to the foliations .X; Y 0 ; : : : ; Y s ; X ? /, i.e.

(18)

Holonomy groups of Lorentzian manifolds i • gkl 2 C 1 .U / with

@ i .gkl / @x

• a 1-form on U with d

D

77

@ i j .gkl / @ym

@ ; @ @x @y i l

d

D 0 for i 6D j and @ @ D 0 for i 6D j . j i ;

@yk

@yl

Furthermore, these coordinates can be chosen in a way such that: @ 0 D 0. D ıkl and @z 1. gkl i 2. The initial condition gkl .p/ D ıkl holds. 3. satisfies:

(a) On Xp? holds D dx. (b) On the curve z with the coordinates .0; : : : ; 0; z/ holds D dx, and the 1-form j„? is closed. z

(c) Let Szi WD fq 2 Yi z j x.q/ D 0g. Then for all q 2 Szi , it holds that jTq Szi D 0. Adapted coordinates with 1, 2, and 3 are uniquely determined by its values on the initial manifold Xp? . The 1-form is uniquely determined by the three conditions 3 (a), 3 (b) and 3 (c) and the relation for Y D

@ j @yl

1 @ @ @ @ @ @ @ D R ..Y // ; ; ; Y @x @z @yki @x @z @yki @z

with i 6D j or i D 0 or Y

(19)

@ . @x

In case that g acts irreducible, the unique form of (18) reduces to the form (17). In [18], Theorem 5.3 is used to find equivalent conditions for an indecomposable, nonirreducible Lorentzian manifold to have holonomy of type 1, 2, 3, or 4. However, these equivalent conditions do not give examples of metrics for any of the possible groups. We will explain in the next section how to get these. 5.2 Metrics that realise all types of Lorentzian holonomy. For a given Riemannian holonomy group G, it is not difficult to construct an indecomposable, non-irreducible Lorentzian manifold having G as orthogonal component of its holonomy – provided we have a Riemannian manifold with holonomy G. In fact, the following is true [49], [52]. Proposition 5.4. Let .N; g/ be a n-dimensional Riemannian manifold with holonomy group G and let f 2 C 1 .R N / a smooth function on M also depending on the parameter x, and ' a smooth real function of the parameter z. Then the Lorentzian manifold .M WD R N R; h D 2dxdz C f dz 2 C e2' g/ has holonomy .R G/ Ë Rn if f is sufficiently generic, and G Ë Rn if f does not depend on x.

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This gives a construction method for any Lorentzian holonomy group of uncoupled type 1 or 2. This procedure was used in physics literature to construct examples of Lorentzian manifolds in special cases [33]. Although in [11] some examples of metrics with holonomy of coupled types 3 and 4 were constructed in order to verify that there are metrics of this type – which we will explain at the end of this section – after the classification of possible screen holonomies, the following question arose: Given a Riemannian holonomy group G with Lie algebra g having a non-trivial center z, and given an epimorphism ' W z ! Rl for 0 < l < n, does there exist a Lorentzian manifold with holonomy algebra of type 3 or 4 defined by '? In [38] this question was set in the affirmative by providing a unified construction of local polynomial metrics realising all possible indecomposable, non-irreducible holonomy algebras of Lorentzian manifolds. We will now sketch this method. Let g so.n/ be the holonomy algebra of a Riemannian manifold. As seen above, we have an orthogonal decomposition Rn D Rn0 ˚ Rnn0 where g acts trivially on Rn0 and Rnn0 decomposes further, Rnn0 D Rn1 ˚ ˚ Rns ; where g acts irreducible on the Rni inducing the decomposition of g into the direct sum of ideals g D g1 ˚ ˚ gs ; such that gi acts trivially on Rnj for i 6D j . Hence, gi so.ni / is an irreducible subalgebra for 1 i s. Moreover, the Lie algebras gi are the holonomy algebras of Riemannian manifolds, and B.g/ D B.g1 / ˚ ˚ B.gs /: Obviously, g so.n n0 / does not annihilate any proper subspace of Rnn0 . If h is an indecomposable subalgebra of p with orthogonal part g having center z and of coupled type 3 defined by an epimorphism ' W z ! R then we denote h by h.g; '/. If h is of coupled type 4 defined by an epimorphisms W z ! Rp for 0 < p < n we denote h by h.g; ; p/. Note that in the latter case we have 0 < p n0 < n. Moreover, as all the gi act irreducibly and thus have at most a one-dimensional center, it is 0 < p s. First, for a weak-Berger algebra g so.n/ – which has to be a Riemannian holonomy algebra by Theorems 4.5 and 4.16 – one fixes a basis e1 ; : : : ; en of Rn , orthonormal w.r.t. the scalar product h ; i on Rn and adapted to the above decomposition of Rn , and weak curvature endomorphisms QA 2 B.g/ for A D 1; : : : ; N such that fQA gAD1:::N span B.g/. Now one defines the following polynomials on RnC1 , ui .y1 ; : : : ; yn ; z/ WD

n N X X

˝ ˛ 1 QA .ek /el C QA .el /ek ; ei yk yl z A : (20) 3.A 1/Š „ ƒ‚ … AD1 k;lD1 i DWQAkl

79


Note that QA .ei / D 0 for i D 1; : : : ; n0 and that i i QAkl D QAlk

i k l and QAkl C QAli C QAik D 0:

Then we define the following Lorentzian metric on RnC2 , h D 2dxdz C f dz 2 C 2

n X

ui dyi dz C

iD1

n X

dyk2 ;

(21)

kD1

where f is a function on RnC2 to be specified. If h is of type 3 defined by an epimorphism ' W z ! R, i.e. h D h.g; '/, first we extend ' to the whole of g by setting it to zero on g0 , i.e. we set '.Z z C U / WD '.Z/ for Z 2 z and U 2 g0 . Then, for A D 1; : : : ; N and i D n0 C 1; : : : ; n we define the numbers 'Ai D

1 '.Q z A .ei //: .A 1/Š

If h is of type 4 defined by an epimorphism W z ! Rp , i.e. h D h.g; ; p/, again we extend to an epimorphism z to the whole of g as above, and define the following numbers, ˝ ˛ 1 z .QA .ei //; eb ; Aib WD .A 1/Š for A D 1; : : : ; N , i D n0 C 1; : : : ; n and b D 1; : : : ; p. Then in [38] we proved the following. Theorem 5.5. Let h so.1; n C 1/ be indecomposable and non-irreducible with a Riemannian holonomy algebra g as orthogonal part. If h is given by the left-handside of the following table, then the holonomy algebra in the origin 0 2 RnC2 of the Lorentzian metric h given in (20) is equal to h if the function f is defined as in the right-hand side of the table: f

h Type 1: h D .R ˚ g/ Ë Rn

x2 C

iD1

n0 P

Type 2: h D g Ë Rn Type 3: h D h.g; '/ Type 4: h D h.g; ; p/

n0 P

2x

N P

iD1

n P

AD1 iDn0 C1

2

N P

n P

p P

AD1 iDn0 C1 bD1

yi2

yi2

'Ai yi z A1 C

Aib yi yb z

A1

n0 P kD1

C

yk2 n0 P

kDpC1

yk2

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Obviously, this theorem implies the ‘if’-direction of the main classification result of Theorem 1.2 in the introduction. We will now explain the idea of the proof of this theorem. The metric h given in (20) with a function f given as in the theorem is analytic, hence its holonomy at 0 2 RnC2 is generated by the derivatives of the curvature tensor at 0. But the metric is constructed in a way such that the only non-vanishing so.n/-parts of the curvature and its derivatives satisfy at 0 2 RnC2 :

(22) pr so.n/ .r@z : : : r@z R/ .@i ; @z / D QA .ei /; „ ƒ‚ … .A1/times

@ and @i for @y@ i . Since for A D 1; : : : ; N , i D n0 C 1; : : : ; n, and writing @z for @z Q1 ; : : : ; QN span B.g/, the derivatives of the curvature will span g, because this is a weak-Berger algebra. Hence, the orthogonal part g of h we started with is the orthogonal part of hol0 .RnC2 ; h/. A more detailed analysis also shows that (22) implies that Rnn0 is contained in hol0 .RnC2 ; h/. But, more importantly, one gets the following formulas for the curvature and its derivatives involving derivatives of the function f :

1 @2 f 2 pr R R .@x ; @z / D ; 2 .@x/2

with @x WD

1 @AC1 f R .@i ; @z / D ; pr R r@A1 z 2 @x@yi .@z/A1

pr R r@A1 R .@a ; @z / z

@ ; @x

for i D n0 C 1; : : : n;

n0 @AC1 f 1X D eb ; 2 @ya @yb .@z /A1

for a D 1; : : : ; n0 ;

bD1

in which A D 1; : : : ; N . For the different choices of f as given in the theorem we can use either the first equality to see hol0 .RnC2 ; h/ D .R ˚ g/ Ë Rn is of type 1, or the second equality for hol0 .RnC2 ; h/ D h.g; '/ is of type 3. For ˛ D 0 we use the last equality for all types, for ˛ 0 we use the last equality to see that hol0 .RnC2 ; h/ D h.g; ; p/ is of type 4. 5.3 Some examples. The method of Theorem 5.5 works for any Riemannian holonomy algebra, as soon as one is able to calculate B.g/. Sometimes it is not necessary to calculate the whole of B.g/ but a sub-module which is sufficient to generate the Lie algebra g. This could be the sub-module R.g/ (c.f. Lemma 3.4). For instance, in [54] we considered a Riemannian symmetric space G=K with g D k ˚ m. The curvature endomorphisms of k satisfy K.k/ D R Œ ; , where Œ ; is the commutator of g. Since k is the holonomy algebra of this space we get k D spanfŒX; Y j X; Y 2 mg. Hence, for a basis X1 ; : : : ; Xn of m, the Qj WD ad.Xj / are spanning the sub-module R.k/ in B.k/ and generate the whole Lie algebra k. In this situation, the polynomials ui defined in (20) can be written in terms of the basis

81


Xi and the Killing form of g, .y1 ; : : : ; yn ; z/ u.G;K/ i n X 1 WD B ŒXj ; Xk ; ŒXl ; Xi C B ŒXj ; Xl ; ŒXk ; Xi yk yl z j ; 3.j 1/Š j;k;lD1

where Œ ; is the commutator in g and B the Killing form. In this his way one obtains a Lorentzian manifold with the isotropy group K of a symmetric space G=K as screen holonomy. For non-symmetric Riemannian holonomy algebras, K.g/ can be very big and thus the calculations complicated. As sketched in [54], another way is to use other, easier sub-modules of B.g/. This methods works if g is simple, since any sub-module of B.g/ generates a non-trivial ideal in g which has to be equal to g if g is g simple. For example, in the case of the exceptional Lie algebra g2 so.V /, with V D R7 , the g2 -module Hom.V; g2 / which contains B.g2 / splits into the direct sum of VŒ1;1 , ˇ20 V and V , where VŒ1;1 is the 64-dimensional g2 -module of highest weight .1; 1/, and ˇ20 V is the 27-dimensional module of highest weight .2; 0/. Since B.g2 / is the kernel of the skew-symmetrisation Hom.V; g2 / VŒ1;1 ˚ ˇ20 V ˚ V

/ ƒ3 V

ˇ20 V ˚ V ˚ R,

a dimension analysis shows that B.g/ must contain VŒ1;1 . Thus, by choosing a basis of VŒ1;1 a metric of the form (21) with coefficients as in (20) can be defined and one obtains a Lorentzian manifold with screen holonomy G2 . Using such methods we are able to construct metrics having the Lie algebras g2 Ë R7 so.1; 8/ and spin.7/ËR8 so.1; 9/ as holonomy but without using Riemannian manifolds with holonomy g2 or spin.7/. This was done in [38]. We start with the Lie subalgebra g2 so.7/. The vector subspace g2 so.7/ is spanned by the following matrices (using the conventions of [7]): A1 A5 A9 A13

D E12 E34 ; D E14 E23 ; D E16 E25 ; D E27 E35 ;

A2 A6 A10 A14

D E12 E56 ; A3 D E13 C E24 ; A4 D E13 E67 ; D E14 E57 ; A7 D E15 C E26 ; A8 D E15 C E47 ; D E16 C E37 ; A11 D E17 E36 ; A12 D E17 E45 ; D E27 C E46 ;

where Eij 2 so.7/ (i < j ) is the skew-symmetric matrix such that .Eij /ij D 1, .Eij /j i D 1 and .Eij /kl D 0 for other k and l. Consider the linear map Q 2 Hom.R7 ; g2 / defined as Q.e1 / D A6 ; Q.e5 / D A4 ;

Q.e2 / D A4 C A5 ; Q.e6 / D A5 C A6 ;

Q.e3 / D A1 C A7 ; Q.e7 / D A7 :

Q.e4 / D A1 ;

82


It can be checked that Q 2 B.g2 /. Moreover, the elements A1 ; A4 ; A5 ; A6 ; A7 2 g2 generate the Lie algebra g2 . Now, the holonomy algebra of the metric g D 2dxdz C

7 X

dyi2 C 2

iD1

7 X

ui dyi z;

iD1

where u1 D 23 .2y2 y3 C y1 y4 C 2y2 y4 C 2y3 y5 C y5 y7 /; u2 D 23 .y1 y3 y2 y3 y1 y4 C 2y3 y6 C y6 y7 /; u3 D 23 .y1 y2 C y22 y3 y4 .y4 /2 y1 y5 y2 y6 /; u4 D 23 .y12 y1 y2 C y32 C y3 y4 /; u5 D 23 .y1 y3 2y1 y7 y6 y7 /; u6 D 23 .y2 y3 2y2 y7 y5 y7 /; u7 D 23 .y1 y5 C y2 y6 C 2y5 y6 /; at the point 0 2 R9 is g2 Ë R7 so.1; 8/. Now we consider the Lie subalgebra spin.7/ so.8/. The vector subspace spin.7/ so.8/ is spanned by the following matrices (again see [7]): A1 D E12 C E34 ;

A2 D E13 E24 ;

A3 D E14 C E23 ;

A4 D E56 C E78 ;

A5 D E57 C E68 ;

A6 D E58 C E67 ;

A7 D E15 C E26 ;

A8 D E12 C E56 ;

A9 D E16 C E25 ;

A10 D E37 E48 ; A11 D E38 C E47 ;

A12 D E17 C E28 ;

A13 D E18 E27 ;

A14 D E35 C E46 ; A15 D E36 E45 ;

A16 D E18 C E36 ;

A17 D E17 C E35 ;

A18 D E26 E48 ; A19 D E25 C E38 ;

A20 D E23 C E67 ;

A21 D E24 C E57 : Consider the linear map Q 2 Hom.R8 ; spin.7// defined as Q.e1 / D 0;

Q.e2 / D A14 ;

Q.e3 / D 0;

Q.e4 / D A21 ;

Q.e5 / D A20 ; Q.e6 / D A21 A18 ; Q.e7 / D A15 A16 ; Q.e7 / D A14 A17 : It can be checked that Q 2 B.spin.7//. Moreover, the elements A14 , A15 A16 , A17 , A18 , A20 , A21 in spin.7/ generate the Lie algebra spin.7/. Then the holonomy algebra of the metric g D 2dxdz C

8 X iD1

.dyi /2 C 2

8 X iD1

ui dyi dz;


83

where u1 D 43 y7 y8 ;

u2 D 23 .y42 C y3 y5 C y4 y6 y62 /;

u3 D 43 y2 y5 ;

u4 D 23 .y2 y4 2y2 y6 y5 y7 C 2y6 y8 /;

u5 D 23 .y2 y3 C 2y4 y7 C y6 y7 /;

u6 D 23 .y2 y4 C y2 y6 C y5 y7 y4 y8 /;

u7 D 23 .y4 y5 2y5 y6 C y1 y8 /; u8 D 23 .y4 y6 C y1 y7 /; at the point 0 2 R9 is spin.7/ Ë R8 so.1; 9/. Some other examples with irreducible screen holonomy so.3/ so.5/ given by this Riemannian symmetric pair have been constructed. The first example of such a manifold was given in [44], although with another purpose. One considers the one-form on R5 , 5 X uk dyk ; D kD1

p with u1 D u3 D 2 3 y2 y3 2y4 y5 , u5 D 2 3 y2 y5 C 2y3 y4 , u2 D u4 D 0. Now one defines the Lorentzian metric on R7 by y32

4y42

y52 ,

p

h WD 2dxdz C f dz 2 C dz C

5 X

dyk2

kD1 @f where f is a function on R7 with @y 6D 0. The holonomy of this manifold equals i 5 to .R ˚ so.3; R// Ë R or if f does not depend on x equal to so.3; R/ Ë R5 where so.3; R/ so.5; R/ is the irreducible representation defined by the Riemannian symmetric pair: the Lie algebra sl.3; R/ can be decomposed into vector spaces sl.3; R/ D so.3; R/ ˚ sym0 .3; R/, where sym0 .3; R/ denote the trace free symmetric matrices. This representation is equal to the holonomy representation of the Riemannian symmetric space SL.3; R/= SO.3; R/. Another example of this type having the same holonomy was constructed in [52] by setting u1 D 4y1 y2 , u2 D 4y1 y2 , u3 D y1 yp 4 y2 y4 C y1 y3 y2 y3 C p 3.y4 y5 y3 y5 /, u4 D y1 y4 y2 y4 Cy1 y3 Cy2 y3 C 3.y4 y5 Cy3 y5 / and u5 D 0. Recently in [38] another such example was constructed by defining u1 D 23 ..y3 /2 C p p 4.y4 /2 C.y5 /2 /, u2 D 2 3 3 ..y3 /2 .y5 /2 /, u3 D 23 .y1 y3 3y2 y3 3y4 y5 .y5 /2 /, p u4 D 83 y1 y4 and u5 D 23 .y1 y5 C 3y2 y5 C 3y3 y4 C y3 y5 /. These examples also have so.3/ so.5/ as screen holonomy. We do not know whether the three examples are locally isometric. On the other hand one can construct a manifold with the same holonomy but with different geometric properties, by the construction given in Proposition 5.4. Let g be the Riemannian metric on SL.3; R/= SO.3; R/ and consider the Lorentzian manifold (23) M WD R2 .SL.3; R/= SO.3; R// ; h WD 2dxdz C f dz 2 C g

If f is sufficient general this manifold is indecomposable and has holonomy algebra so.3/ Ë R5 or .R ˚ so.3// Ë R5 . But, unlike the other examples, its curvature

84


restricted to @? x does not vanish because R @i ; @j ; @k ; @l equals to the curvature of SL.3; R/= SO.3; R/ (see next section). Finally, we want to return to the question of closedness of holonomy groups. In [11] Lorentzian manifolds with indecomposable, non-irreducible holonomy of coupled type 3 and 4 are constructed which have a non-closed holonomy group. These examples use a dense immersion of the real line into the 2-torus. They are constructed similar to our construction method. Consider the metric h D 2dxdz

4 X

dyi2 C 2x .y1 y2 C ˛ y3 y4 / dz 2

iD1

C 2 y22 y1 dy1 y12 y2 dy2 C y42 y3 dy3 y32 y4 dy4 dz on R6 depending on the parameter ˛. For this metric one can show that it is of coupled type 3 defined by an epimorphism ' W z ! R, its orthogonal part is the torus T 2 , and that the kernel of ' defines a closed subgroup in T 2 if and only if ˛ is rational. Hence, for ˛ irrational, the holonomy group of h is not closed in SO0 .1; 5/. Similarly, the metric h D 2dxdz

5 X

dyi2 C dz 2

iD1

C 2 y22 y1 dy1 y12 y2 dy2 C y42 y3 dy3 y32 y4 dy4 C z .y1 y2 C ˛ y3 y4 / dy5 dz on R7 has a holonomy group of coupled type 4, with T 2 as orthogonal part, and which is non-closed if ˛ is irrational. 5.4 pp-waves and generalisations. In this section we want to present some results obtained in [53] and [54] about pp-waves and some holonomy-related generalisations. A Lorentzian manifold with parallel light-like vector field is called Brinkmann wave. A Brinkmann wave admits coordinates as in Proposition 5.2. A Brinkmann-wave is called pp-wave if its curvature tensor R satisfies the trace condition tr .3;5/.4;6/ .R ˝ R/ D 0. R. Schimming [58] proved that an .n C 2/-dimensional pp-waves admits coordinates .x; .yi /niD1 ; z/ such that 2

h D 2 dxdz C f dz C

n X i1

dyi2

with

@f @x

D 0,

(24)

and that a Brinkmann wave .M; h/ with parallel light-like vector field X is a pp-wave if and only if one of the following conditions – in which denotes the 1-form h.X; / – is satisfied: ƒ.1;2;3/ . ˝ R/ D 0;

(25)

or ƒ.1;2/.3;4/ . ˝ % ˝ / D R

for a symmetric tensor % with X

,

(26)


85

or tr .1;5/.4;8/ .R ˝ R/ D ' ˝ ˝ ˝

for a function '.

(27)

In [53] we gave another equivalence for the definition which seems to be simpler than any of the trace conditions and which allows for generalisations. Proposition 5.6. A Brinkmann-wave .M; h/ with parallel light-like vector field X and induced parallel distributions „ and „? is a pp-wave if and only if its curvature tensor satisfies (28) R.U; V / W „? ! „ for all U; V 2 TM; or equivalently

R.Y1 ; Y2 / D 0 for all Y1 ; Y2 2 „? :

(29)

From this description one obtains easily that a pp-wave is Ricci-isotropic, which means that the image of the Ricci-endomorphism is totally light-like, and has vanishing scalar curvature. But it also enables us to introduce a first generalisation of pp-waved by supposing (28) but only the existence of a recurrent light-like vector field. Assuming that the abbreviation ‘pp’ stands for ‘plane fronted with parallel rays’ we call them pr-waves, ‘plane fronted with recurrent rays’. Definition 5.7. A Lorentzian manifold with recurrent light-like vector field X is called pr-wave if R.U; V / W „? ! „ for all U; V 2 TM; (30) or equivalently R.Y1 ; Y2 / D 0 for all Y1 ; Y2 2 X ? . Since X is not parallel, all the trace conditions which were true for a pp-wave, fail to hold for a pr-wave. But in [53] an equivalence similar to (25) is proved. Lemma 5.8. A Lorentzian manifold .M; h/ with recurrent light-like vector field X is a pr-wave if and only if ƒ.1;2;3/ . ˝ R/ D 0, where denotes the 1-form h.X; /. Similar to a pp-wave, a Lorentzian manifold .M; h/ is a pr-wave if and only if there are coordinates .x; .yi /niD1 ; z/ such that h D 2 dxdz C f dz 2 C

n X

dyi2

with f 2 C 1 .M /.

(31)

i1

Regarding the vanishing of the screen holonomy the following result can be obtained by the description of Proposition 5.6 and the definition of a pr-wave. Proposition 5.9. A Lorentzian manifold .M; h/ with recurrent light-like vector field is a pr-wave if and only if the following equivalent conditions are satisfied: (1) The screen holonomy of .M; h/ is trivial. (2) .M; h/ has solvable holonomy contained in R Ë Rn .

86


In addition, .M; h/ is a pp-wave if and only if its holonomy is Abelian, i.e. contained in Rn . Finally, in [53] we have proved that a pr-wave is a pp-wave if and only if it is Ricci-isotropic. On the one hand, there are very important subclasses of pp-waves. The first are z the plane waves which are pp-waves with quasi-recurrent curvature, i.e. rR D ˝ R z a .4; 0/-tensor. For plane waves the function f in the local where D h.X; / and R P form of the metric is of the form f D ni;j D1 aij yi yj where the aij are functions of z. A subclass of plane waves are the Lorentzian symmetric spaces with solvable transvection group, the so-called Cahen–Wallach spaces (see [24], also [11]). For P these the function f satisfies f D ni;j D1 aij yi yj , where the aij are constants. On the other hand, pp-waves can be generalised in the following way [54]. Definition 5.10. A Lorentzian manifold .M; h/ with recurrent light-like vector field X has light-like hypersurface curvature if R.U; V / W „? ! „

for all U; V 2 „? ;

(32)

where „ and „? are the light-like distributions defined by X . Of course, (32) is equivalent to the fact the .4; 0/-curvature tensor vanishes on „ ? „? „? „ ? . The chosen name can be explained by the following considerations. We have seen in Section 5.1 that the parallel distributions „ „? TM define a foliation of M into light-like hypersurface Xp? with tangent bundles T Xp? D „? jXp? . Since the distribution „? is parallel, i.e. rU W .„.?/ / ! .„.?/ / for every U 2 TM , the LeviCivita connection r of .M; h/ defines a connection on the hypersurfaces Xp? , denoted by rV W .T X ? ˝ T X ? / ! .T X ? /. Then we get the following equivalences. p

p

p

Proposition 5.11. A Lorentzian manifold with recurrent light-like vector field X has light-like hypersurface curvature if and only if every light-like hypersurface Xp? , defined V satisfies one of the following equivalent by X and equipped with induced connection r, conditions: V is the curvature of r, V then for any U; V; W 2 T X ? the tangent vector (1) If R V R.U; V /W is light-like. (2) The holonomy of rV is solvable and contained in R Ë Rn .

p

If in addition X is parallel, then the holonomy of rV is Abelian and contained in Rn . In [16] the quantities assigned to the hypersurfaces Xp? are used to describe the holonomy of a Lorentzian manifold further, in particular to decide to which type in Theorem 1.1 the holonomy algebra belongs. This approach makes use of a screen distribution which is complementary and orthogonal to „ in „? (see also [9], or recently [5] and [31]). Such a screen distribution can always be chosen, but since it


87

requires a choice we prefer to work with an analogue to the screen bundle introduced in Section 2.3 which can be defined without making such a choice. For a light-like hypersurface Xp? through p 2 M we define the restricted screen bundle over XP? as V We obtain SV WD SjXp? , which is equipped with a covariant derivative defined by r. another equivalence in terms of the screen bundle. Proposition 5.12. A Lorentzian manifold with recurrent light-like vector field X has light-like hypersurface curvature if and only if over every light-like hypersurface Xp? V defined by X the connection r S on the restricted screen bundle SV is flat. For the case where the vector field X is parallel we obtain the following equivalent trace condition. Proposition 5.13. A Brinkmann wave .M; h/ has light-like hypersurface curvature if and only if kRk2 D 0, where kRk2 is the square of the norm of the curvature tensor, defined by kRk2 WD tr .1;5/.2;6/.3;7/.4;8/ .R ˝ R/. Regarding the description in local coordinates one can show that .M; h/ has lightlike hypersurface curvature if and only if there are coordinates .x; .yi /niD 1 ; z/ such that n n X X 2 h D 2 dxdz C f dz C ui dyi dz C dyi2 (33) iD1

iD1

i with @u D 0 and f 2 C 1 .M /. If, in addition, @x P .M; h/ is a Brinkmann wave, then f does not depend on x. Moreover, if z D niD1 ui dyi is closed, then .M; h/ is a prwave, i.e. has trivial screen holonomy. Of course, there are also Schimming coordinates (ui D 0, compare Proposition 5.2), in which the gij have to be the coefficients of a z-dependent family of flat Riemannian metrics. From these coordinate description one sees that any metric constructed by the method of Theorem 5.5 has light-like hypersurface curvature. Hence, although the curvature restrictions on these manifolds are quite strong, we obtain the following remarkable conclusion.

Proposition 5.14. Any indecomposable, non-irreducible subalgebra of so.1; n C 1/ with a Riemannian holonomy algebra as orthogonal part can be realised as holonomy algebra of a Lorentzian manifold with light-like hypersurface curvature. Concluding this section, we want to have a look at Ricci isotropy and Ricci flatness of manifolds with light-like hypersurface curvature. If z is the family of one-forms and f the function appearing in (33), then .M; h/ is Ricci-isotropic if and only if d d z D 0

(34)

for all z, itP is Ricci-flat, if and only if in addition f D 0, where d is the co-differential and D niD1 @y@ i the Laplacian with respect to the flat Riemannian metric on Rn .

88


5.5 Holonomy of space-times. For a space-time, i.e. a 4-dimensional Lorentzian manifold of signature . C CC/, there are 14 types of holonomy groups. This was discovered by J. F. Schell [57] and R. Shaw [62] (see also [40] and [41]). These 14 types can be derived by the following case study, in which we will also give examples of metrics realising these groups. Let H be the connected holonomy group of a 4-dimensional Lorentzian manifold. 1. H acts irreducibly, i.e. H D SO0 .1; 3/. It can be realised by the 4-dimensional de Sitter space S 1;3 . 2. H acts indecomposably, but non-irreducibly. Then H is one of the following:: (a) H D .RC SO.2// Ë R2 . (b) H D SO.2/ Ë R2 . (c) H is of type 3, i.e. H D LËR2 with L given by the graph of an epimorphism ' W so.2/ ! R. (d) H D R2 , i.e. the holonomy of a 4-dimensional pp-wave. (e) H D R Ë R2 , i.e. the holonomy of an 4-dimensional pr-wave. In all these cases the previous section gives examples of metrics realising H . 3. H acts decomposably. Then H is one of the following: (a) H D SO.2/, i.e. the holonomy of the product of the 2-sphere S 2 with the 2-dimensional Minkowski space R1;1 . (b) H D SO.1; 1/, i.e. the holonomy of the product of the 2-dimensional de Sitter space S 1;1 with the flat R2 . (c) H D SO.3/, i.e. the holonomy of the product of .R; dt 2 / with the 3-sphere S 3 . (d) H D SO.1; 2/, i.e. the holonomy of the product of the line R with the 3-dimensional de Sitter space S 1;2 . (e) H D SO.1; 1/SO.2/, i.e. the holonomy of the product of the 2-dimensional de Sitter space S 1;1 with the 2-sphere S 2 . (f) H D R Ë R. This is the holonomy of the product of R with a 3-dimensional Lorentzian manifold with a recurrent but not parallel light-like vector field, i.e. with a 3-dimensional pr-wave metric. The latter is of the form h D 2dxdz C g.y/dy 2 C f .x; y; z/dz 2 . (g) H D R. This is the holonomy of the product of R with a 3-dimensional Lorentzian manifold with a parallel light-like vector field, i.e. with a 3dimensional pp-wave metric. The latter is of the form h D 2dxdz C g.y/dy 2 C f .y; z/dz 2 . (h) H is trivial, i.e. the holonomy of the flat Minkowski space R1;3 . We should point out that there is another type of subgroup in SO.1; 3/, which is a oneparameter subgroup of SO.1; 1/ SO.2/, not equal to either of the factors. But this


89

cannot be a holonomy of a Lorentzian manifold because it does not satisfy the de Rham– Wu decomposition of Theorem 2.1. This is explained in [15, Section 10.J], where also the question is asked whether there is a space-time with holonomy of coupled type 3, in [15] denoted by B 3 . This question is answered affirmatively by A. Ikemakhen in [45, Section 4.2.3] by the metric h D 2dxdz dy12 C dy22 C 4˛y1 y2 dy1 dz C 2xy1 dz 2 ; and by the general method given in [38] described in Theorem 5.5 in Section 5.2.

6 Applications and outlook 6.1 Parallel spinors on Lorentzian manifolds. Now we want to draw the conclusions for the existence of parallel spinor fields on Lorentzian manifolds. The existence of a parallel spinor field on a Lorentzian spin manifold .M; h/ implies the existence of a parallel vector field in the following way: To a spinor field ', one may associate a vector field X' , defined by the equation h.V' ; U / D hU '; 'i for any U 2 TM , where h ; i is the inner product on the spin bundle and is the Clifford multiplication. X' sometimes is referred to as Dirac current. Now, the vector field associated to a spinor in this way is light-like or time-like. If the spinor field is parallel, so is the vector field. In the case where it is time-like, the manifold splits by the de-Rham decomposition theorem into a factor .R; dt 2 / and Riemannian factors which are flat or irreducible with a parallel spinor, i.e. with holonomy f1g, G2 , Spin.7/, Sp.k/ or SU.k/. In the case where the parallel vector field is light-like we have a Lorentzian factor which is indecomposable, but with parallel light-like vector field (and parallel spinor) and flat or irreducible Riemannian manifolds with parallel spinors. Hence, in this case one has to know which indecomposable Lorentzian manifolds admit a parallel spinor. The existence of the light-like parallel vector field forces the holonomy of such a manifold with parallel spinor to be contained in SO.n/ Ë Rn i.e. to be of type 2 or 4. Furthermore, the spin representation of the orthogonal part g so.n/ of h must admit a trivial sub-representation. In fact, the dimension of the space of parallel spinor fields is equal to the dimension of the space of spinors which are annihilated by g [49]. But for the coupled type 4, the orthogonal part g has to have a non-trivial center. Due to the decomposition of g into irreducible acting ideals, this center is a sum of one or more so.2/’s, i.e. at least one irreducible acting ideal is equal to u.p/, as so.2/ D RJ with J 2 D Id. But a direct calculation shows that u.p/ cannot annihilate a spinor. Hence we obtain the following consequence. Corollary 6.1. Let .M; h/ be an indecomposable Lorentzian spin manifold of dimension n C 2 > 2 with holonomy group H admitting a parallel spinor field. Then it is H D G ËRn where G is the holonomy group of an n-dimensional Riemannian manifold with parallel spinor, i.e. G is a product of f1g, SU.p/, Sp.q/, G2 or Spin.7/. This generalises a result of R. L. Bryant in [23] (see also [33]) where it is shown up to n 9 that the maximal subalgebras of the parabolic algebra admitting a trivial

90


sub-representation of the spin representation are of type (Riemannian holonomy)ËRn . Combining Corollary 6.1 with the de Rham–Wu decomposition theorem we obtain Theorem 1.3 of the introduction. 6.2 Holonomy of indecomposable, non-irreducible Einstein manifolds. From the description of the curvature endomorphisms in Theorem 3.7 we get a result for the holonomy of indecomposable, non-irreducible Lorentzian Einstein manifolds. For h p so.1; n C 1/ let R D R.; L; T; R0 ; Q/ 2 K.h/ be a curvature endomorphisms defined by 2 R, L 2 .Rn / , T 2 End.Rn / such that T D T , R0 2 K.g/, and Q 2 B.g/ as in Theorem 3.7. Then the Ricci-trace Ric D tr .1;4/ R is given by Ric.X; Z/ D ; Ric.U; V / D Ric0 .U; V /; where Ric0 D tr .1;4/ R0 P Ric.U; Z/ D L.U / niD1 hQ.Ei /U; Ei i

(35)

Ric.Z; Z/ D tr.T /; for U; V 2 E WD span.E1 ; : : : ; En / and X and Z as in Section 3.1. Evaluating these formulas we get the following consequence. Theorem 6.2. Let .M; h/ be an indecomposable non-irreducible Lorentzian Einstein manifold. Then the holonomy of .M; h/ is of uncoupled type 1 or 2. If the Einstein constant of .M; h/ is non-zero, then the holonomy of .M; h/ is of type 1. Proof. First of all we see that for an Einstein manifold with Einstein constant , the curvature tensor is given by R D R.; L; T; R0 D RjE Ê ; Q D R.Z; // But for all the types, except type 1, Theorem 3.7 states that D 0. Hence, only Einstein metrics of type 1 may not be Ricci-flat. To exclude the coupled types assume that .M; h/ is Einstein and its holonomy algebra h is of type 3 or 4, and thus Ricci-flat. Let g D g1 ˚ ˚gr be the decomposition of the orthogonal part g of h as in Theorem 3.6. Since h is of type 3 or 4, 'jga ¤ 0 (respectively, jga ¤ 0 for some a, 1 a r. Consequently, the center za of ga is non-trivial. Since ga so.na / is irreducible, we see that na D 2m, ga u.m/, and za D R J where J is the complex structure, i.e. za 6 su.m/. Lemma 6.3. Let g u.m/ be a subalgebra and Q 2 B.g/. Then 2m X

hQ.Ei /U; Ei i D tr J .Q.J U //;

iD1

where U 2 span.E1 ; : : : ; E2m /, J is the complex structure on span.E1 ; : : : ; E2m /, and tr J denotes the complex trace.


91

Proof. Assume that EmC1 D JE1 ; : : : ; E2m D JEm . Then we have 2m X

hQ.Ei /U; Ei i D

iD1

D D

m X iD1 m X iD1 m X

hQ.Ei /U; Ei i C hQ.Ei /U; Ei i hQ.Ei /U; Ei i C

m X

hQ.JEi /U; JEi i

iD1 m X

hQ.JEi /J U; Ei i

iD1 m X

iD1

hQ.Ei /JEi ; J U i

iD1

C

m X

hQ.J U /Ei ; JEi i

iD1

D

m X

hQ.J U /Ei ; E2i i

iD1

D tr J .Q.J U //; which proves the lemma. Let R D P1 ı R.P . /; P . // ı P be a curvature endomorphism of h coming from the curvature R and the parallel displacement P of .M; h/. All these curvature endomorphism generate h. Let Ric be the trace of such an R and Rich the Ricci tensor of .M; h/. Then 0 D P Rich D Ric : If h is of type 4, then 0 D Ric.U; Z/ D tr J .Q.J U //, U 2 R2m . Hence, Q.J U / 2 su.m/, i.e. such R’s cannot generate h. We get a contradiction. Suppose that h is of type 3, then 0 D Ric.U; Z/ D '.Q.U // C tr J .Q.J U //;

for all U 2 R2m :

If '.Q.U // ¤ 0, then pr za Q.U / ¤ f0g. This implies tr J .Q.J U // ¤ 0 and thus pr za Q.J U / ¤ f0g. Now, let U ¤ 0 be in the orthogonal complement to ker.pr za ıQ/, then pr za Q.U / ¤ f0g. Since J U is orthogonal to U , we have J U 2 ker.pr za ıQ/, which is a contradiction. Corollary 6.4. If a Lorentzian Einstein manifold .M; h/ admits a light-like parallel vector field, then it is Ricci flat. Proof. If we suppose that .M; h/ is Einstein but not Ricci-flat, the Wu decomposition of .M; h/ of Theorem 2.4 consists of non-Ricci-flat Einstein factors, in particular there is no flat factor. Hence there is an non-irreducible factor with the parallel vector field, and with holonomy of type 1. But this is a contradiction.

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Corollary 6.5. If a Lorentzian manifold .M; h/ is indecomposable, non-irreducible and Einstein and not of type 1 .and thus Ricci flat/, then in the decomposition of Theorem 3.6 for the orthogonal part of h each ga is from the list so.k/, su.p/, sp.q/, g2 , spin.7/, or the holonomy of a non-Kählerian Riemannian symmetric space, or trivial. Proof. Let .M; h/ be an indecomposable, non-irreducible Lorentzian Einstein manifold with parallel light-like vector field and thus with holonomy h of type 2. Let ga be an irreducible component of the orthogonal part. For type 2 it is L D 0, and as in the proof of the theorem we get that ga u.p/ implies that ga su.p/. Then the statement follows from the classification of the orthogonal component in Section 4. Unfortunately, we cannot yet exclude that ga is be the holonomy of a non-Kählerian Riemannian symmetric space. If ga is the holonomy of a Riemannian symmetric space, the Ricci-flatness implies that K.ga / D f0g. But we do not know if this implies that B.ga / D 0. However, Theorem 6.2 generalises a result in [44, Theorem 11], which states that if the orthogonal part of the holonomy algebra is Abelian, and the holonomy is of one of the coupled types, then the Lorentzian manifold cannot be Einstein. 6.3 Open problems and outlook on higher signatures. From the holonomy classification presented in this article many questions arise. First of all, a more geometric proof of the classification result would be desirable, similar to the proof of the Riemannian Berger list in [63]. On the other hand, the metrics which were constructed in order to realise the candidates of holonomy groups are only locally given analytic metrics. Hence, the next task is the construction of global geometric models with these holonomy groups. In particular, the construction of metrics with certain topological properties such as completeness or global hyperbolicity. First attempts in this direction have been made in [8] using a cylinder construction introduced in [6]. It would also be interesting to know the holonomy groups of Lorentzian homogeneous spaces. Widely open is the classification problem of holonomy groups in signatures other than Riemannian and Lorentzian apart from some results in certain signatures. In [45] a similar distinction into different types as in Theorem 1.1 was given for indecomposable, non-irreducible subalgebras of so.2; n C 2/. In [35] we studied the analogue of the orthogonal part of an indecomposable, non-irreducible subalgebra in so.2; n C 2/. The surprising result was that, unlike to the Lorentzian case, there is no additional condition on the subalgebra g so.n/ induced by the Bianchi-identity and replacing the weakBerger property. Instead, any subalgebra of so.n/ can be realised as this part of a holonomy algebra. Furthermore, in [36] indecomposable, non-irreducible holonomy algebras of pseudo-Kählerian manifolds of index 2, i.e. holonomy algebras contained in u.1; n C 1/ so.2; 2n C 2/, were classified. In [12] some partial results for the holonomy algebras of pseudo-Riemannian manifolds of signature .n; n/ were obtained.


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Hypersymplectic manifolds Andrew Dancer and Andrew Swann

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

2 The split quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

3

Hypersymplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

4

Hypersymplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Toric constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6

Cut constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

1 Introduction Hypersymplectic geometry is a cousin of hyperkähler geometry, corresponding to working over the algebra of split quaternions rather than the ordinary quaternions. The origins of hyperkähler geometry go back to its appearance on Berger’s list of geometries associated to reduced holonomy. Since the late 1970s it has been the focus of intensive research, not least because of its importance in physics through supersymmetry. Hyperkähler manifolds are also examples of Ricci-flat Kähler (i.e. Calabi–Yau) manifolds. Many constructions in symplectic geometry turn out to have hyperkähler analogues; in particular the hyperkähler version of the Marsden–Weinstein quotient construction has so far proved to be the most powerful method of constructing hyperkähler spaces. Hypersymplectic manifolds, which were introduced by Hitchin [20], are also Ricciflat Kähler, but in split signature .2n; 2n/. They have also found application in the physics of supersymmetry [22], [9], [27]. As in the hyperkähler case, techniques of symplectic geometry may be applied in the hypersymplectic setting. Indeed in some respects (particularly some aspects of moment map geometry) the hypersymplectic case is closer to symplectic than to hyperkähler geometry. In this article we shall review the definitions of hypersymplectic structures, and describe some constructions for them, including in particular some motivated by symplectic geometry. We shall try to compare hypersymplectic with symplectic and hyperkähler geometry. In particular, we shall describe how ideas of toric geometry may be used to produce examples starting from the action of an Abelian group on flat space.

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Some material in this article (especially in the later sections) is due to the authors; much of this is presented in more detail in [11] (see also [10]) but some results are new to this article.

2 The split quaternions As in [10], we find it helpful to define hypersymplectic structures in terms of the algebra B of split quaternions. This is a four-dimensional real vector space with basis f1; i; s; tg satisfying i 2 D 1; s 2 D 1 D t 2 ; i s D t D si: This contrasts with the ordinary quaternion algebra H with basis 1; i; j; k where i; j; k still anticommute but j 2 D k 2 D 1. We define conjugation in the algebra B by pN D x iy su t v where p D x C iy C su C t v. Elements of B satisfying pN D p are called imaginary. Now B carries a natural indefinite inner product given by hp; qi D Re pq. N Note that kpk2 D x 2 C y 2 u2 v 2 , so in fact we have a metric of signature .2; 2/. This norm is multiplicative, that is, kpqk2 D kpk2 kqk2 , but the presence of elements of length zero means that B contains zero divisors. Using the multiplication rules for B, one finds p 2 D 1

if and only if

p 2 D C1 if and only if

p D iy C su C t v; y 2 u2 v 2 D 1 p D iy C su C t v; y 2 u2 v 2 D 1 or p D ˙1:

We can compare this with the quaternion algebra where there is a two-sphere fp D iy C j u C ku W y 2 C u2 C v 2 D 1g of elements with p 2 D 1. This fact underlies the twistor approach to hyperkähler manifolds. The right B-module Bn Š R4n inherits the inner product h; i D Re N T of signature .2n; 2n/. The automorphism group of .Bn ; h ; i/ is Sp.n; B/ D f A 2 Mn .B/ W ANT A D 1 g which is a Lie group isomorphic to Sp.2n; R/, the symmetries of a symplectic vector space .R2n ; !/. In particular, Sp.1; B/ Š SL.2; R/ is the pseudo-sphere of B D R2;2 . The Lie algebra of Sp.n; B/ is sp.n; B/ D f A 2 Mn .B/ W A C ANT D 0 g; so sp.1; B/ D Im B. Using the complex structure 7! i , we may identify Bn with C n;n via D z C ws. In this context we see Sp.n; B/ as a subgroup of U.n; n/ and note that it contains a compact n-dimensional torus T n D fdiag.e i1 ; : : : ; e in /g Mn .B/. Remark 2.1. The symmetry groups Sp.n; B/ Š Sp.2n; R/ for Bn and Sp.n/ for Hn both complexify to Sp.2n; C/. Indeed they are the split and compact real forms

99

Hypersymplectic manifolds

respectively of this complex group. Hence hypersymplectic and hyperkähler geometry may be viewed intuitively as the two different real geometries which complexify to complex symplectic geometry. This is the viewpoint adopted by Hitchin [20].

3 Hypersymplectic manifolds Let us identify R4n with Bn so Sp.n; B/ GL.4n; R/. We consider an Sp.n; B/structure on a manifold M of dimension 4n, that is, a subbundle of the frame bundle of M that is a principal Sp.n; B/-bundle. Such a structure on M defines a metric g of signature .2n; 2n/. The right action of i, s and t on Bn define endomorphisms I , S and T of TM satisfying I 2 D 1;

S 2 D 1 D T 2;

IS D T D SI

(3.1)

and the compatibility equations g.IX; I Y / D g.X; Y /;

g.SX; SY / D g.X; Y / D g.TX; T Y /;

(3.2)

for sections X; Y of TM . We obtain three 2-forms !I , !S and !T given by !I .X; Y / D g.IX; Y /;

!S .X; Y / D g.SX; Y /;

!T .X; Y / D g.TX; Y /:

The manifold M is said to be hypersymplectic if the 2-forms !I , !S and !T are all closed: d!I D 0; d!S D 0; d!T D 0: Adapting a computation of Atiyah & Hitchin [4] for hyperkähler manifolds, one finds that this implies that the endomorphisms I , S and T are all integrable. This means firstly that locally there are complex coordinates realising I . The integrability of S means that M is locally a product MCS MS where for " D ˙1, TM"S is the "eigenspace of S on TM . (For this reason, integrable endomorphisms whose square is the identity are often referred to as product structures). Note that the submanifolds M"S are totally isotropic with respect to g. Observe also that as I anticommutes with S , it will interchange the tangent spaces to the two factors. We therefore have an example of a complex product structure in the terminology of Andrada and Salamon [3]. We obtain an S 1 family of such splittings by considering the integrable endomorphisms S D S cos C T sin . As pointed out by Hitchin [20], this is an analogue for hypersymplectic manifolds of the twistor construction for hyperkähler manifolds. An alternative twistor construction for hypersymplectic spaces which is a more immediate analogue of the hyperkähler case has been studied in [8]. Here the twistor space is the product of the hypersymplectic manifold with the hyperboloid y 2 u2 v 2 D 1 parametrising the complex structures, as discussed in §2. As in the hyperkähler case, we can define an integrable complex structure on the twistor space. A more sophisticated version of the latter construction makes use of spaces of null geodesics to produce a complex twistor space covered by rational curves [25], [21].

100


For a hypersymplectic manifold M , one finds that I , S and T are parallel with respect to the Levi-Civita connection r of g, so the holonomy group of M reduces to Sp.n; B/. Note that .g; I / defines a signature .2n; 2n/ Kähler structure on M . As the Sp.n; B/ action on Bn and the Sp.n/ action on Hn have the same complexification, one may adapt computations from hyperkähler geometry, to show that hypersymplectic manifolds are Ricci-flat. Hence hypersymplectic manifolds are examples of Calabi– Yau manifolds of signature .2n; 2n/; these are often called neutral Calabi–Yau. Such spaces are of interest in physics as they provide examples, as in the positive definite case, of sigma-models with supersymmetry [22], [9], [5]. A neutral Calabi–Yau will give (2,1) supersymmetry, while a hypersymplectic manifold will give twisted (4,1) supersymmetry. The basic example of a hypersymplectic manifold is Bn . Identifying Bn with C n;n as above one has I.z; w/ D .zi; wi /, S.z; w/ D .w; z/ and one finds that g D Re !I D

1 2i

n X

dzk d zNk dwk d wN k ;

kD1 n X

.dzk ^ d zN k C dwk ^ d wN k /;

kD1

!S C i !T D

n X

dwk ^ d zN k :

kD1

Note that !S C i !T is a holomorphic .2; 0/-form with respect to I (this is true in general on hypersymplectic manifolds; cf. the hyperkähler case where !J C i !K is an I -holomorphic (2,0)-form). Many examples of hypersymplectic structures are known on Lie groups. Kamada [23] was lead to examples of this in his investigation of hypersymplectic structures on compact 4-manifolds. Such manifolds must be compact complex surfaces (for the complex structure I ) with trivial canonical bundle, as !S Ci !T provides a trivialisation. In fact Kamada was able to further show that they must be tori or primary Kodaira surfaces (cf. also Petean [28]). Kamada classified hypersymplectic structures on the Kodaira surfaces, and showed that some were non-flat. (Note that such surfaces have odd first Betti number so cannot admit a Riemannian Kähler structure). The primary Kodaira surfaces are T 2 -bundles over T 2 , but may also be regarded as nilmanifolds nG for G a 2-step nilpotent Lie group. Examples on 2-step nilmanifolds in higher dimensions were obtained in [15], while [2] produces examples on 3-step nilmanifolds. In the non-compact realm, hypersymplectic structures on solvable Lie groups have been studied in [1], [2], [3] (in the last case from the perspective of complex product structures); in particular the four-dimensional examples have been classified. There are also homogeneous symmetric examples [10], [24]. Like their hyperkähler cousins, hypersymplectic structures have also been studied in the integrable systems literature. Dunajski [14] has studied hypersymplectic 4manifolds with a one-parameter group of symmetries. He shows that this gives a


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special class of Lorentzian Einstein–Weyl structures in dimension 3 characterised by solutions of an integrable PDE system of hydrodynamic type. We briefly mention here a link with another type of neutral signature geometry, that of para-quaternionic Kähler manifolds. These bear the same relationship to hypersymplectic manifolds that quaternionic Kähler manifolds do to hyperkähler ones. More precisely, a para-quaternionic Kähler manifold of dimension 4n 8 has an Sp.n; B/ Sp.1; B/ structure, that is a rank 3 subbundle of End.TM / locally generated by elements I , S, T satisfying (3.1), (3.2). Moreover we require that this subbundle should be preserved by the Levi-Civita connection. But I; S; T individually are neither parallel nor globally defined in general. Para-quaternionic Kähler manifolds are quite closely analogous to quaternionic Kähler ones. Their curvature tensor decomposes as the sum of a hypersymplectic curvature tensor and a scalar multiple of the curvature tensor of the basic model space BP n D

Sp.n C 1; B/ : Sp.n; B/ Sp.1; B/

Moreover the Swann bundle construction [29], which builds a hyperkähler metric on the total space of a .R4 f0g/=Z2 -bundle over a quaternionic Kähler manifold, can be modified so as to construct a hypersymplectic structure on the total space of a bundle over a para-quaternionic Kähler base. Recent results on these geometries were surveyed in [10].

4 Hypersymplectic quotients Given a symplectic manifold .M; !/ with group action G preserving the symplectic structure, one can in good cases form a quotient symplectic manifold by the Marsden– Weinstein construction. One looks for a moment map, a map W M ! g that is equivariant with respect to the given action on M and the coadjoint action on g , and satisfies d.Y /./ D !.X ; Y /: (4.1) for all Y 2 TM and 2 g. (Here X denotes the vector field on M associated to by differentiating the action). Now if G acts freely and properly on 1 .0/ then 1 .0/=G is a symplectic manifold. (Note that 0 may be replaced here by any central element of g ). The key point is that the restriction of ! to 1 .0/ at p has kernel equal to the space Gp D f X .p/ W 2 g g which is tangent to the orbits, so we get a nondegenerate form on the quotient 1 .0/=G. Note also that (4.1) shows that ker d is the symplectic orthogonal to Gp (equivalently, in the Kähler case, the metric orthogonal of I Gp ). Hence if G acts freely on 1 .0/, then is of maximal rank at all points of 1 .0/, and hence 1 .0/ and so (if G acts properly) 1 .0/=G are smooth. For hyperkähler or hypersymplectic manifolds, we consider moment maps for each of the triple of 2-forms !I , !J , !K (respectively !I , !S , !T ). We combine these into a hyperkähler moment map W M ! Im H ˝ g or a hypersymplectic moment map

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W M ! Im B ˝ g . It is convenient to split the moment map into real and complex parts I and J C iK (respectively S C iT ). In the hyperkähler situation, the work of Hitchin, Karlhede, Lindström and Roˇcek [19] shows, as in the symplectic case, that 1 .0/=G is hyperkähler provided G acts freely and properly on 1 .0/. We discuss the smoothness argument briefly. Now ker d is the metric orthogonal of I Gp CJ Gp CK Gp . As G acts on 1 .0/, we see Gp is tangent to 1 .0/ hence orthogonal to I Gp ; J Gp and K Gp . Hence these three spaces are mutually orthogonal and so their sum is direct. Hence the rank of d equals dim M dim ker d D 3 dim Gp , and if the action is free this equals 3 dim G so the rank is maximal and 1 .0/=G is smooth. Hitchin [20] has considered hypersymplectic quotients. It is now harder to avoid degeneracies of the hypersymplectic structure in the quotient; more precisely the degeneracy spaces of the fundamental 2-forms restricted to 1 .0/ are: ker !I j1 .0/ D G CS.G \ G ? / C T .G \ G ? / and cyclically. Therefore the conditions we need to guarantee that the hypersymplectic quotient is a smooth hypersymplectic manifold are: (F) G should act freely and properly on 1 .0/, (S) the rank of d should be 3 dim g at each point of 1 .0/; equivalently dim.I Gp CJ Gp CK Gp / D 3 dim g

for all p 2 1 .0/;

(D) Gp \ Gp? D f0g at each point p of 1 .0/, Conditions (F) and (S) guarantee that the quotient M is a smooth manifold. Note that, unlike in the hyperkähler case, (S) does not follow automatically from (F), as the metric g is now indefinite so orthogonality of the sum I Gp CJ Gp CK Gp does not imply that the sum is direct. When (F) and (S) are satisfied, the symplectic forms !I , !S and !T descend to closed two-forms on M . Now condition (D) is, from above, just the statement that the induced forms define a non-degenerate hypersymplectic structure. Note that when (F) is satisfied, smoothness of the quotient follows from nondegeneracy (D). For if we have IX C J Y C KZ D 0 for X; Y; Z 2 Gp , then taking the inner product with I U for an arbitrary U 2 Gp gives g.U; X / D g.I U; IX / D 0 so nondegeneracy implies X D 0, and similarly Y D Z D 0. Hence the sum I Gp CJ Gp CK Gp is direct, as required. Remark 4.1. Conditions (F) and (S) are usually fairly easy to arrange, but (D) is more difficult. Therefore many of the examples of hypersymplectic quotients are smooth manifolds but have a hypersymplectic structure only defined on the complement of a degeneracy locus. Some noncompact examples where the hypersymplectic structure is defined everywhere appear in [11]. We also note that in the case of a free circle action with Killing field X , conditions (S) and (D) reduce to the condition that g.X; X / is nowhere zero on 1 .0/.


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5 Toric constructions In Kähler geometry a particularly beautiful set of examples are the toric varieties, Kähler manifolds of real dimension 2n whose geometry is invariant under the Hamiltonian action of an n-dimensional torus with generic orbits of dimension n. Geometric questions about toric varieties usually reduce to combinatorics, making them a popular choice of examples for detailed study. Delzant [13] and Guillemin [17] have shown how to construct toric varieties as symplectic quotients of flat space C d by a torus action. Inspired by this, Bielawski and Dancer studied hyperkähler quotients of flat Hd by the action of a torus. In [7] these were called toric hyperkähler spaces but more recent authors such as M. Harada and N. Proudfoot ([18], for example) use the more vivid terminology hypertoric spaces. In the hypersymplectic situation it is thus natural to look at reductions of Bd D C d;d by a torus action. In §2, we already noted that there is a d -dimensional torus T d that acts on C d;d by .zk ; wk / 7! .e ik zk ; e ik wk / and this preserves the hypersymplectic structure described in §3. Following [17], a subtorus of N of T d may be described as follows. Let fe1 ; : : : ; ed g denote the standard basis for Rd . Consider a linear map ˇ W Rd ! Rn ;

ˇ.ek / D uk ;

for some vectors u1 ; : : : ; ud spanning Rn . Then n D ker ˇ is a linear subspace of Rd . Regarding the latter as the Lie algebra of T d , we view n as the Lie algebra of an Abelian subgroup N of T d . This subgroup is closed, and hence compact, if the vectors uk are integral, i.e., lie in the standard lattice Zn Rn . More precisely, we take N to be the kernel of exp ıˇ ı exp1 W T d ! T n . This is well-defined because of the integrality of the uk . We may write the moment maps for this action of N on C d;d as follows: I .z; w/ D

d X kD1

1 .jzk j2 2

S C iT .z; w/ D

d X

C jwk j2 /˛k C c1 ;

i zNk wk ˛k C c2 C i c3 ;

kD1

where ˛k is the orthogonal projection of ek to n. The vectors cj lie in n, so we may P / / choose scalars .j such that cj D dkD1 .j ˛k . k k Remark 5.1. This differs from the hyperkähler moment map in that in the second equation zk is conjugated, and, more importantly, in the first equation we have a sum of squares of absolute values rather than a difference of squares. In this respect the hypersymplectic case is closer to the Kähler situation, where the moment map for the P torus action on C d is just 12 dkD1 jzj2k ˛k .

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The hypersymplectic moment map therefore fails to be surjective, by contrast with the hyperkähler case. Rather the image will be the intersection of a collection of cones. Roughly speaking, the cones play the role in hypersymplectic geometry that half-spaces do in Kähler (or more generally symplectic) geometry. Hence intersections of cones are the hypersymplectic analogues of polyhedra in symplectic geometry. Example 5.2. As a very simple example take d D 1 and N D T d , so n D 0, i.e. consider the T 1 -action on B D C 1;1 . We have the inequality jzj2 C jwj2 > 2 jzwj, N and in fact we find the image of D I i C S s C T t is f .a; b/ 2 R C W a c1 > jb .c2 C i c3 /j gI

(5.1)

a solid cone in R3 . The T 1 -action is free except where a c1 D 0 D b .c2 C i c3 /. Moreover the fibre of over .a; b/ consists of two T 1 -orbits when we have strict inequality in (5.1) and of a unique T 1 -orbit if we have equality in (5.1). Explicitly, we can take the absolute value of the complex moment map equation and eliminate jzj using the real equation to find r q .a c1 /2 jb .c2 C i c3 /j2

jwj D C a c1 ˙

Now for each of the two choices for jwj, the complex equation determines .z; w/ up to the T 1 action. Note in particular that the fibre of may be disconnected, in contrast to the symplectic or hyperkähler situation. In particular, the hypersymplectic quotient in this example at a level in the interior of the cone is a pair of points. Let us now return to the general case of the moment map for N T d . Using the description of n as the kernel of ˇ, and dualising, we obtain a short exact sequence ˇ

0 ! Rn ! Rd ! n ! 0; where W n ! Rd is the inclusion. We have ˛k D ek

and

ˇ .a/ D

d X

ha; uk iek ;

kD1

where we identify Rd with its dual via an inner product with respect to which the ek are orthonormal. From our description above of the moment maps, and the fact that the kernel of 1 is the image of ˇ , we see that .z; w/ lies in 1 .0/ D I1 .0/ \ 1 S .0/ \ T .0/ if n n and only if there exist a 2 R and b 2 C such that .1/ ha; uk i D 12 .jzk j2 C jwk j2 / C k ;

hb; uk i D i zNk wk C

.c/ ; k

(5.2) (5.3)


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for k D 1; : : : ; d and .c/ D .2/ C i .3/ . k k k n As u1 ; : : : ; ud span R , we see that .a; b/ is determined by .z; w/ 2 1 .0/. Moreover, .a; b/ is invariant under the T d -action on .z; w/. We denote the hypersymplectic quotient 1 .0/=N by M . When the conditions (F),(S),(D) of section 4 are satisfied, M is a hypersymplectic space of dimension 4n with a residual action of the torus T n D T d =N . Hence M is a hypersymplectic analogue of toric varieties and of the hypertoric varieties of [7]. We have a map W M ! R3n ; .z; w/ D .a; b/; where a and b are as in equations (5.2) and (5.3). When the quotient M is a smooth hypersymplectic manifold is simply the moment map for the action of T n . However note that is defined as a T n -invariant map even when conditions (F), (S) and (D) are not satisfied. Let us define ak D ha; uk i .1/ ; k

bk D hb; uk i .c/ ; k

where .c/ D .2/ C i .3/ . Motivated by the four-dimensional Example 5.2, we k k k introduce the solid convex cones Kk , their boundaries Wk and vertices Vk given by Kk D f .a; b/ 2 Rn C n W ak > jbk j g; Wk D f .a; b/ 2 Rn C n W ak D jbk j g; Vk D f .a; b/ 2 Rn C n W ak D 0 D jbk j g: For a given x D .z; w/, let J D f k W x 2 Vk g;

L D f ` W x 2 W` g:

Proposition 5.3. The image of the moment map is the convex set KD

d \

Kk R3n :

iD1

The induced map Q W M=T n ! K is finite-to-one with the preimage of .a; b/ containing 2d jLj orbits of T n . The convex set K is the hypersymplectic analogue of the Delzant polytope for toric varieties. In the hyperkähler case, the analogous space is just all of R3n , as the moment map is now surjective. We can use to obtain some information about the topology of the hypersymplectic quotient M . Theorem 5.4. Let M D 1 .0/=N with N 6 T d given by integral vectors u1 ; : : : ; ud 2 Rn . Then

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1. M is connected if and only if for each k D 1; : : : ; d , Wk \ K is nonempty. 2. M is compact if and only if the convex polyhedra f s 2 Rn W hs; uk i > k ; k D 1; : : : ; d g are bounded for each choice of 1 ; : : : ; d 2 R. We now examine conditions (F), (S) and (D). It turns out that these may be analysed in terms of the combinatorics of the moment map. Freeness may be determined using the techniques of Delzant [13] and Guillemin [17] for Kähler metrics on toric varieties. Since (F) and (D) imply (S), the following result suffices to determine when we obtain smooth hypersymplectic structures on M . Proposition 5.5. The freeness condition (F) is satisfied at each .z; w/ 2 1 .0/ if and only if at each x 2 K the vectors fuk W x 2 Vk g are contained in a Z-basis for the integral lattice Zn Rn . The non-degeneracy condition (D) fails at some point .z; w/ 2 1 .0/ if and only if P there exist scalars 1 ; : : : ; d not all zero and a vector s 2 Rn such that dkD1 k uk D 0 and 4 k2 .ak2 jbk j2 / D hs; uk i for k D 1; : : : ; d , where .z; w/ D .a; b/. The smoothness condition (S) in the presence of a locally free action of N may be stated in terms of injectivity of certain linear maps ƒ.a;b/ depending on .a; b/ 2 K. To be precise, for a subset P f1; : : : ; d g, let RP be the subspace of Rd spanned by ek for k 2 P . Now write nL;J for the kernel of the map RLnJ ! Rn = Im.ˇjRJ / induced by ˇ. Then condition (S) holds only if ƒ.a;b/ W nL;J ˝.R C/ ! CLnJ ;

ƒ.a;b/ .ck ; dk / D .ak dk C bk ck /

is injective for all .a; b/ 2 K. Although these conditions may seem rather technical, they have some useful immediate consequences. For example, this last version of condition (S) holds trivially at points where L is empty. This is true of points of the combinatorial interior of K, which is defined to be the set CInt.K/ D K

d [

Wk :

kD1

Moreover, one can show using Proposition 5.5 that the degeneracy locus over the combinatorial interior has codimension at least one. Theorem 5.6. If the combinatorial interior of K is non-empty, then a dense open subset of M D 1 .0/=N carries a smooth hypersymplectic structure.


107

On the other hand, there are simple situations in which conditions (S) and (D) fail. If more than 3n of the Wk ’s meet at x 2 K then the smoothness condition (S) cannot hold at the common point. Similarly, if n C 1 of the Wk ’s meet at an x 2 K then condition (D) fails. This indicates that one can expect examples where the quotient is smooth, but the induced hypersymplectic structure has degeneracies. In fact, one can show that if the quotient M is compact, then the degeneracy locus is always non-empty. This occurs for example when taking the quotient by the diagonal circle in T d . Example 5.7. Non-trivial non-compact examples of this construction without degeneracies may be given in all dimensions as follows. Take d D n C 1, put uk D ek for k D 1; : : : ; n and let unC1 D e1 C C en . Take all the .i/ to be zero apart from k .1/ nC1 D < 0. Then K is the set f.a; b/ 2 Rn C n W ak jbk j .k D 1; : : : ; n/g; where here ak ; bk are just the kth coordinates of a; b. Note that as is strictly ˇ ˇ P positive, P K lies in the interior of KnC1 D f.a; b/ 2 Rn C n W C dkD1 ak ˇ dkD1 bk ˇ. In particular, WnC1 does not meet K. One may now directly check that conditions (F), (S) and (D) hold for this configuration of nC1 cones in R3n and that resulting hypersymplectic quotient M is smooth and non-degenerate. Topologically M is a disjoint union of two copies of R4n , however the induced metric is not flat. Example 5.8. Finally, we consider an example of a quotient of Bd by a noncompact group. Our example is a hypersymplectic analogue of the construction of the Taub-NUT metric as a hyperkähler quotient in [6, Addendum E]. We consider the R-action on C 2;2 given by: .z1 ; z2 ; w1 ; w2 / 7! .e it z1 ; z2 C t; e it w1 ; w2 /: The action is hypersymplectic with moment map 1 I W .z; w/ 7! .jz1 j2 C jw1 j2 / C Im z2 C c1 ; 2 S C iT W .z; w/ 7! iz1 wN 1 wN2 C c2 C i c3 ; for arbitrary constants c1 ; c2 ; c3 . Considering the quotient 1 .0/=R, we observe that setting the real part of z2 to zero gives a slice for the R-action, and now each choice of .z1 ; w1 / gives a unique solution to the equations. Hence the hypersymplectic quotient is diffeomorphic to R4 (like the Taub-NUT space). However as the normsquare of the Killing field for the action is 2.jz1 j2 jw1 j2 C 1/ we see the degeneracy locus of the symplectic structure is nonempty. Remark 5.9. One can also perform an analysis in the spirit of this section for the hypersymplectic Rd -actions given by zk ! 7 cosh. k /zk C sinh. k /wN k ; 7 sinh. k /zNk C cosh. k /wk ; wk !

108


or equivalently, in the variables k D zNk C wk ; k D zNk wk , by k 7! e k k ;

k 7! e k k :

The moment map for the action of N Rd is now given by I W .z; w/ 7!

d i X .N k k k Nk /˛k C c1 4 kD1

S C iT W .z; w/ 7!

d 1X k k ˛k C c2 C i c3 : 2 kD1

6 Cut constructions In the mid-90s Lerman [26] introduced a cut construction for symplectic manifolds with circle action. In [12] we described an analogous construction for hyperkähler manifolds. There is also an analogue in hypersymplectic geometry, as we now describe. The general idea of the cut construction is to take the product of the original S 1 manifold M with another (rather simple) space X with the same type of geometric structure and also with a circle action, such that the moment reduction of X is a point. The reduction of M X by the antidiagonal circle action therefore gives a new space Mcut of the same dimension as M , inheriting a circle action from the diagonal action on M X. The precise way in which M and Mcut are related can be understood in terms of the moment map geometry of the circle action on X . For Lerman’s symplectic cut one takes X D C. Now the moment map W X ! R is just the map z 7! jzj2 , which expresses X as a trivial circle bundle over the nonnegative half-line with point fibre over the origin. The moment map for the antidiagonal circle action on M X is just , where is the moment map for the circle action on M . This has the effect that the cut space Mcut at level of a symplectic manifold M with moment map may be formed from M by discarding the set fm W .m/ < g and collapsing circle fibres on the boundary 1 ./ of the remaining set. (Of course, we may also do the same construction with the inequality reversed, by considering the diagonal rather than antidiagonal action). As an example, if M is flat C n we may obtain as Mcut either CP n or the blowup of C n at a point, depending on the choice of direction in the inequality. In the hyperkähler case, we naturally take X D H. Now the moment map HK W H 7! R3 is given by the Hopf fibration on distance spheres. In contrast to the symplectic case, HK is a surjection and gives a nontrivial circle fibration away from the origin. At the origin, as before, the fibre is a point. The upshot is that the hyperkähler analogue of the cut (i.e. the hyperkähler quotient of M H at level ) still involves collapsing circle fibres on 1 ./. However no part of the manifold is now discarded. We therefore refer to the space produced by this construction as the modification Mmod rather than the cut space. Moreover the complements M 1 ./


109

and Mmod .1 ./=S 1 / are not diffeomorphic, but rather related by having a third space sitting over them as the total space of circle bundles. The modification process may be iterated. For example starting with M D H we obtain the Gibbons–Hawking multi-instanton spaces [16]. In the hypersymplectic case the natural choice for X is C 1;1 . As explained in Example 5.2, the associated moment map HS W X ! R3 D R C has image the cone f.a; b/ W a > jbjg. Moreover HS induces a 2:1 map from C 1;1 =S 1 onto the cone, branched over the boundary. Given a hypersymplectic manifold M with circle action and moment map , its cut at level will be the reduced space O 1 ./=S 1 , where O D HS is the moment map for the antidiagonal action on M C 1;1 . We have a diagram M

p

O 1 ./ ! Mcut ;

where p is the quotient map for the circle action and is projection from O 1 ./ M C 1;1 to M . Now the image of is the subset of M : M D fm 2 M W R .m/ R > jC .m/ C j g; that is, the preimage under of a cone with vertex . So, as in the symplectic case, but unlike the hyperkähler case, we are indeed removing part of the hypersymplectic manifold, that lying over the exterior of the cone. Observe also that the fibre of over a point in M where inequality is strict is two circles. Also, is injective over 1 ./ and has fibre equal to a single circle over other points of M where equality holds. As in the hyperkähler case, the circle fibrations in will be non-trivial. In some cases, depending on the shape of .M /, and the choice of cone vertex .R ; C / the hypersymplectic cut may give a compactification of .M / and even of M , as in the symplectic case. However the hypersymplectic structure may degenerate on a locus within the compactification. For example, we may consider hypersymplectic cuts of C 1;1 . These may be viewed as hypersymplectic quotients of C 2;2 by a circle subgroup of T 2 . In the terminology of §5 we are taking d D 2 and n D 1, so the combinatorial data involves two cones in R3 . If these cones point in opposite directions their intersection is compact and we obtain a compact hypersymplectic quotient with nonempty degeneracy locus. On the other hand if we choose the cones to point in the same direction we will obtain a noncompact quotient. Indeed we may in this situation obtain a quotient with empty degeneracy locus by taking one cone to be inside the other.

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[3] Andrada, A., and Salamon, S., Complex product structures on Lie algebras. Forum Math. 17 (2005), 261–295. 99, 100 [4] Atiyah, M. F., and Hitchin, N. J., The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures, Rice University, Princeton University Press, Princeton, New York 1988. 99 [5] Barrett, J., Gibbons, G. W., Perry, M. J., Pope, C. N., and Ruback, P. J., Kleinian geometry and the N D 2 superstring. Int. J. Mod. Phys. A 9 (1994), 1457–1493. 100 [6] Besse, A. L., Einstein manifolds. Ergeb. Math. Grenzgeb. 10, Springer-Verlag, Berlin 1987. 107 [7] Bielawski, R., and Dancer, A. S., The geometry and topology of toric hyperkähler manifolds. Comm. Anal. Geom. 8 (4) (2000), 727–760. 103, 105 [8] Blair, D. E., Davidov, J., and Muskarov, O., Hyperbolic twistor spaces. Rocky Mountain J. Math. 35 (2005), 1437–1465. 99 [9] Carvalho, M., Helayël-Neto, J. A., and de Oliveira, M. W., Locally product structures and supersymmetry. Lett. Math. Phys. 64 (2) (2003), 93–104. 97, 100 [10] Dancer, A., Jørgensen, H. R. and Swann, A., Metric geometries over the split quaternions. Rend. Sem. Mat. Pol. Torino 63 (2005), 119–139. 98, 100, 101 [11] Dancer, A., and Swann, A., Toric hypersymplectic quotients. Amer. Math. Soc. 359 (2007) 1265–1284. 98, 102 [12] Dancer, A., and Swann, A., Modifying hyperkähler manifolds with circle symmetry. Asian J. Math. 10 (4) (2006), 815–826. 108 [13] Delzant, T., Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France 116 (3) (1988), 315–339. 103, 106 [14] Dunajski, M., A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type. J. Geom. Phys. 51 (2004), 126–137. 100 [15] Fino, A., Pedersen, H., Poon, Y.-S., and Weye Sørensen, M., Neutral Calabi-Yau structures on Kodaira manifolds Comm. Math. Phys. (2) 248 (2004), 255–268. 100 [16] Gibbons, G. W., and Hawking, S. W., Gravitational multi-instantons. Phys. Lett. B 78 (1978), 430–432. 109 [17] Guillemin, V., Kaehler structures on toric varieties J. Differential Geom. 40 (2) (1994), 285–309. 103, 106 [18] Harada, M., and Proudfoot, N., Properties of the residual circle action on a hypertoric variety. Pacific J. Math. 214 (2) (2004), 263–284 103 [19] Hitchin, N. J., Karlhede, A., Lindström, U., and Roˇcek, M., HyperKähler metrics and supersymmetry. Comm. Math. Phys. 108 (1987), 535–589. 102 [20] Hitchin, N. J., Hypersymplectic quotients. Acta Acad. Sci. Tauriensis 124 (1990), no. suppl., 169–180. 97, 99, 102 [21] Jørgensen, H. R., Contact structures and Einstein metrics of split signature. Ph.D. thesis, University of Southern Denmark, Odense, 2005. 99 [22] Hull, C. M., Actions for .2; 1/ sigma models and strings. Nuclear Phys. B 509 (1–2) (1998), 252–272. 97, 100


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[23] Kamada, H., Neutral hyper-Kähler structures on primary Kodaira surfaces. Tsukuba J. Math. 23 (2) (1999), 321–332. 100 [24] Kath, I., and Olbrich, M., New examples of indefinite hyper-Kähler symmetric spaces. J. Geom. Phys. 57 (8) (2007), 1697–1711. 100 [25] LeBrun, C. R., H -space with a cosmological constant. Proc. Roy. Soc. London Ser. A 380 (1778) (1982), 171–185. 99 [26] Lerman, E., Symplectic cuts. Math. Res. Lett. 2 (3) (1995), 247–258. 108 [27] Okubo, S., Introduction to octonion and other non-associative algebras in physics. Montroll Memorial Lecture Ser. Math. Phys. 2, Cambridge University Press, Cambridge 1995. 97 [28] Petean, J., Indefinite Kähler-Einstein metrics on compact complex surfaces. Comm. Math. Phys. 189 (1997) 227–235. 100 [29] Swann, A. F., Hyperkähler and quaternionic Kähler geometry. Math. Ann. 289 (1991), 421–450. 101

Anti-self-dual conformal structures in neutral signature Maciej Dunajski and Simon West

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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Local geometry in neutral signature . . . . . . . . 2.1 Conformal compactification . . . . . . . . . 2.2 Spinors . . . . . . . . . . . . . . . . . . . . 2.3 ˛- and ˇ-planes . . . . . . . . . . . . . . . . 2.4 Anti-self-dual conformal structures in spinors

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Integrable systems and Lax pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.1 Curvature restrictions and their Lax pairs . . . . . . . . . . . . . . . . . . . . . 120

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Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.1 Non-null case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2 Null case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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Global results . . . . . . . . . . . . . . . . . . . 6.1 Topological restrictions . . . . . . . . . . . 6.2 Tod’s scalar-flat Kähler metrics on S 2 S 2 6.3 Compact neutral hyper-Kähler metrics . . . 6.4 Ooguri–Vafa metrics . . . . . . . . . . . .

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1 Introduction We begin with some well-known facts from Riemannian geometry. Given an oriented Riemannian 4-manifold .M; g/, the Hodge- operator is an involution on 2-forms. This induces a decomposition ƒ2 D ƒ2C ˚ ƒ2 (1) of 2-forms into self-dual and anti-self-dual components, which only depends on the conformal class Œg. Now choose g 2 Œg. The Riemann tensor has the index symmetry Rabcd D RŒabŒcd so can be thought of as a map R W ƒ2 ! ƒ2 . This map decomposes

114 under (1) as follows:

M. Dunajski and S. West

1

0 B CC C B B RDB B B @

s 12

C C

s 12

C C C C: C C A

(2)

The C˙ terms are the self-dual and anti-self-dual parts of the Weyl tensor, the terms are the tracefree Ricci curvature, and s is the scalar curvature which acts by scalar multiplication. The Weyl tensor is conformally invariant, so can be thought of as being defined by the conformal structure Œg. An anti-self-dual conformal structure is one with CC D 0. Such structures have a global twistor correspondence [3] which has been studied intensively; they have also been studied from a purely analytic point of view using elliptic techniques [51]. What happens in other signatures? In Lorentzian signature .C C C/, the Hodge- is not an involution (it squares to 1 instead of 1) and there is no decomposition of 2-forms. In neutral .C C / signature, the Hodge- is an involution, and there is a decomposition exactly as in the Riemannian case, depending on Œg. Thus anti-selfdual conformal structures exist in neutral signature. This article is devoted to their properties. At the level of PDEs, the difference between neutral and Riemannian is that in the neutral case the gauge-fixed anti-self-duality equations are ultrahyperbolic, whereas in the Riemannian case they are elliptic. This results in profound differences, both locally and globally. Roughly speaking, the neutral case is far less rigid than the Riemannian case. For instance, any Riemannian anti-self-dual conformal structure must be analytic by the twistor construction. In the neutral case there is no general twistor construction, and in fact neutral conformal structures are not necessarily analytic. This lack of analyticity provides scope for rich local behaviour, as wave like solutions exists. Assuming symmetries in the form of Killing vectors, one often finds that the equations reduce to integrable systems. Different integrable systems can be obtained by combining symmetries with geometric conditions for a metric in a conformal class. The story here in some sense parallels the case of the self-dual Yang–Mills equations in neutral signature, where imposing symmetries leads to many well-known integrable systems [37]. The subject of this review is the interplay between the ultrahyperbolic differential equations, and the anti-self-duality condition. We shall make a historical digression, and note that both concepts arouse separately in mid 1930s. Indeed, the ultrahyperbolic wave equation appears naturally in integral geometry, where the X-ray transform introduced in 1938 by John [25] can be used to construct all its smooth solutions. This takes a smooth function on RP 3 (a compactification of R3 ) and integrates it over an oriented geodesic. The resulting function is defined on the Grassmannian Gr2 .R4 / of two-planes in R4 and satisfies the wave equation for a flat metric in .C C / signature. To see it explicitly consider a smooth function

Anti-self-dual conformal structures in neutral signature

115

f W R3 ! R which satisfies R suitable decay conditions at infinity. For any oriented line L R3 define .L/ D L f , or Z 1 .x; y; w; z/ D f .xs C z; ys w; s/ds; (3) 1

where we have chosen an explicit parametrisation of all lines which are not perpendicular to the x3 axis. The dimension of the space of oriented lines is 4. This is greater than the dimension of R3 , and one does not expect to be arbitrary. Differentiating under the integral sign shows that must satisfy the wave equation in neutral signature @2 @2 C D 0: @x@w @y@z

(4)

John has demonstrated that equation (4) is the only condition constraining the range of the integral transform in this case, and that all smooth solutions to (4) arise by (3) from some f . One can regard the X -ray transform as the predecessor of twistor theory. In this context RP 3 should be regarded as a totally real submanifold of a twistor space CP 3 . In fact Woodhouse [56] showed that any local solution of (4) can be generated from a function on the real twistor space of R2;2 . The twistor space is the set of totally null self-dual 2-planes and is three-dimensional, so we are again dealing with a function of three variables. To obtain the value at a point p, one integrates f over all the planes through p. This was motivated by the Penrose transform with neutral reality conditions. It is less well known that the ASD equation on Riemann curvature dates back to the same period as the work of John (at least 40 years before the seminal work of Penrose [45] and Atyiah–Hitchin–Singer [3]). It arose in the context of Wave Geometry – a subject developed in Hiroshima during the 1930s. Wave Geometry postulates the existence of a privileged spinor field which in the modern super-symmetric context would be called a Killing spinor. The integrability conditions come down to the ASD condition on a Riemannian curvature of the underlying complex space time. This condition implies vacuum Einstein equations. The Institute at Hiroshima where Wave Geometry had been developed was completely destroyed by the atomic bomb in 1945. Two of the survivors wrote up the results in a book [40]. In particular in [50] it was shown that local coordinates can be found such that the metric takes the form gD

@2 @2 @2 @2 dxdw C dydz C dydw C dxdz @x@w @y@z @y@w @x@z

(5)

and ASD vacuum condition reduces to a single PDE for one function : @2 @2 @2 @2 D 1: @x@w @y@z @x@z @y@w

(6)

This is nowadays known as the first heavenly equation after Plebanski who rediscovered it in 1975 [47]. If .; x; y; w; z/ are all real, the resulting metric has neutral signature. The flat metric corresponds to D wx C zy. Setting D wx C zy C

.x; y; w; z/

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we see that up to the linear terms in the heavenly equation reduces to the ultrahyperbolic wave equation (4). Later we shall see that the twistor method of solving (6) is a non-linear version of John’s X-Ray transform. This concludes our historical digression. The article is structured as follows. In Section 2 we introduce the local theory of neutral anti-self-dual conformal structures. It is convenient to use spinors, which for us will be a local tool to make the geometric structures more transparent. In Section 3 we explain how neutral ASD conformal structures are related to Lax pairs and hence integrable systems. We review various curvature restrictions on a metric in a conformal class (Ricci-flat, scalar flat Kähler etc), and show how these can be characterised in terms of their Lax pair. Section 4 is devoted to symmetries; in this section we make contact with many well known integrable systems. We discuss twistor theory in Section 5, explaining the differences between the Riemannian and neutral case, and describing various twistor methods of generating neutral ASD conformal structures. Despite the ultrahyperbolic nature of the equations, some strong global results have been obtained in recent years using a variety of techniques. We discuss these in Section 6. The subject of neutral anti-self-dual conformal structures is rather diverse. We hope to present a coherent overview, but the different strands will not all be woven together. Despite this, we hope the article serves a useful purpose as a path through the literature.

2 Local geometry in neutral signature 2.1 Conformal compactification. We shall start off by describing a conformal compactification of the flat neutral metric. Let R2;2 denote R4 with a flat .C C / metric. Its natural compactification is a projective quadric in RP 5 . To describe it explicitly consider Œx; y as homogeneous coordinates on RP 5 , and set Q D jxj2 jyj2 . Here .x; y/ are vectors on R3 with its natural inner product. The cone Q D 0 is projectively invariant, and the freedom .x; y/ .cx; cy/, where c ¤ 0 is fixed to set jxj D jyj D 1 which is S 2 S 2 . We need to quotient this by the antipodal map .x; y/ ! .x; y/ to obtain the conformal compactification1 R2;2 D .S 2 S 2 /=Z2 : Parametrising the double cover of this compactification by stereographic coordinates we find that the flat metric jd xj2 jd yj2 on R3;3 yields the metric g0 D 4

d d N dd N 4 N 2 .1 C / N 2 .1 C /

(7)

on S 2 S 2 . To obtain the flat metric on R2:2 we would instead consider the intersection of the zero locus of Q in R3;3; with a null hypersurface x0 y0 D 1. The metric g0 is conformally flat and scalar flat, as the scalar curvature is the difference between curvatures on both factors. It is also Kähler with respect to the natural complex structures on CP 1 CP 1 with holomorphic coordinates .; /. In 1

This compactification can be identified with the Grassmannian Gr2 .R4 / arising in the John transform (3).


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Section 6.2 we shall see that g0 admits nontrivial scalar-flat Kähler deformations [53] globally defined on S 2 S 2 . 2.2 Spinors. It is often convenient in four dimensions to use spinors, and the neutral signature case is no exception. The relevant Lie group isomorphism in neutral signature is SO.2; 2/ Š SL.2; R/ SL.2; R/=Z2 : (8) We shall assume that the neutral four manifold .M; g/ has a spin structure. Therefore there exist real two-dimensional vector bundles S; S 0 (spin-bundles) over M equipped with parallel symplectic structures ; 0 such that T M Š S ˝ S 0 is a canonical bundle isomorphism, and g.v1 ˝ w1 ; v2 ˝ w2 / D .v1 ; v2 / 0 .w1 ; w2 / for v1 ; v2 2 .S/ and w1 ; w2 2 .S 0 /. The two-component spinor notation [46] will used in the paper. The spin bundles S and S 0 inherit connections from the Levi-Civita connection such that , 0 are covariant constant. We use the standard convention in which spinor indices are capital letters, unprimed for sections of S and primed for 0 sections of S 0 . For example A denotes a section of S , the dual of S , and A a section of S 0 . The symplectic structures on spin spaces AB and A0 B 0 (such that 01 D 00 10 D 1) are used to raise and lower indices. For example given a section A of S we define a section of S by A WD B BA . 0 0 Spin dyads .oA ; A / and .oA ; A / span S and S 0 respectively. We denote a normalised null tetrad of vector fields on M by e 000 e 010 : eAA0 D e 100 e 110 This tetrad is determined by the choice of spin dyads in the sense that 0

oA oA eAA0 D e 000 ;

0

A oA eAA0 D e 100 ;

0

0

oA A eAA0 D e 010 ;

A A eAA0 D e 110 :

0

The dual tetrad of one-forms by e AA determine the metric by 0

0

0

0

0

0

g D AB A0 B 0 e AA ˝ e BB D 2.e 00 ˇ e 11 e 10 ˇ e 01 /

(9)

where ˇ is the symmetric tensor product. With indices, the above formula2 for g becomes gab D AB A0 B 0 . 0 0 The local basis † AB and † A B of spaces of ASD and SD two-forms are defined by 0 0 0 0 0 0 e AA ^ e BB D AB † A B C A B † AB : (10) Note that we drop the prime on 0 when using indices, since it is already distinguished from by the primed indices. 2

118

M. Dunajski and S. West 0

0

A vector V can be decomposed as V AA eAA0 , where V AA are the components of V 0 in the basis. Its norm is given by det.V AA /, which is unchanged under multiplication 0 of the matrix V AA by elements of SL.2; R/ on the left and right 0

0

0

A BB 0A V AA ! ƒB V ƒB 0 ;

0

0

ƒ 2 SL.2; R/; ƒ 2 SL.2; R/

giving (8). The quotient by Z2 comes from the fact that multiplication on the left and 0 right by 1 leaves V AA unchanged. Spinor notation is particularly useful for describing null structures. A vector V is 0 0 0 null when det.V AA / D 0, so V AA D A A by linear algebra. In invariant language, this says that a vector V is null iff V D ˝ where ; are sections of S; S 0 . The decomposition of a 2-form into self-dual and anti-self-dual parts is straightforward in spinor notation. Let FAA0 BB 0 be a 2-form in indices. Now FAA0 BB 0 D F.AB/.A0 B 0 / C FŒABŒA0 B 0 C F.AB/ŒA0 B 0 C FŒAB.A0 B 0 / D F.AB/.A0 B 0 / C cAB A0 B 0 C AB A0 B 0 C A0 B 0 AB : Here we have used the fact that in two dimensions there is a unique anti-symmetric matrix up to scale, so whenever an anti-symmetrized pair of spinor indices occurs we can substitute a multiple of AB or A0 B 0 in their place. Now observe that the first two terms are incompatible with F being a 2-form, i.e. FAA0 BB 0 D FBB 0 AA0 . So we obtain FAA0 BB 0 D AB A0 B 0 C A0 B 0 AB ; (11) where AB and A0 B 0 are symmetric. This is precisely the decomposition of F into self-dual and anti-self dual parts. Which is which depends on the choice of volume form; we choose A0 B 0 AB to be the self-dual part. Invariantly, we have ƒ2C Š S 0 ˇ S 0 ;

ƒ2 Š S ˇ S :

(12) 0

2.3 ˛- and ˇ-planes. Suppose at a point x 2 M we are given a spinor A 2 Sx0 . 0 0 A two-plane …x is defined by all vectors of the form V AA D A A , with varying A A0 B 0 0 0 A B 2 S. Now suppose V; W 2 …x . Then g.V; W / D A B AB D 0 since A0 B 0 is antisymmetric. Therefore we say the two-plane is totally null. Furthermore, the 2-form VŒa Wb is proportional to A0 B 0 AB ; i.e. the two-plane is self-dual. In summary, a spinor in S defines a totally null self-dual two-plane, which is called an ˛-plane. Similarly a spinor in S defines a totally null anti-self-dual two-plane, called a ˇ-plane. 2.4 Anti-self-dual conformal structures in spinors. A neutral conformal structure Œg is an equivalence class of neutral signature metrics, with the equivalence relation g e f g for any function f . Another way of viewing such a structure is as a linebundle valued neutral metric; we will not need this description because in most cases we will be working with particular metrics within a conformal class.


119

Choose a g 2 Œg. Then there is a Riemann tensor, which possesses certain symmetries under permutation of indices. In the same way that we deduced (11) for the decomposition of a 2-form in spinors, the Riemann tensor decomposes as [46] RAA0 BB 0 C C 0 DD 0 D CABCD A0 B 0 C 0 D 0 C CA0 B 0 C 0 D 0 AB CD C ABC 0 D 0 A0 B 0 CD C A0 B 0 CD AB C 0 D 0 s C .AC BD A0 B 0 C 0 D 0 C AB CD A0 D 0 B 0 C 0 /: 12 This is the spinor version of (2). Here CA0 B 0 C 0 D 0 , CABCD are totally symmetric, and correspond to CC , C in (2). The spinor A0 B 0 CD is symmetric in its pairs of indices, and corresponds to in (2). An anti-self-dual conformal structure is one for which CA0 B 0 C 0 D 0 D 0. In the next section we explain the geometric significance of this condition in more detail. It is appropriate here to recall the Petrov–Penrose classification [46] of the algebraic type of a Weyl tensor. In split signature this applies separately to CABCD and CA0 B 0 C 0 D 0 . In our case CA0 B 0 C 0 D 0 D 0 and we are concerned with the algebraic type of CABCD . One can form a real polynomial of fourth order P .x/ by defining A D .1; x/ and setting P .x/ D A B C D CABCD . The Petrov–Penrose classification refers to the position of roots of this polynomial, for example if there are four repeated roots then we say CABCD is type N. If there is a repeated root the metric is called algebraically special indexalgebraically special. There are additional complications in the split signature case [32] arising from the fact that real polynomials may not have real roots.

3 Integrable systems and Lax pairs In this section we show how anti-self-dual conformal structures are related to integrable systems and Lax pairs. Let g 2 Œg and let r denote the Levi-Civita connection on M. This connection induces spin connections on spin bundles which we also denote r. Let 0 us consider S 0 . The connection coefficients AA0 BC0 of r are defined by 0

0

0

0

rAA0 C D eAA0 . C / C AA0 BC0 B ; 0

0

where A is a section of S 0 in coordinates determined by the basis eAA0 . The AA0 BC0 symbols can be calculated in terms of the Levi-Civita connection symbols. They can 0 0 0 0 also be read off directly from the Cartan equations d e AA D e BA ^B A Ce AB ^B 0 A , 0 0 0 where B 0 C D AA0 BC0 e AA . See [46] for details. Now given a connection on a vector bundle, one can lift a vector field on the base to a horizontal vector field on the total 0 space. We follow standard notation and denote the local coordinates of S 0 by A . Then the horizontal lifts eQ AA0 of eAA0 are given explicitly by 0

eQ AA0 WD eAA0 C AA0 BC0 B Now we can state a seminal result of Penrose:

0

@ : @ C 0

120


Theorem 1 ([45]). Given a neutral metric g, define a two-dimensional distribution on S 0 by D D spanfL0 ; L1 g, where 0

LA WD A eQ AA0 :

(13)

Then D is integrable iff g is anti-self-dual. So when g is ASD, S 0 is foliated by surfaces. Since the LA are homogeneous in 0 the A coordinates, D defines a distribution on PS 0 , the projective version of S 0 . 0 0 The push down of D from a point A D A in a fibre of S 0 to the base is the 0 ˛-plane defined by A , as explained in Section 2.3. So the content of Theorem 1 is that g is ASD iff any ˛-plane is tangent to an ˛-surface, i.e. a surface that is totally null and self-dual at every point. Any such ˛-surface lifts to a unique surface in PS 0 , or a one parameter family of surfaces in S 0 . 3.1 Curvature restrictions and their Lax pairs. A more recent interpretation of Theorem 1 is to regard LA as a Lax pair for the ASD conformal structure. Working 0 0 on PS 0 , with inhomogeneous fibre coordinate D 1 = 0 , the condition that D commutes is the compatibility condition for the pair of linear equations L0 f D .eQ 000 C eQ 010 /f D 0 L1 f D .eQ 100 C eQ 110 /f D 0 to have a solution f for all 2 R, where f is a function on PS 0 . In integrable systems language, is the spectral parameter. Here we describe various conditions that one can place on a metric g 2 Œg on top of anti-self-duality. This provides a more direct link with integrable systems as in each case described below one can choose a spin frame, and local coordinates to reduce the special ASD condition to an integrable scalar PDE with corresponding Lax pair. 3.1.1 Pseudo-hyperhermitian structures. This is the neutral analogue of Riemannian hyperhermitian geometry. The significant point for us is that in four dimensions, pseudo-hyperhermitian metrics (defined below) are necessarily anti-self-dual. Consider a structure .M; I; S; T /, where M is a four-dimensional manifold and I; S; T are anti-commuting endomorphisms of the tangent bundle satisfying S 2 D T 2 D 1;

I 2 D 1;

ST D T S D 1:

(14)

This is called the algebra of para-quaternions [24] or split quaternions [12]. Consider the hyperboloid of almost complex structures on M given by aI CbS CcT , for .a; b; c/ satisfying a2 b 2 c 2 D 1. If each of these almost complex structures is integrable, we call .M; I; S; T / a pseudo-hypercomplex manifold. So far we have not introduced a metric. A natural restriction on a metric given a pseudo-hypercomplex structure is to require it to be hermitian with respect to each of the complex structures. This is equivalent to the requirement: g.X; Y / D g.IX; I Y / D g.SX; S Y / D g.TX; T Y /;

(15)


121

for all vectors X; Y . A metric satisfying (15) must be neutral. To see this consider the endomorphism S , which squares to the identity. Its eigenspaces decompose into C1 and 1 parts. Any eigenvector must be null from (15). So choosing an eigenbasis one can find 4 null vectors, from which it follows that the metric is neutral. Given a pseudo-hypercomplex manifold, we call a metric satisfying (15) a pseudohyperhermitian metric. Given a local pseudo-hypercomplex structure in four dimensions one can construct many pseudo-hyperhermitian metrics for it as follows. Take a vector field V and let .V; I V; S V; T V / be an orthonormal basis in which the metric has diagonal components .1; 1; 1; 1/. The fact that these vectors are linearly independent follows from (14). It is easy to check that (15) holds for any two vectors in the above basis, and hence by linearity for any .X; Y /. By varying the length of V one obtains a different conformal class. However, even the conformal class is not uniquely specified. To see this take a vector W that is null for the metric specified by V , and form a new metric by the same procedure using W . Then W is not null in this new metric, so this metric must be in a different conformal class. As mentioned above, it turns out that pseudo-hyperhermitian metrics are necessarily anti-self-dual. One way to formulate this is via the Lax pair formalism as follows: Theorem 2 ([14]). Let eAA0 be four independent vector fields on a four-dimensional real manifold M. Put L0 D e 000 C e 010 ;

L1 D e 100 C e 110 :

If ŒL0 ; L1 D 0

(16)

for every value of a parameter , then g given by (9) a pseudo-hyperhermitian metric on M. Given any four-dimensional pseudo-hyperhermitian metric there exists a null tetrad such that (16) holds. Interpreting as the projective primed spin coordinate as in Section 3, we see that a pseudo-hyperhermitian metric must be ASD from Theorem 1. Theorem 2 characterises pseudo-hyperhermitian metrics as those which possess a Lax pair containing no @ terms. We shall now discuss the local formulation of the pseudo-hyperhermitian condition as a PDE. Expanding equation (16) in powers of gives ŒeA00 ; eB00 D 0;

ŒeA00 ; eB10 C ŒeA10 ; eB00 D 0;

ŒeA10 ; eB10 D 0:

(17)

It follows from (17), using the Frobenius theorem and the Poincaré lemma that one can choose coordinates .p A ; w A / (A D 0; 1) in which eAA0 take the form eA00 D

@ ; @p A

eA10 D

@ @‚B @ ; @w A @p A @p B

122


where ‚B are a pair of functions satisfying a system of coupled non-linear ultrahyperbolic PDEs. @‚B @2 ‚C @2 ‚C C D 0: (18) @pA @w A @p A @pA @pB Note the indices here are not spinor indices, they are simply a convenient way of labelling coordinates and the functions ‚A . We raise and lower them using the standard antisymmetric matrix AB , for example pA WD p B BA , and the summation convention is used. 3.1.2 Scalar-flat Kähler structures. Let .M; g/ be an ASD four manifold and let J be a (pseudo-)complex structure such that the corresponding fundamental two-form is closed. This ASD Kähler condition implies that g is scalar flat, and conversely all scalar flat Kähler four manifolds are ASD [13]. In this section we shall show that in the scalar-flat Kähler case the spin frames can be chosen so that the Lax pair (13) consists of volume-preserving vector fields on M together with two functions on M. The following theorem has been obtained in a joint work of Maciej Przanowski and the first author. We shall formulate and prove it in the holomorphic category which will allow both neutral and Riemannian real slices. Theorem 3. Let eAA0 D .e 000 ; e 010 ; e 100 ; e 110 / be four independent holomorphic vector fields on a four-dimensional complex manifold M and let f1 ; f2 W M ! C be two holomorphic function. Finally, let be a nonzero holomorphic four-form. Put L0 D e 000 C e 010 f0 2

@ ; @

L1 D e 100 C e 110 f1 2

@ : @

(19)

Suppose that for every 2 CP 1 ŒL0 ; L1 D 0;

LLA D 0;

(20)

where LV denotes the Lie derivative. Then eO AA0 D c 1 eAA0 ;

where c 2 WD .e 000 ; e 010 ; e 100 ; e 110 /;

is a null-tetrad for an ASD Kähler metric. Every such metric locally arises in this way. Proof. First assume that there exists a tetrad eAA0 and two functions fA D .f0 ; f1 / such that equations (20) are satisfied. For convenience write down equations ŒL0 ; L1 D 0 in full ŒeA00 ; eB00 D 0; (21) ŒeA00 ; eB10 C ŒeA10 ; eB00 D 0;

(22)

ŒeA10 ; eB10 D AB f C e C10 ;

(23)

e

A

00 fA

D 0;

(24)

e A 10 fA D 0:

(25)


123

Define the almost complex structure J by J.eA10 / D ieA10 ;

J.eA00 / D i eA00 :

Equations (21), and (23) imply that this complex structure is integrable. Let g be a metric corresponding to eO AA0 by (9). To complete this part of the proof we need to show that a fundamental two-form ! defined by !.X; Y / D g.X; J Y / is closed. First observe that @ . .L0 ; L1 ; ; //jD0 : !D @ It is therefore enough to prove that † D .L0 ; L1 ; : :; : / is closed for each fixed . We shall establish this fact using equations (20), and d D 0. Let us calculate d † D d. .L0 ; L1 ; ; // D d.L0 . .L1 ; ; ; /// D LL0 . .L1 ; ; ; // L0 .d .L1 ; ; ; // D ŒL0 ; L1 C L1 LL0 . / L0 .L1 d / D L0 .LL1 L0 .L1 d. /// D 0: Therefore ! is closed which in the case of integrable J also implies r! D 0 [31]. Converse. The metric g is Kähler, therefore there exist local coordinates .w A ; wQ A / and a complex valued function D .w A ; wQ A / such that g is given by gD

@2 dw A d wQ B : @w A @wQ B

(26)

Choose a spin frame .oA0 ; A0 / such that the tetrad of vector fields eAA0 is 0

eA00 D oA eAA0 D

@ ; @w A

0

eA10 D A eAA0 D

@ @2 : A B @w @wQ @wQ B

The null tetrad for the metric (26) is eO AA0 D G 1 eAA0 , where G D det.g/ D

1 @2 @2 : 2 @wA @wQ B @w A @wQ B

The Lax pair (13) is LA D

@ @ @ @2 A B C lA : A @w @w @wQ @wQ B @

Consider the Lie bracket @2 @3 @ @2 @ A C l A B C A B @w @wQ @wA @wQ B @wQ @wQ C @w @wQ @wQ B A 2 A @l @l A @ @ @l C C l : A @w A @w A @wQ B @wQ B @ @

ŒL0 ; L1 D 2

(27)

124


The ASD condition is equivalent to integrability of the distribution LA , therefore ŒLA ; LB D AB ˛ C LC for some ˛ C . The lack of @=@w A term in the Lie bracket above implies ˛ C D 0. Analysing other terms we deduce the existence of f D f .w A ; wQ A / 2 ker such that lA D 2 @f =@w A , and

@2 @ @2 @w A @wQ B @wQ C @wA @wQ B

D

@2 @f : @wA @w A @wQ C

(28)

The real-analytic .C C / slices are obtained if eAA0 ; ; f1 ; f2 are all real. In this case we alter our definition of J by J.eA10 / D eA10 ;

J.eA00 / D eA00 :

Therefore J 2 D 1, and g is pseudo-Kähler. In the Euclidean case the quadratic-form g and the complex structure 0

0

J D i.e A0 ˝ eA00 e A1 ˝ eA10 / are real but the vector fields eAA0 are complex. As a corollary from the last theorem we can deduce a formulation of the scalarflat Kähler condition [44]. Scalar-flat Kähler metric are locally given by (26) where .w A ; wQ A / is a solution to a 4th order PDE (which we write as a system of two second order PDEs ): @2 @ ln G @f D ; (29) @w A @w A @wQ B @wQ B f D

@2 @2 f D 0: A B @w @wQ @wA @wQ B

(30)

Moreover (29,30) arise as an integrability condition for the linear system L0 ‰ D L1 ‰ D 0, where ‰ D ‰.w A ; wQ A ; / and LA D

@ @f @ @ @2 C 2 A : A A B @w @w @wQ @wQ B @w @

(31)

To see this note that in the proof of Theorem 3 we have demonstrated that f 2 ker . In the adopted coordinate system D

@2 @2 ; A B @w @wQ @wA @wQ B

which gives (33). Solving the algebraic system (28) for @f =@w A yields (30).


125

3.1.3 Null-Kähler structures. A null-Kähler structure on a real four-manifold M consists of an inner product g of signature .CC/ and a real rank-two endomorphism N W T M ! T M parallel with respect to this inner product such that N2 D 0

and

g.NX; Y / C g.X; N Y / D 0

for all X; Y 2 T M. The isomorphism ƒ2 C .M/ Š Sym2 .S 0 / between the bundle of self-dual two-forms and the symmetric tensor product of two spin bundles implies that the existence of a null-Kähler structure is in four dimensions equivalent to the existence of a parallel real spinor. The Bianchi identity implies the vanishing of the curvature scalar. In [8] and [15] it was shown that null-Kähler structures are locally given by one arbitrary function of four variables, and admit a canonical form3 g D dwdx C dzdy ‚xx dz 2 ‚yy dw 2 C 2‚xy dwdz;

(32)

with N D dw ˝ @=@y dz ˝ @=@x. Further conditions can be imposed on the curvature of g to obtain non-linear PDEs for the potential function ‚. Define f WD ‚wx C ‚zy C ‚xx ‚yy ‚2xy :

(33)

• The Einstein condition implies that f D xP .w; z/ C yQ.w; z/ C R.w; z/; where P; Q and R are arbitrary functions of .w; z/. In fact the number of the arbitrary functions can be reduced down to one by redefinition of ‚ and the coordinates. This is the hyper-heavenly equation of Plebański and Robinson [48] for non-expanding metrics of type ŒN [Any]. (Recall that .M; g/ is called hyper-heavenly if the self-dual Weyl spinor is algebraically special). • The conformal anti-self-duality (ASD) condition implies a 4th order PDE for ‚ f D 0;

(34)

where is the Laplace–Beltrami operator defined by the metric g. This equation is integrable: It admits a Lax pair L0 D .@w ‚xy @y C ‚yy @x / @y C fy @ ; L1 D .@z C ‚xx @y ‚xy @x / C @x C fx @ : and its solutions can in principle be found by twistor methods [15], or the dressing approach [7]. 3 The local form (32) is a special case of Walker’s canonical form of a neutral metric which admits a two-dimensional distribution which is parallel and null [54]. Imposing more restrictions on Walker’s metric leads to examples of conformally Osserman structures, i.e. metrics for which the eigenvalues of the operator a Y a ! Cbcd X b Y c X d are constant on the unit pseudo-sphere fX 2 T M; g.X; X / D ˙1g. These metrics are all SD or ASD according to [6].

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• Imposing both conformal ASD and Einstein condition implies (possibly after a redefinition of ‚) that f D 0, which yields the celebrated second heavenly equation of Plebański [47]: ‚wx C ‚zy C ‚xx ‚yy ‚2xy D 0:

(35)

ASD ; NNN ww NNN w w NNN w w NN& w ww Pseudo-hyper-Kähler Null-Kähler GG GG pp8 GG ppp p p GG p G# ppp Einstein 3.1.4 Pseudo-hyper-Kähler structures. Suppose we are given a pseudo-hypercomplex structure as defined in the previous section, i.e. a two-dimensional hyperboloid of integrable complex structures. In the previous section we defined a pseudo-hyperhermitian metric to be a metric that is hermitian with respect to each complex structure in the family. If we further require that the 2-forms !I . ; / D g. ; I /;

!S . ; / D g. ; S /; !T . ; / D g. ; T /;

(36)

be closed, we call say g is pseudo-hyper-Kähler . These define three symplectic forms, and Hitchin has termed such structures hypersymplectic4 [23]. It follows from similar arguments to those in standard Riemannian Kähler geometry that .I; S; T / are covariant constant, and hence so are !I ; !S ; !T . As in the Riemannian case, pseudo-hyper-Kähler metrics are equivalent to Ricci-flat anti-self-dual metrics. One can deduce this by showing that the 2-forms (36) are selfdual, and since they are also covariant constant there exists a basis of covariant constant primed spinors. Then using the spinor Ricci identities one can deduce anti-self-duality and Ricci-flatness. See for details. The Lax pair formulation for a pseudo-hyper-Kähler metric is as follows: Theorem 4 ([1], [36]). Let eAA0 be four independent vector fields on a four-dimensional real manifold M, and be a 4-form. Put L0 D e 000 C e 010 ;

L1 D e 100 C e 110 :

If ŒL0 ; L1 D 0

(37)

LLA D 0;

(38)

1

for every 2 RP , and 1

then f eAA0 is a null tetrad for a pseudo-hyper-Kähler metric on M, where f 2 D

.e 000 ; e 00 ; e 100 ; e 110 /. Given any four-dimensional pseudo-hyper-Kähler metric such a null tetrad and 4-form exists. 4

Other terminology includes neutral hyper-Kähler [28] and hyper-para-Kähler [24].


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The extra volume preserving condition (38) distinguishes this from Theorem 2. Alternatively Theorem 4 arises as a special case of Theorem 3 with fA D 0. The Heavenly equations. It was shown by Plebański [47] that one can always put a pseudo-hyper-Kähler metric into the form (5), where satisfies the first Heavenly equation (6). The function can be interpreted as the Kähler potential for one of the complex structures. Plebański also gave the alternative local form. The metric is given by (32), and the potential ‚ satisfies the second Heavenly equation (35). The Heavenly equations are non-linear ultrahyperbolic equations. These formulations are convenient for understanding local properties of pseudo-hyper-Kähler metrics, as they only depend on a single function satisfying a single PDE.

4 Symmetries By a symmetry of a metric, we mean a conformal Killing vector, i.e. a vector field K satisfying LK g D c g; (39) where c is a function. If c vanishes, K is called a pure Killing vector, otherwise it is called a conformal Killing vector. If c is a nonzero constant K is called a homothety. If we are dealing with a conformal structure Œg, a symmetry is a vector field K satisfying (39) for some g 2 Œg. Then .39/ will be satisfied for any g 2 Œg, where the function c will depend on the choice of g 2 Œg. Such a K is referred to as a conformal Killing vector for the conformal structure. In neutral signature there are two types of Killing vectors: non-null and null. Unlike in the Lorentzian case where non-null vectors can be timelike or spacelike, there is essentially only one type of non-null vector in neutral signature. Note that a null vector for g 2 Œg is null for all g 2 Œg, so nullness of a vector with respect to a conformal structure makes sense. 4.1 Non-null case. Given a neutral four-dimensional ASD conformal structure .M; Œg/ with a non-null conformal Killing vector K, the three-dimensional space W of trajectories of K inherits a conformal structure Œh of signature .C C /, due to (39). The ASD condition on Œg results in extra geometrical structure on .W ; h/; it becomes a Lorentzian Einstein–Weyl space. This is called the Jones–Tod construction, and is described in Section 4.1.2. The next section is an summary of Einstein–Weyl geometry. 4.1.1 Einstein–Weyl geometry. Let W be a three-dimensional manifold. Given a conformal structure Œh of signature5 .2; 1/, a connection D is said to preserve Œh if Dh D ! ˝ h; 5

(40)

The formalism in this section works in general dimension and signature but we specialize to the case we encounter later.

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for some h 2 Œh, and a 1-form !. It is clear that if (40) holds for a single h 2 Œh it holds for all, where ! will depend on the particular h 2 Œh. (40) is a natural condition; it is the requirement that null geodesics of any h 2 Œh are also geodesics of D. Given D we can define its Riemann and Ricci curvature tensors W ij kl , Wij in the usual way. The notion of a curvature scalar must be modified, because there is no distinguished metric in the conformal class to contract Wij with. Given some h 2 Œh we can form W D hij Wij . Under a conformal transformation h ! 2 h, W transforms as W ! 2 W . This is because Wij unaffected by any conformal rescaling, being formed entirely out of the connection D. W is an example of a conformally weighted function, with weight 2. One can now define a conformally invariant analogue of the Einstein equation as follows: W.ij / 13 W hij D 0: (41) These are the Einstein–Weyl equations. Notice that the left-hand side is well defined tensor (i.e. weight 0), since the weights of W and hij cancel. Equation (41) is the Einstein–Weyl equation for .D; Œh/. It says that given any h 2 Œh, the Ricci tensor of W is tracefree when one defines the trace using h. Notice also that Wij is not necessarily symmetric, unlike the Ricci-tensor for a Levi-Civita connection. In the special case that D is the Levi-Civita connection of some metric h 2 Œh, (41) reduces to the Einstein equation. This happens when ! is exact, because under h ! 2 h, we get ! ! !C2d.ln/, so if ! is exact a suitable choice of will transform it to 0, giving Dh D 0 in (40). All Einstein metrics in 2C1 or 3 dimensions are spaces of constant curvature. The Einstein–Weyl condition allows non-trivial degrees of freedom. The general solution to (41) depends on four arbitrary functions of two variables. In what follows, we refer to an Einstein–Weyl structure by .h; !/. The connection D is fully determined by this data using .40/. 4.1.2 Reduction by a non-null Killing vector; the Jones–Tod construction. The Jones–Tod construction relates ASD conformal structures in four dimensions to Einstein–Weyl structures in three dimensions. In neutral signature it can be formulated as follows: Theorem 5 ([26]). Let (M; Œg) be a neutral ASD four manifold with a non-null conformal Killing vector K. An Einstein–Weyl structure on the space W of trajectories of K is defined by h WD jKj2 g jKj4 K ˇ K;

! D 2jKj2 g .K ^ d K/;

(42)

where jKj2 WD g.K; K/, K WD g.K; /, and g is the Hodge- of g. All EW structures arise in this way. Conversely, let .h; !/ be a three-dimensional Lorentzian EW structure on W , and let .V; / be a function and a 1-form on W satisfying the generalised monopole equation h d V C 12 !V D d; (43)


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where h is the Hodge- of h. Then g D V 2 h .d C /2 is a neutral ASD metric with non-null Killing vector @ . This is a local theorem, so we may assume W is a manifold. A vector in W is a vector field Lie-derived along the corresponding trajectory in M, and one applies the formulae (42) to this vector field to obtain .Œh; !/ on W . In the Riemannian case it has been successfully applied globally in certain nice cases [33]. When one performs a conformal transformation of g, one obtains a conformal transformation of h and the required transformation of !, so this is a theorem about conformal structures, though we have phrased it in terms of particular metrics. The Jones–Tod construction was originally discovered using twistor theory in [26]; since then other purely differential-geometric proofs have appeared [27], [11]; although these are in Riemannian signature the arguments carry over to the neutral case. In Section 5 we explain the twistorial argument that originally motivated the theorem. 4.1.3 Integrable systems and the Calderbank–Pedersen construction. Applying the Jones–Tod correspondence to the special ASD conditions discussed in Section 2 will yield special integrable systems in 2 C 1 dimensions. In each case of interest we shall assume that the symmetry preserves the special geometric structure in four dimensions. This will give rise to special Einstein–Weyl backgrounds, together with general solutions of the generalised monopole equation (43) on these backgrounds. We can then seek special monopoles such that the resulting ASD structure is conformal to pseudo-hyper-Kähler. An elegant framework for this is provided by the Calderbank–Pedersen construction [11]. In this construction self-dual complex (or null) structures on M correspond to shear-free geodesic congruences (SFGC) on W . This gives rise to a classification of three-dimensional EW spaces according to the properties of associated congruences. Below we shall list the resulting reductions and integrable systems. In each case we shall specify the properties of the associated congruence without going into the details of the Calderbank–Pedersen correspondence. Scalar-flat Kähler with symmetry. The SU.1/-Toda equation. Let .M; g/ be a scalar-flat Kähler metric in neutral signature, with a symmetry K Lie deriving the Kähler form !. One can follow the steps of LeBrun [33] to reduce the problem to a pair of coupled PDEs: the SU.1/-Toda equation and its linearisation. The key step in the construction is to use the moment map for K as one of the coordinates, i.e. define a function t W M ! R by dt D K !. Then x; y arise as isothermal coordinates on two-dimensional surfaces orthogonal to K and dt . The metric takes the form g D V .e u .dx 2 C dy 2 / dt 2 /

1 .d C /2 ; V

(44)

130


where the function u satisfies the SU.1/-Toda equation .e u / t t uxx uyy D 0;

(45)

and V is a solution to its linearization – the generalised monopole equation (43). The corresponding EW space from the Jones–Tod construction is h D e u .dx 2 C dy 2 / dt 2 ; ! D 2u t dt:

(46)

It was shown in [52] that the EW spaces that can be put in the form (45) are precisely those possessing a shear-free twist-free geodesic congruence. Given the Toda EW space, any solution to the monopole equation will yield a .C C / scalar flat Kähler metric. The special solution V D cu t , where c is a constant, will lead to a pseudohyper-Kähler metric with symmetry. In [35] solutions to (45) were used to construct neutral ASD Ricci flat metrics without symmetries. ASD null-Kähler with symmetry. The dKP equation. Let .M; g; N / be an ASD null Kähler structure with a Killing vector K such that LK N D 0. In [15] it was demonstrated that there exist smooth real valued functions H D H.x; y; t / and W D W .x; y; t/ such that g D Wx .dy 2 4dxdt 4Hx dt 2 / Wx1 .d Wx dy 2Wy dt /2

(47)

is an ASD null-Kähler metric on a circle bundle M ! W if Hyy Hxt C Hx Hxx D 0;

(48)

Wyy Wxt C .Hx Wx /x D 0:

(49)

All real analytic ASD null-Kähler metrics with symmetry arise from this construction. With definition u D Hx the x derivative of equation (48) becomes .u t uux /x D uyy ; which is the dispersionless Kadomtsev–Petviashvili equation originally used in [17]. The corresponding Einstein–Weyl structure is h D dy 2 4dxdt 4udt 2 ; ! D 4ux dt: This EW structure possesses a covariant constant null vector with weight 12 , and in fact every such EW structure with this property can be put into the above form. The covariant constancy is with respect to a derivative on weighted vectors that preserves their weight. Details can be found in [17]. The linear equation (49) is a (derivative of) the generalised monopole equation from the Jones–Tod construction. Given a dKP Einstein–Weyl structure, any solution to this monopole equation will yield and ASD Null Kähler structure in four dimensions. The special monopole V D Hx =2 will yield a pseudo-hyper-Kähler structure with symmetry whose self-dual derivative is null.


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Pseudo-hypercomplex with symmetry. The hyper-CR equation. Let us assume that a pseudo-hypercomplex four manifold admits a symmetry which Lie derives all (pseudo) complex structures. This implies [16] that the EW structure is locally given by h D .dy C udt /2 4.dx C wdt /dt;

! D ux dy C .uux C 2uy /dt;

where u.x; y; t / and w.x; y; t / satisfy a system of quasi-linear PDEs u t C wy C uwx wux D 0;

uy C wx D 0:

(50)

The corresponding pseudo-hypercomplex metric will arise form any solution to this coupled system, and its linearisation (the generalised monopole (43)). The special monopole V D ux =2 leads to pseudo-hyper-Kähler metric with triholomorphic homothety. 4.2 Null case. Given a neutral four-dimensional ASD conformal structure .M; Œg/ with a null conformal Killing vector K, the three-dimensional space of trajectories of K inherits a degenerate conformal structure of signature .C 0/, and the Jones–Tod construction does not hold. The situation was investigated in detail in [18] and [10]. It was shown that K defines a pair of totally null foliations of M, one by ˛-surface and one by ˇ-surfaces; these foliations intersect along integral curves of K which are null 0 0 geodesics. In spinors, if K a D A oA then an ˛-plane distribution is defined by oA , A and a ˇ-plane distribution by , and it follows from the Killing equation that these distributions are integrable. The main result from [18] is that there is a canonically defined projective structure on the two-dimensional space of ˇ-surfaces U which arises as a quotient of M by a distribution A eAA0 . A more general framework where the distribution A eAA0 is still 0 0 integrable, but A oA is not a symmetry for any oA 2 .S 0 / was recently developed by Calderbank [10] and extended by Nakata [42]. A projective structure is an equivalence class of connections, where two connections are equivalent if they have the same unparameterized geodesics. In Section 5 we will explain the twistor theory that led to the observation that projective structures are involved, and give a new example of a twistor construction. It turns out that one can explicitly write down all ASD conformal structures with null conformal Killing vectors in terms of their underlying projective structures as follows: Theorem 6 ([18]). Let .M; Œg; K/ be a smooth neutral signature ASD conformal structure with null conformal Killing vector. Then there exist local coordinates .; x; y; z/ and g 2 Œg such that K D @ and g has one of the following two forms, according to whether the twist K ^ d K vanishes or not .K WD g.K; //: 1. K ^ d K D 0. g D .d C .zA3 Q/dy/.dy ˇdx/ .dz .z.ˇy C A1 C ˇA2 C ˇ 2 A3 //dx .z.A2 C 2ˇA3 / C P /dy/dx;

(51)

132


where A1 , A2 , A3 , ˇ, Q,P are arbitrary functions of .x; y/. 2. K ^ d K ¤ 0. g D .d C A3 @z Gdy C .A2 @z G C 2A3 .z@z G G/ @z @y G/dx/ .dy zdx/ @2z Gdx.dz .A0 C zA1 C z 2 A2 C z 3 A3 /dx/;

(52)

where A0 ; A1 ; A2 ; A3 are arbitrary functions of .x; y/, and G is a function of .x; y; z/ satisfying the following PDE: .@x C z@y C .A0 C zA1 C z 2 A2 C z 3 A3 /@z /@2z G D 0:

(53)

The functions Ai .x; y/ in the metrics (51) and (52) determine projective structures on the two-dimensional space U in the following way. A two projective structure in two dimensions is equivalent to a second-order ODE

d 2y dy D A3 .x; y/ 2 dx dx

3

dy C A2 .x; y/ dx

2

dy C A1 .x; y/ C A0 .x; y/; (54) dx

obtained by choosing local coordinates .x; y/ and eliminating the affine parameter from the geodesic equation. The Ai functions can be expressed in terms of combinations of connection coefficients that are invariant under projective transformation. In (52) all the Ai ; i D 0; 1; 2; 3 functions occur explicitly in the metric. In (51) the function A0 does not explicitly occur. It is determined by the following equation: A0 D ˇx C ˇˇy ˇA1 ˇ 2 A2 ˇ 3 A3 :

(55)

If the projective structure is flat, i.e.Ai D 0 and ˇ D P D 0 then (51) is Ricci flat [47], and in fact this is the most general ASD Ricci flat metric with a null Killing vector which preserves the pseudo-hyper-Kähler structure [4]. More generally, if the projective structure comes from a Riemannian metric on U then there will always exist a (pseudo-)Kähler structure in the conformal class Œg if G D z 2 =2C.x; y/z Cı.x; y/ for certain ; ı [9]. It is interesting that integrable systems are not involved in the null case, given their ubiquity in the non-null case.

5 Twistor theory In Riemannian signature, given an ASD conformal structure .M; Œg/ in four dimensions one can form a 2-sphere bundle over it, and endow this with an integrable complex structure by virtue of anti-self-duality [3]. The resulting complex manifold P T is called the twistor space. The original manifold is the moduli space of rational curves in P T preserved under a certain anti-holomorphic involution, and one can recover the conformal structure by looking at how the rational curves intersect one another. Hence the .M; Œg/ is completely encoded in P T and its anti-holomorphic involution. The


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important feature of a successful twistor construction is that the original geometry becomes encoded in the holomorphic geometry of the twistor space, and can be recovered from this. Neutral signature ASD conformal structures cannot be encoded purely in holomorphic geometry as in the Riemannian case. This is not surprising as generically they are not analytic. However, there is a recent twistor construction due to LeBrun–Mason [34] in the neutral case that uses a mixture of holomorphic and smooth ingredients; we review this in Section 5.2. Let us now review the differences in Riemannian and neutral signature. In the Riemannian case, if one expresses the metric in terms of a null tetrad as in (9) then the basis vectors eAA0 must be complex, as there are no real null vectors. The spin bundles are complex two-dimensional vector bundles S, S0 , with an isomorphism TC M Š S ˝ S0 , at least locally. One then takes the projective bundle P S0 , which has CP 1 fibres. Even if S0 does not exist globally, the bundle P S0 does exist globally, since the Z2 obstruction to existence of a spin bundle is eliminated on projectivizing. Concretely, P S0 is the bundle of complex self-dual totally null 2-planes; from this description it clearly exists globally. 0 Now one can form the LA vectors as in Theorem 1, where now A are complex (the homogeneous fibre coordinates of P S0 ). The connection coefficients in the expression for eQ AA0 will now be complex, and satisfy certain Hermiticity properties that we need not go into. The LA span a complex two-dimensional distribution on the complexified tangent space of P S0 , and the Riemannian version of Theorem 1 is that this distribution is complex integrable iff the metric is ASD. Together with @N , where is the inhomogeneous fibre coordinate on P S0 , we obtain a complex three-dimensional distribution x D 0. If the metric is ASD, … is complex integrable and defines …, satisfying … \ … a complex structure on P S0 . This construction works globally. It was discovered by Atiyah, Hitchin and Singer [3]. In the neutral case one can complexify the real spin bundles S, S 0 and obtain TC M Š SC ˝ SC0 as in the Riemannian case. One can define a complex distribution 0 … distribution on PSC0 , by allowing A in Theorem 1 to be complex. The key point 0 is that the vectors LA become totally real when A is real. So on the hypersurface x D 0, so does not PS 0 PSC0 , the distribution spanf…; @N g no longer satisfies … \ … A0 define an almost complex structure. When is not real, the distribution spanned by LA and @N does define an almost complex structure, which is integrable when g is ASD. We obtain two non-compact regions in PSC0 , each of which possesses an integrable complex structure, separated by a hypersurface PS 0 . This is more complicated than the Riemannian case, where the end result is simply a complex manifold. Nevertheless, the construction is reversible in a precise sense given by Theorem of LeBrun–Mason which we review in Section 5.2 (Theorem 7). Before describing the work of LeBrun–Mason we review the analytic case, where one can complexify and work in the holomorphic category.

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5.1 The analytic case. In this section we work locally. Standard references for this material are [55], [22]. Suppose a neutral four-dimensional ASD conformal structure .M; Œg/ is analytic in some coordinate system. Then we can complexify by letting the coordinates become complex variables, and we obtain a holomorphic conformal structure .MC ; ŒgC /. If each coordinate is defined in some connected open set on R, then one thickens this slightly on both sides of the axis to obtain a region in C on which the complex coordinate is defined. The holomorphic conformal structure is obtained by picking a real metric g and allowing the coordinates to be complex to obtain gC . Then ŒgC is the equivalent class of gC up to multiplication by nonzero holomorphic functions. From Theorem 1, which is valid equally for holomorphic metrics, we deduce that given any holomorphic ˛-plane at a point, there is a holomorphic ˛-surface through that point. Assuming we are working in a suitably convex neighbourhood so that the space of such ˛-surfaces is Hausdorff, we define P T to be this space. P T is a three-dimensional complex manifold, since the space of ˛-planes at a point is complex one-dimensional and each surface is of complex codimension two in MC . This is summarised in the double fibration picture p

q

PSC0 ! P T ;

MC

(56)

where q is the quotient by the twistor distribution LA . If we had started with a Riemannian metric this would lead to the same twistor space, locally, as the Atiyah–Hitchin–Singer construction described above, though we shall not demonstrate this here. A point x 2 M, corresponds to an embedded CP 1 P T , since there is a CP 1 of ˛-surfaces through x. By varying the point x 2 M we obtain a four complex parameter family of CP 1 ’s. P T inherits an anti-holomorphic involution . To describe , note that there is an anti-holomorphic involution of MC that fixes real points, i.e. points of M MC . This is just the map from a coordinate to its complex conjugate, so we can arrange our complexification regions in which the coordinates are defined so that maps the regions to themselves. Now will map holomorphic ˛-surfaces to holomorphic ˛-surfaces, so gives an anti-holomorphic involution on P T . One way to see this is to note that ˛-surfaces are totally geodesic as the geodesic shear free condition 0

0

A B rAA0 B 0 D 0 is equivalent to CA0 B 0 C 0 D 0 , and consider the holomorphic geodesic equation. Using the fact that the connection coefficients are real, one can show that the involution will map the null geodesics in an ˛-surface to other null geodesics in another ˛-surface. The ˛-surfaces fixed by this are the real ˛-surfaces in M. In terms of P T , this last fact means that fixes an equator of each of the four complex parameter family of embedded CP 1 ’s. Moreover, an ˛-surface through a real point gets mapped to one through that same point since the point is fixed by . So the CP 1 ’s that are fixed by are a four real parameter family corresponding to M, we call these real CP 1 ’s.


135

How does one recover the neutral conformal structure from the data .P T ; /? As described above, M is the moduli space of CP 1 ’s fixed by . Now a vector at a point in M corresponds to a holomorphic section of the normal bundle O.1/ ˚ O.1/ of the corresponding real CP 1 in P T , such that the section ‘points’ to another real CP 1 . We define a vector to be null if this holomorphic section has a zero. Since vanishing of a section of O.1/ ˚ O.1/ is a quadratic condition, this gives a conformal structure. One can prove that this conformal structure is ASD fairly easily, by showing that the required ˛-surfaces must exist in terms of the holomorphic geometry. Moreover, special conditions on a gC 2 ŒgC can be encoded into the holomorphic geometry of the twistor space: • Holomorphic fibration W P T ! CP 1 corresponds to hyper-hermitian conformal structures [5], [14]. • Preferred section of 1=2 which vanishes at exactly two points on each twistor line corresponds to scalar-flat Kähler gC [49]. • Preferred section of 1=4 corresponds to ASD null-Kähler gC [15]. • Holomorphic fibration W P T ! CP 1 and holomorphic isomorphism O.4/ Š correspond to hyper-Kähler gC [45], [3], [22]. Here is a holomorphic canonical bundle of P T , and O.4/ is a power of the tautological bundle on the base of . To obtain a real metrics the structures above must be preserved by an anti-holomorphic involutions fixing a real equator of each rational curve in P T . It is worth saying a few words about the construction of solutions of integrable systems using the twistor correspondence. It is shown in Section 4 that a number of well-known integrable systems 2 C 1 dimensions are special cases of ASD conformal structures. Analytic solutions to these integrable systems therefore correspond6 to twistor spaces P T . There will be extra conditions on P T , depending on the special case in question. However, solutions to the integrable systems are not always analytic. 5.1.1 Symmetries and twistor spaces. In Section 4 we discussed the appearance of Einstein–Weyl structures and projective structures in the cases of a non-null and null Killing vector respectively. In both cases twistor theory was the key factor in revealing these correspondences. We shall now explain this briefly. In [22], Hitchin gave three twistor correspondences. He considered complex manifolds containing embedded CP 1 ’s with normal bundles O.1/, O.2/ and O.1/ ˚ O.1/ respectively. Kodaira deformation theory guarantees a local moduli space of embedded CP 1 ’s, whose complex dimension is the dimension of the space of holomorphic sections of the corresponding normal bundle, i.e. 2, 3, 4 respectively. By examining how nearby curves intersect, he 6

This correspondence is not one-one due to coordinate freedom.

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deduced that the moduli space inherits a holomorphic projective structure, Einstein– Weyl structure, or ASD conformal structure respectively. He also showed that the construction is reversible in each case. Now given a four-dimensional holomorphic ASD conformal structure, its twistor space is the space of ˛-surfaces, as described in Section 5.1. A conformal Killing vector preserves the conformal structure, so preserves ˛-surfaces, giving a holomorphic vector field on the twistor space. If the Killing vector is non-null then the vector field on twistor space P T is nonvanishing. This is because the Killing vector is transverse to any ˛-surface, as it is non-null. In this case one can quotient the three-dimensional twistor space by the induced vector field, and it can be shown [26] that the resulting two-dimensional complex manifold contains CP 1 ’s with normal bundle O.2/. Using Hitchin’s results, this corresponds to a three-dimensional Einstein–Weyl structure. This the twistorial version of the Jones–Tod construction, Theorem 5. If the Killing vector is null then the induced vector field on the twistor space P T vanishes on a hypersurface. This is because at each point, the Killing vector is tangent to a single ˛ surface. Hence it preserves a foliation by ˛-surfaces, and vanishes at the hypersurface in twistor space corresponding to this foliation. However, one can show [18] that it is possible to continue the vector field on twistor space to a one-dimensional distribution Ky that is nowhere vanishing. Quotienting P T by this distribution gives a two-dimensional complex manifold Z containing CP 1 ’s with normal bundle O.1/. Using Hitchin’s results, this corresponds to a two-dimensional projective structure. This is the twistorial version of the correspondence described in Section 4.2. The situation is illustrated by the following diagram. M

˛-surface

PT

ˇ-surface

ˇ1

Ky

˛

ˇ2 ˇ2

ˇ1

˛

ˇ2

ˇ2

ˇ1

ˇ1 U

Z

Figure 1. Relationship between M, U , P T and Z.


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In M, a one parameter family of ˇ-surface is shown, each of which intersects a one parameter family of ˛-surfaces, also shown. The ˇ-surfaces correspond to a projective structure geodesic in U , shown at the bottom left. The ˇ-surfaces in M correspond to surfaces in P T , as discussed above. These surfaces intersect at the dotted line, which corresponds to the one parameter family of ˛-surfaces in M. When we quotient P T by Ky to get Z, the surfaces become twistor lines in Z, and the dotted line becomes a point at which the twistor lines intersect; this is shown on the bottom right. This family of twistor lines intersecting at a point corresponds to the geodesic of the projective structure. Example7 . Here we give an explicit construction of the twistor space of an analytic neutral ASD conformal structure with a null Killing vector, from the reduced projective structure twistor space. We take Z to be the total space of O.1/. This is the twistor space of the flat projective structure. Now suppose we are given a 1-form ! on U . We shall complexify the setup and regard ! as holomorphic a holomorphic connection on a holomorphic line bundle B ! U . This gives rise to a holomorphic line bundle E ! Z, where the vector space over z 2 Z is the space of parallel sections of B over the geodesic in U corresponding to z. The twistor lines in Z are the two-parameter family of embedded CP 1 ’s, each corresponding to the set of geodesics through a single point in U . We denote the twistor line corresponding to a point x 2 U by x. O Now E restricted to a twistor line xO is trivial, because to specify a parallel section of B through any geodesic through x, one need only know its value at x. This is a simple analogue of the Ward correspondence relating solutions of the anti-self-dualYang–Mills equations on C 4 to vector bundles over the total space of O.1/ ˚ O.1/ that are trivial on twistor lines. The situation here is simpler since there are no PDEs involved; this is because there are no integrability conditions for a space of parallel sections to exist on a line. As with the Ward correspondence, the construction is reversible, i.e. given a holomorphic line bundle trivial on twistor lines one can find a connection on U to which it corresponds in the manner described above. We will not prove this here, it is simply a case of mimicking the argument for the Ward correspondence [55]. Now to create the twistor space P T , we must tensor E with a line bundle L so that E ˝ L restricts to O.1/ on the twistor lines in Z. Then the total space of E ˝ L will have embedded CP 1 ’s with normal bundle O.1/ ˚ O.1/, so will be a twistor space for an ASD conformal structure. For L we choose the pull back of O.1/ to the total space of Z. Let us now make the above explicit. Let , Q be the inhomogeneous coordinate on the two patches U0 , U1 of CP 1 . The total space of O.1/ can be coordinatized as follows. Let be the fibre coordinate over U0 , and Q the fibre coordinate over U1 . The line bundle transition relation on the overlap is Q D 1 . Now suppose we have a line bundle E ! Z D O.1/, that is trivial on holomorphic sections of Z ! CP 1 . Let , Q be the fibre coordinates on the two patches, satisfying a transition relation Q D F .; /, where F .; / is holomorphic and nonvanishing on the overlap, i.e. for 2 C f0g, 2 C. In sheaf terms, F is an element of 7

We thank Paul Tod for his help with this example.

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H 1 .O.1/; O /. Now the short exact sequence 0 ! Z ! O ! O ! 0

(57)

gives rise to a long exact sequence, part of which is: ! H 1 .O.1/; Z/ ! H 1 .O.1/; O/ ! H 1 .O.1/; O / ! H 2 .O.1/; Z/ ! (58) The first term in (58) vanishes and the final term is Z, by topological considerations. The final term gives the Chern class of the line bundle determined by the element of H 1 .O.1/; O /. This vanishes for E, since it is trivial on twistor lines. The third arrow in (57) is the exponential map. Together these facts imply that F can be written F .; / D e f .;/ , where f .; / is a holomorphic function on the overlap that may have zeros. After twisting by L, we obtain the following transition function for E ˝ L, again using , Q as fibre coordinates: Q D

1 f .;/ : e

(59)

To find the conformal structure we must find the four parameter family of twistor lines in E ˝ L. The two parameter family in O.1/ is given in one patch by ./ D X C Y , Q D X C Y Q . Restricting to one of these we can split f : and in the other by . Q / Q f .; X C Y / D h.X; Y; / h.X; Y; 1=/;

(60)

where h and hQ are functions on U CP 1 holomorphic in and 1= respectively. For fixed .X; Y / there is then a further two parameter family of twistor lines, given by ./ D e h.X;Y;/ .W Z/

(61)

Q Q Q / Q D e h.X;Y; . Q / .W Z/:

(62)

in one patch, and

It is easy to check that (59) is satisfied by (61) and (62). One must now calculate the conformal structure on the moduli space of lines parametrised by X a D .X; Y; W; Z/ by determining the quadratic condition for a section of the normal bundle to a twistor line to vanish. The sections of the normal bundle to xO P T correspond to tangent vectors in Tx M, and sections with one zero will determine null vectors and therefore the conformal structure. Using the identity .@X @Y /f D 0 together with (60) we deduce (by Liouville theorem or using power series) that

@h @h D B.X; Y / A.X; Y /; @Y @X

for some analytic functions A,B.

(63)


139

Now take the variation of ./ and ./ for a small change ıX a to obtain 0 D ı D ıY C ıX; @h @h h 0 D ı D e ıX ıY .W Z/ C e h .ıW ıZ/: @X @Y

(64) (65)

Substituting D ıY =ıX from the first expression to the second, using (63) and multiplying the resulting expression by ıX we find that the conformal structure is represented by the following metric: g D dXd W C d YdZ .W dX C Zd Y /.A.X; Y /dX C B.X; Y /d Y /:

(66)

This conformal structure possesses the null conformal Killing vector K D W @W C Z@Z , which is twisting. The global holomorphic vector field on P T induced by K is @ D Q @Q where the equality holds on the intersection of the two coordinate patches. This vanishes on the hypersurface defined by D 0 in one patch and Q D 0 in the other, which intersects each twistor line at a single point, as we expect from the argument in Section 5.1.1. The 1-form ! D AdX C Bd Y in g is the inverse Ward transform of F 2 H 1 .O.1/; O /. To compare with (52) one must transform to coordinates .; x; y; z/ in which K D @ . Dividing by a conformal factor W , transforming with .; x; y; z/ D .logW; Y; X; Z=W /, and then translating to eliminate an arbitrary one function of .x; y/ gives g D .d C f .x; y/dx/.dy zdx/ dzdx; (67) which is a special case of (52) with flat projective structure, and G D z 2 =2 zC.x; y/, where f D @y C . If we take the coordinates to be real we obtain a neutral metric. The twistor space P T fibres over Z D O.1/ and this fibres over CP 1 , so P T fibers over CP 1 and (67) is pseudo-hyperhermitian. To construct an example of a conformal structure with non-twisting null Killing vector one uses an affine line bundle over Z D O.1/; see [18] for details. 5.2 LeBrun–Mason construction. Here we describe recent work of LeBrun and Mason in which a general, global twistor construction is given for neutral metrics. We will only be able to give a crude paraphrase, and refer the reader to the original paper [34] for details. Note that their paper uses the opposite duality conventions to ours; they use self-dual conformal structures with integrable ˇ-plane distributions. We described above how a neutral ASD conformal structure .M; Œg/ gives rise to a complex structure on CP 1 bundle over M, which degenerates on a hypersurface. The following theorem of LeBrun–Mason is a converse to this, and is the closest one can come to a general twistor construction in the neutral case: Theorem 7 ([34]). Let M be a smooth connected 4-manifold, and let $ W X ! M be a smooth CP 1 -bundle. Let % W X ! X be an involution which commutes with $ , and has as fixed-point set X% an S 1 -bundle over M which disconnects X into two

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closed 2-disk bundles X˙ with common boundary X% . Suppose that … TC X is a distribution of complex 3-planes on X such that x 1. % … D …; 2. 3. 4. 5.

the restriction of … to XC is smooth and involutive, x D 0 on X X% , …\… … \ ker $ is the .0; 1/ tangent space of the CP 1 fibers of $ , the restriction of … to a fiber of X has first Chern class 4 with respect to the complex orientation.

Then E D …\T X% is an integrable distribution of real 2-planes on X% , and M admits a unique smooth split-signature ASD conformal structure Œg for which the ˛-surfaces are the projections via $ of the integral manifolds of E. This theorem provides a global twistor construction for neutral ASD four manifolds, whereas the analytic construction of the last section only works locally. At first sight the theorem does not seem like a promising method of generating ASD conformal structures, since the conditions required on the CP 1 bundle over M are complicated, and it is not clear how one might construct examples. This obstacle is overcome in [34] by deforming a simple example (another example was given by Nakata [41]). Consider the conformally flat neutral metric g0 given by (7) on M D S 2 S 2 that is just the difference of the standard sphere metrics on each factor. The underlying manifold M can be realised as the space of CP 1 ’s embedded in CP 3 that are invariant under the complex conjugate involution, which we call the real CP 1 ’s; these real CP 1 ’s are the fibres of the bundle X. The involution % of X is induced by the complex conjugate involution of CP 3 . The fixed point set X% consists of the invariant equators of the real CP 1 ’s, and is therefore a circle bundle over M. The closed disc bundles X˙ are obtained by slicing the real CP 1 ’s at their invariant equator, and throwing away one of the open halves. The fixed point set of the complex conjugate involution is the standard embedding of RP 3 , and this is the space of ˛-surfaces in M D S 2 S 2 . Taking all the real CP 1 ’s through a point p 2 RP 3 gives an ˛-surface. To obtain the … from Theorem 7, take XC and construct a map f to CP 3 as follows. On the interior of XC , f is a diffeomorphism onto CP 3 RP 3 . The boundary @XC gets mapped by f to RP 3 , by taking a point in @XC , i.e. a holomorphic disc and a point p on RP 3 lying on the intersection of the disc with RP 3 , to the point p. Let f1;0 W TC XC ! T 1;0 CP 3 be the .1; 0/ part of the derivative of f . Then the … of Theorem 7 is defined on XC by … D kerf1;0 TC XC : Note that f maps the five-dimensional boundary @XC to the three-dimensional space RP 3 ; this means that on the boundary … restricts to the complexification of a real twoplane distribution, direct summed with the complexification of the direction into the disc. The … here agrees with the one described at the beginning of Section 5, defined in terms of the twistor distribution LA on PSC0 .


141

The idea for creating new spaces satisfying the conditions of Theorem 7 is to deform the standard embedding of RP 3 CP 3 slightly. One forms the bundle XC by taking the same S 2 S 2 family of real holomorphic discs, now with boundary on the deformed embedding of RP 3 . The complex distribution … is formed in the same way as in the conformally flat case described above. One then patches two copies of XC together to form the bundle X, and it is shown that this satisfies the conditions of the Theorem. It is also shown that the resulting conformal structure on S 2 S 2 has the property that all null geodesics are embedded circles; conformal structures with this property are termed Zollfrei. It turns out that all ASD conformal structures close enough in a suitable sense to the conformally flat one are Zollfrei, and in fact the twistor description gives a complete understanding of ASD conformal structures near the standard one. The embedded RP 3 is the real twistor space, i.e. the space of ˛-surfaces in M, and a significant portion of [34] is devoted to showing that the for a space-time oriented Zollfrei 4-manifold the real twistor space must be RP 3 , making contact with the picture of a deformed RP 3 CP 3 . We mention that there is another twistor-like construction of smooth ASD conformal structures with avoids the holomorphic methods altogether [20]. In this approach one views the real twistor curves in RP 3 as solutions to a system of two second order nonlinear ODEs. The ODEs have to satisfy certain conditions (expressed in terms of point invariants) if their solution spaces are equipped with ASD conformal structures.

6 Global results In the last section we outlined the global twistor construction for neutral ASD four manifolds due to LeBrun–Mason, which they used to construct Zollfrei metrics on S 2 S 2 . In this section we review the known explicit constructions of globally defined neutral ASD conformal structures on various compact and non-compact manifolds. 6.1 Topological restrictions. Existence of a neutral metric on a four manifold M imposes topological restrictions on M. A neutral inner product on a four-dimensional vector V space splits V into a direct sum V D VC ˚ V , where the inner product is positive definite on VC and negative definite on V . So a neutral metric g on a four manifold M splits the tangent bundle T M D TC M ˚ T M;

(68)

where T˙ are two-dimensional subbundles of T M. Conversely, given such a splitting one can construct neutral metrics on M by taking a difference of positive definite metrics on the vector bundles TC M and T M. If M admits a non-vanishing 2-plane field E (a real two-dimensional distribution), then a splitting of the form (68) can be found by taking E to define TC M, choosing a Riemannian metric, and letting T M be the orthogonal complement. So a four manifold M admits a neutral metric iff it admits a 2-plane field.

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The topological conditions for existence of a 2-plane field were discovered by Hirzebruch and Hopf and are as follows: Theorem 8 ([21]). A compact smooth four-manifold M admits a field of 2-planes iff ŒM and ŒM satisfy a pair of conditions 3 ŒM C 2ŒM 2 .M/; 3 ŒM 2ŒM 2 .M/; where ŒM D f M .w; w/ 2 Z W w are arbitrary elements in H 2 .M; Z/=Torg: Here ŒM, ŒM are the signature and Euler characteristic respectively, and M is the intersection form on H 2 .M; Z/=Tor. A neutral metric implies that the structure group of the tangent bundle can be reduced to O.2; 2/, by choosing orthonormal bases in each patch. In fact O.2; 2/ has four connected components, so there are various different orientability requirements one can impose. The simplest is to require the structure group to reduce to the identity component SOC .2; 2/. It is shown in [39] that this is equivalent to the existence of a field of oriented 2-planes, i.e. an orientable two-dimensional subbundle of the tangent bundle. The topological restrictions imposed by this were discovered by Atiyah: Theorem 9 ([2]). Let M be a compact oriented smooth manifold of dimension 4, such that there exists a field of oriented 2-planes on M. Then ŒM 0 mod 2;

ŒM ŒM mod 4:

(69)

In fact Matsushita showed [38] that for a simply-connected 4-manifold, (69) are actually sufficient for the existence of an oriented field of 2-planes. A more subtle problem is to determine topological obstructions arising from existence of an ASD neutral metric. This deserves further study. 6.2 Tod’s scalar-flat Kähler metrics on S 2 S 2 . Consider S 2 S 2 with the conformally flat metric described in Sections 2.1 and 5.2, i.e. the difference of the standard sphere metrics on each factor. Thinking of each sphere as CP 1 and letting and be non-homogeneous coordinates for the spheres, this metric is given by (7). As we have already said, g0 is scalar flat, indefinite Kähler. The obvious complex structure J gives a closed two form and WD g0 .J; /. Moreover g0 clearly has a high degree of symmetry, since the 2-sphere metrics have rotational symmetry. In [53], Tod found deformations of g0 preserving the scalar-flat Kähler property, by using the explicit expression (44) for neutral scalar-flat Kähler metrics with symmetry. Take the explicit solution 1 t2 eu D 4 .1 C x 2 C y 2 /2


143

to (45), which can be obtained by demanding u D f1 .x; y/ C f2 .t /. There remains a linear equation for V . Setting W D V .1 t 2 / and performing the coordinate transformation t D cos , D x C iy gives g D 4W

sin2 d d N W d 2 .d C /2 ; 2 N W .1 C /

(70)

and W must solve a linear equation. This metric reduces to (7) for W D 1, D 0, with , standard coordinates for the second sphere. Tod shows that on differentiating , one obtains the ultrahyperbolic wave the linear equation for W and setting Q D @W @t equation r12 Q D r22 Q; (71) where r1;2 are the Laplacians on the 2-spheres, and Q is independent of , i.e. is axisymmetric for one of the sphere angles. Equation (71) can be solved using Legendre polynomials, and one obtains non-conformally flat deformations of (7) in this way. In the process one must check that W behaves in such a way that (70) extends over S 2 S 2 . The problem of relating these explicit metrics to the Zollfrei metrics on S 2 S 2 known to exist by results described in Section 5.2 appears to be open. In a recent paper, Kamada [29] rediscovered the above metrics, and showed that a compact neutral scalar-flat Kähler manifold with a Hamiltonian S 1 symmetry must in fact be S 2 S 2 . Here a Hamiltonian S 1 symmetry is an S 1 action preserving the Kähler form, and which possesses a moment map. In the case of S 2 S 2 case, there is always a moment map since the manifold is simply connected. Without the symmetry, there are other neutral scalar-flat Kähler manifolds. For example, take a Riemann surface † with a constant curvature metric g. Then on † †, the metric 1 g 2 g is neutral scalar-flat Kähler, where i are the projections onto the first and second factors. 6.3 Compact neutral hyper-Kähler metrics. The only compact four-dimensional Riemannian hyper-Kähler manifolds are the complex torus with the flat metric and K3 with a Ricci-flat Calabi–Yau metric. In the neutral case, Kamada showed in [28] that a compact pseudo-hyper-Kähler four manifold must be either a complex torus or a primary Kodaira surface. In the complex torus case, the metric need not be flat, in contrast to the Riemannian case. Moreover in both cases one can write down explicit non-flat examples, in contrast to the Riemannian case where no explicit non-flat Calabi–Yau metric is known. To write down explicit examples, consider the following hyper-Kähler metric g D ddy dzdx Q.x; y/dy 2 ;

(72)

for Q and arbitrary function. This is the neutral version of the pp-wave metric of general relativity [47], and is a special case of (51), where the underlying projective structure is flat. It is non-conformally flat for generic Q. Define complex coordinates z1 D C iz, z2 D x C iy on C 2 . By quotienting the z1 - and z2 -planes by lattices one obtains a product of elliptic curves, a special type of complex torus. If we require Q to

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be periodic with respect to the z2 lattice, then (72) descends to a metric on this manifold. Likewise, a primary Kodaira surface can be obtained as a quotient of C 2 by a subgroup of the group of affine transformations, and again by assuming suitable periodicity in Q the metric (72) descends to the quotient. In our framework [18] we compactify the flat projective space R2 to two-dimensional torus U D T 2 with the projective structure coming from the flat metric. Both and z in (72) are taken to be periodic, thus leading to O W M ! U , the holomorphic toric fibration over a torus. Assume the suitable periodicity on the function Q W U ! R. This leads to a commutative diagram: MB BB O Q BB BB T2 B U Q / R: In the framework of [28] and [19] the Kähler structure on M is given by !flat C N O Q/, where .@; !flat / is the flat Kähler structure on the Kodaira surface induced i@@. from C 2 . As remarked in [28], the existence of pseudo-hyper-Kähler metrics on complex tori other than a product of elliptic curves is an open problem. 6.4 Ooguri–Vafa metrics. In [43] Ooguri, Vafa and Yau constructed a class of noncompact neutral hyper-Kähler metrics on cotangent bundles of Riemann surfaces with genus 1, using the Heavenly equation formalism. This is similar to (6), but one takes a different .C C / real section of MC . Instead of using the real coordinates we set w D ;

N y D ;

z D ip;

x D i p; N

; p 2 C

with D i.p N p/ N corresponding to the flat metric. Let † be a Riemann surface N with a local holomorphic coordinate , such that the Kähler metric on † is h N d d . Suppose that p is a local complex coordinate for fibres of the cotangent bundle T †. If ! is the Kähler form for a neutral metric g then gi jN D @i @jN for a function on the cotangent bundle. Then the equation det gi jN D 1 is equivalent to the first Heavenly equation (6), and gives a Ricci-flat ASD neutral metric. The idea in [43] is to suppose that depends only on the globally defined function N X D h p p, N which is the length of the cotangent vector corresponding to p. There is a globally defined holomorphic .2; 0/-form ! D d ^ dp, which is the holomorphic part of the standard symplectic form on the cotangent bundle, so .; p/ are the holomorphic coordinates in the Plebański coordinate system. The heavenly equation reduces to an ODE for .X/ and Ooguri–Vafa show that for solutions of this ODE to exist h must have constant negative curvature, so † has genus greater than one. In this case one can


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solve the ODE to find p p A2 C BX A 2 D 2 A C BX C A ln p ; A2 C BX C A where A; B are arbitrary positive constants. The metric g is well behaved when X ! 0 N (or p ! 0), as in this limit ! ln .X / and g restricts to h N d d N on † and h dpd pN on the fibres. In the limit X ! 1 the metric is flat. To see it one needs to chose a uniformising coordinate on † so that h is a metric on the upper half plane. Then p p make a coordinate transformation 1 D p; 2 D p. The holomorphic two form is p still d 1 ^ d 2 , and the Kähler potential D i.2 N1 1 N2 / B yields the flat metric. Ooguri–Vafa also observed that the pp-wave metric (72) can be put onto T †, by requiring Q.x; y/ to satisfy certain symmetries. Globally defined neutral metrics on non-compact manifolds were also studied by Kamada and Machida in [30], where they obtained many neutral analogues of well known ASD Riemannian metrics.

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A neutral Kähler surface with applications in geometric optics Brendan Guilfoyle and Wilhelm Klingenberg

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

2 The space of oriented affine lines in E3 2.1 The space of oriented lines . . . . 2.2 Coordinates on L . . . . . . . . . 2.3 The Euclidean group acting on L . 2.4 The correspondence space . . . . 2.5 Jacobi fields along a line in E3 . .

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3 The Kähler structure on L . . . . . . . . 3.1 The complex structure . . . . . . 3.2 The symplectic structure . . . . . 3.3 The neutral metric . . . . . . . . . 3.4 The action of the Euclidean group 3.5 Geodesics . . . . . . . . . . . . .

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Line congruences in E3 . . . . . . . . . . . . . 4.1 Line congruences . . . . . . . . . . . . . 4.2 The optical scalars . . . . . . . . . . . . 4.3 Parametric line congruences . . . . . . . 4.4 The optical scalars and the Kähler metric . 4.5 Generalized curvature . . . . . . . . . . .

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Constructing surfaces using line congruences . . . 5.1 Surfaces from line congruences . . . . . . . . 5.2 Normal line congruences . . . . . . . . . . . 5.3 Surfaces given by zero of a function . . . . . 5.4 Example: elliptic and hyperbolic paraboloids

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Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.1 Reflection in a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.2 Example: plane wave scattered off a paraboloid . . . . . . . . . . . . . . . . . 163

7 The focal set of a line congruence . . . . . 7.1 Focal points of a line congruence . . . 7.2 Alternative definition of focal surfaces 7.3 Focal sets and the Kähler metric . . . 7.4 Further geometric properties . . . . . 8

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Reflection off a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.1 The normal congruence of a cylinder . . . . . . . . . . . . . . . . . . . . . . . 167

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B. Guilfoyle and W. Klingenberg

8.2 The focal set of a plane wave reflected off the inside of a cylinder . 8.3 A point source reflected off a cylinder . . . . . . . . . . . . . . . 8.4 Multiple reflections of a point source off a cylinder . . . . . . . . 8.5 The focal set of a point source reflected off the inside of a cylinder 8.6 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . .

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Generalizations . . . . . . . . . . . . . . . . 9.1 Higher dimensions . . . . . . . . . . . 9.2 Geodesics on 3-manifolds other than E3 9.3 Neutral Kähler structures on T N . . . .

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1 Introduction The space L of oriented affine lines in Euclidean 3-space has an abundance of natural geometric structure. In particular, there is a neutral Kähler metric on this 4-manifold and the purpose of this paper is to review recent work on this structure and its relationship with geometric optics [7]–[12]. The fundamental objects of study in geometric optics are 2-parameter families of oriented lines, or line congruences, which we view as surfaces in L. Thus we are lead to consider the geometry of immersed surfaces † L, which we now briefly summarize. Since L can be identified with the tangent space to the 2-sphere, there is the natural bundle map W L ! S 2 . If j† W † ! S 2 is not an immersion, we say that † is flat. Otherwise, † can be described, at least locally, by sections of the canonical bundle. On the other hand, there is a natural symplectic structure on L, and † L is Lagrangian with respect to this symplectic structure iff the line congruence admits a family of orthogonal surfaces in E3 . In geometric optics such surfaces are the wavefronts of the propagating light. In addition, L admits a natural complex structure J obtained by rotation through 90ı about the oriented line in E3 . Thus, L is a complex surface, and there exists a preferred class of line congruences: the holomorphic curves in L. In general, these line congruences are not Lagrangian and so the planes orthogonal to the lines in L are not integrable (in the sense of Frobenius). The complex and symplectic structures turn out to be compatible and, therefore, together can be used to define a Kähler metric. This metric has signature .C C / and so the metric induced on a surface † may be Riemannian, Lorentz or degenerate. In what follows we review the above geometric structures and apply them to some questions in geometric optics in a homogeneous isotropic medium: the theory of light propagation under the assumption that the light travels along straight lines in E3 . We consider two topics in this field: reflection and focal sets. Given a C 1 surface S in E3 we consider reflection of a ray in S. Thus the reflected ray lies in the plane containing the initial ray and the oriented normal at the point of reflection, with the normal bisecting the angle formed by the initial and reflected rays. Thus a surface S in E3 gives rise to a mapping from L to itself which is continuous, but

A neutral Kähler surface with applications in geometric optics

151

not necessarily smooth. This approach to reflection, which we outline, has previously been exploited in [10], [11], [12]. As a wavefront evolves along a line congruence, if there is any focusing, the surface becomes singular. The points at which this occurs are referred to as focal points. Mathematically, the focal set of a generic line congruence is well understood [1], [3], [15]. Special examples of focal sets, also referred to as caustics by some authors, have been studied for many decades [2], [4], [6], [14]. In this work we give a method for computing the focal set of an arbitrary line congruence and relate its properties to the Kähler structure. We compute one case in detail: the focal set formed by the (multiple) reflection off a cylinder. The first such reflected focal set is often referred to as the coffeecup caustic, since its cross-section is commonly observed on the top of a cup of coffee in the presence of a low bright light. We show that the focal sets of the kth reflection exhibit unexpected symmetries and fit well with observation. Finally, we discuss a number of possible generalizations of the neutral Kähler metric. Acknowledgement. The authors would like to thank Helga Baum for support during the development of this research. This work was completed while the authors were research visitors to Humboldt University, Berlin, supported by SFB 647.

2 The space of oriented affine lines in E3 2.1 The space of oriented lines. We start with 3-dimensional Euclidean space E3 and fix standard coordinates .x 1 ; x 2 ; x 3 /. In what follows we combine the first two coordinates to form a single complex coordinate z D x 1 C ix 2 , set t D x 3 and refer to coordinates .z; t / on E3 . Let L be the set of oriented lines, or rays, in Euclidean space E3 . Such a line is uniquely determined by its unit direction vector UE and the vector VE joining the origin to the point on the line that lies closest to the origin. That is, D fVE C r UE 2 E3 j r 2 RI g; where r is an affine parameter along the line. By parallel translation, we move UE to the origin and VE to the head of UE . Thus, we obtain a vector that is tangent to the unit 2-dimensional sphere in E3 . The mapping is one-to-one and so it identifies the space of oriented lines with the tangent bundle of the 2-sphere T S 2 (see Figure 1). L D f.UE ; VE / 2 E3 E3 j jUE j D 1; UE VE D 0g: 2.2 Coordinates on L. The space L is a 4-dimensional manifold and the above identification gives a natural set of local complex coordinates. Let be the local complex coordinate on the unit 2-sphere in E3 obtained by stereographic projection from the south pole.

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VE UE

0

UE

VE

0

Figure 1

In terms of the standard spherical polar angles .; /, we have D tan 2 e i . We N back to .; / using convert from coordinates .; / p N N N N p ; sin D p : cos D 1 N ; sin D 2 N ; cos D C N N 1C

1C

2

2i

This can be extended to complex coordinates .; / on L minus the tangent space over the south pole, as follows. First note that a tangent vector XE to the 2-sphere can always be expressed as a linear combination of the tangent vectors generated by and : @ @ C X : XE D X @ @ In our complex formalism, we have the natural complex tangent vector @ @ @ i 2 D cos . 2 / e i ; @ @ 2 cos 2 sin 2 @ and any real tangent vector can be written as @ @ XE D C N ; @ @N for a complex number . We identify the real tangent vector XE on the 2-sphere (and hence the ray in E3 ) with the two complex numbers .; /. Loosely speaking, determines the direction of the ray, and determines its perpendicular distance vector to the origin – complex representations of the vectors UE and VE . The coordinates .; / do not cover all of L – they omit all of the lines pointing directly downwards. However, the construction can also be carried out using stereographic projection from the north pole, yielding a coordinate system that covers all of L except for the lines pointing directly upwards. Between these two coordinate patches the whole of the space of oriented lines is covered. In what follows we work in the patch that omits the south direction.


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2.3 The Euclidean group acting on L. The Euclidean group of translations and rotations acts upon L and are naturally reflected by the above coordinates. A translation moves the origin and simply alters the coordinates by the addition of a certain quadratic holomorphic section of the bundle W L ! S 2 : ! 0 D ;

! 0 D C ˛ C a ˛ N 2;

for ˛ 2 C and a 2 R. On the other hand, a rotation about the origin is given by the derived action of SO(3) on L: b ˇ @ @ @ ! 0 D ; ; ! 0 0 D 0 2 N N N @ @ @ ˇ C b .ˇ C b/ for b; ˇ 2 C with b bN C ˇ ˇN D 1. 2.4 The correspondence space. Geometric data will be transferred between E3 and L by use of a correspondence space. Definition 2.1. The map ˆ W L R ! E3 is defined to take ..; /; r/ 2 L R to the point in E3 on the oriented line .; / that lies a distance r from the point on the line closest to the origin (see the right of Figure 2).

LR @

1 ? L

@ˆ @ R @

r

ˆ.; ; r/

E3 0 Figure 2

The double fibration on the left gives us the correspondence between the points in L and oriented lines in E3 : we identify a point .; / in L with ˆ ı 11 .; / E3 , which is an oriented line. Similarly, a point p in E3 is identified with the 2-sphere 1 ı ˆ1 .p/ L, which consists of all of the oriented lines through the point p. The map ˆ is of crucial importance when describing surfaces in E3 and has the following coordinate expression: Proposition 2.2 ([7]). If ˆ.; ; r/ D .z.; ; r/; t .; ; r//, then zD

N 2. N 2 / C 2.1 C /r N 2 .1 C /

and t D

2.N C / N C .1 2 N 2 /r ; N 2 .1 C /

(1)

154


where z D x 1 C ix 2 , t D x 3 and .x 1 ; x 2 ; x 3 / are Euclidean coordinates in E3 . 2.5 Jacobi fields along a line in E3 . The identification with T S 2 endows the space L with a differential structure. The classical interpretation of the tangent space to a line in L as the infinitesimally close lines [16] can be given explicitly as follows. The derivative Dˆ identifies tangent vectors to L at a line with the Jacobi fields orthogonal to the line in E3 . Recall that a Jacobi field along a line in E3 is a vector field X along the line which satisfies the equation rP rP X D 0: Choosing an affine parameter r along the line, this has solution X D X1 C rX2 , for constant vector fields X1 and X2 along . The solutions of this equation form a 6dimensional vector space for each oriented line , which we denote by J. /. Let J ? ./ be the 4-dimensional vector space of Jacobi fields along that are orthogonal to . The vector space J ? . / can be identified with the tangent space T L, by introducing frames. Definition 2.3. A null frame at a point p in E3 is a trio of vectors e.0/ , e.C/ , e./ 2 C ˝ Tp E3 such that e.0/ D e.0/ , e.C/ D e./ , e.0/ e.0/ D e.C/ e./ D 1, and e.0/ e.C/ D 0, where the Euclidean inner product is extended bilinearly over C. Orthonormal frames fe.0/ ; e.1/ ; e.2/ g on T E3 and null frames are related by 1 e.C/ D p .e.1/ i e.2/ / 2

and

1 e./ D p .e.1/ C i e.2/ /: 2

Given an oriented line in E3 with direction 2 S 2 , we can construct an adapted null frame so that e.0/ is the direction of the line: @ 2N @ C C 1 C N @z 1 C N @zN p p 2 2 @ 2 N @ D 1 C N @z 1 C N @zN

e.0/ D e.C/

2

1 N @ ; 1 C N @t p 2 N @ : 1 C N @t

Now the derivative of the map ˆ at a line has the following description in terms of adapted null frame: Proposition 2.4 ([8]). The derivative Dˆ W T.;;r/ L R ! Tˆ.;;r/ E3 is p N 2 2N @ 2 e ; D r e.C/ Dˆ.;;r/ N N N 2 .0/ @ 1 C 1 C .1 C / p 2 @ @ D D e.0/ ; e.C/ ; Dˆ.;;r/ Dˆ.;;r/ N @ @r 1 C


155

Thus the push-forward of the tangent vectors to L define vector fields along the associated line in E3 which are linear in the affine parameter r, i.e. Jacobi fields along the line. Projecting parallel to the line we get the identification between the tangent space to L with the orthogonal Jacobi fields, namely h W T L ! J ? . / where p p N @ 2 2 @ 2 h e.C/ ; h e.C/ : D r D N N @ @ 1 C 1 C 1 C N

3 The Kähler structure on L 3.1 The complex structure. The existence of complex coordinates on L implies that there is a complex structure J on L [13]. That is, a map J W T L ! T L with J 2 D Id which satisfies a certain integrability condition. In fact, this map can be defined as follows. Given an oriented line in E3 consider the map R W J ? . / ! J ? . / given by rotation of the Jacobi field through 90ı in a positive sense about the oriented line . Now we use the identification of the tangent space T L with J ? . / to define the complex structure: J D h1 ı R ı h. This clearly satisfies J 2 D Id and the integrability condition is equivalent to the existence of complex coordinates, which we now establish. Proposition 3.1. The coordinates .; / are holomorphic with respect to the complex structure J defined above. Proof. Rotation through 90ı about the oriented line sends e.C/ to i e.C/ . Thus p N @ 2 2 1 J e.C/ D h ı R r @ 1 C N 1 C N p N 2 @ 2 1 Dh i r e.C/ D i ; N N @ 1 C 1 C and

@ J @

1

Dh

p

ı R

2

1 C N

1

e.C/ D h

i

p 2 1 C N

e.C/ D i

@ : @

Thus and define the eigenspaces of J – they are holomorphic coordinates with respect to J . 3.2 The symplectic structure. The complex structure can be supplemented with a natural symplectic structure . This is a closed non-degenerate 2-form on L which we define by ˝ ˛ ˝ ˛ .;/ .X; Y / D h.X/; r.0/ h.Y / h.Y /; r.0/ h.X/ ; where X; Y 2 T.;/ L, h ; i is the Euclidean metric on E3 and r.0/ is the covariant derivative in the direction.

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In terms of our local coordinates, the symplectic form is D

2 N 2 .1 C /

N 2. N / N N d ^ d C d N ^ d C d ^ d : 1 C N

It follows that this 2-form is closed and non-degenerate. In fact, it is globally exact, being the exterior derivative of a 1-form D d‚. This global 1-form, locally pulled back in the above coordinates, is ‚D

2d N 2d N C : N 2 N 2 .1 C / .1 C /

The symplectic form is compatible with the complex structure, in that ˝ ˛ ˝ ˛ .;/ .J .X/; J .Y // D h.J .X//; r.0/ h.J .Y // h.J .Y //; r.0/ h.J .X// ˝ ˛ ˝ ˛ D R .h.X//; R .r.0/ h.Y // R .h.Y //; R .r.0/ h.X// ˝ ˛ ˝ ˛ D h.X/; r.0/ h.Y / h.Y /; r.0/ h.X/ D .;/ .X; Y /: 3.3 The neutral metric. Given a symplectic form and compatible complex structure J , one can form an inner product by G. ; / D .J ; /. While this will give a symmetric non-degenerate 2-tensor (i.e. a metric), this tensor may not be positive definite. There are only two possibilities: either it is positive definite or else it has split (or neutral) signature .C C /. In our case, we have the latter: Proposition 3.2. The metric G defined as above on L is a neutral metric which is conformally flat, with zero scalar curvature, but is not Einstein. Proof. The local coordinate expression for the metric can be computed to be GD

2i N 2 .1 C /

N 2. N / N N dd d d N C d d ; 1 C N

from which the stated results follow. This metric can be given the following geometric interpretation: Proposition 3.3. The length of X 2 T L with respect to G is the angular momentum about of the line determined by the Jacobi field associated to X. Proof. A direct computation using Dˆ shows that the length of X D X1 C rX2 is the oriented area of the parallelogram spanned by X1 and X2 , i.e. G.X; X/ D .X1 X2 / e.0/ . The Kähler potential of the metric is ‡D

N 2i. N / : N 1 C


157

3.4 The action of the Euclidean group. Since the above construction of G is invariant under Euclidean motions, it is clear that the isometry group of G contains the Euclidean group of translations and rotations. In fact G admits no other continuous isometries: Theorem 3.4 ([8]). The identity component of the isometry group of the metric G on L is isomorphic to the identity component of the Euclidean isometry group. This follows from the following proposition: Proposition 3.5 ([8]). The Killing vectors of G form a 6-parameter Lie algebra given by @ @ @ N @ C K C K C KN ; K D K N @ @ @ N @ with

K D ˛ C 2ai C ˛ N 2;

N 2; K D 2.ai C ˛/ N C ˇ C b ˇ

where ˛; ˇ 2 C and a; b 2 R. The Killing vectors given by ˛ and a generate infinitesimal rotations while those given by ˇ and b generate infinitesimal translations. A Kähler structure with this property on the space of oriented lines in En exists only when n D 3 or n D 7 [18] – see the discussion later on generalizations. 3.5 Geodesics. Any curve in L gives a 1-parameter family of oriented lines in E3 . Classically, these are referred to as ruled surfaces, and the ruled surfaces that correspond to the geodesics of G are: Theorem 3.6 ([8]). A geodesic of G is either a plane or a helicoid in E3 , the former in the case when the geodesic is null, the latter when it is non-null.

4 Line congruences in E3 4.1 Line congruences. In its simplest form, geometric optics models the propagation of light through a homogeneous isotropic medium by a 2-parameter family of rays. Thus we introduce the following definition: Definition 4.1. A line congruence is a 2-parameter family of oriented lines in E3 . From our perspective a line congruence is a surface † in L. For example, a point source corresponds to the 2-parameter family of oriented lines that contain the source point, which thus defines a 2-sphere in L. The dual picture of light propagation is to consider the wavefronts, or surfaces that are orthogonal to a given set of rays. However, not every line congruence has such orthogonal surfaces – indeed, most do not. To explain this we consider the first order properties of †, which can described by two complex functions, the optical scalars: ; W † R ! C. The real part ‚ and the imaginary part of are the divergence and twist of the congruence, while is the shear.

158


4.2 The optical scalars Definition 4.2. A null frame adapted to † L is a null frame fe0 ; eC ; e g which, for each 2 †, we have e0 tangent to in E3 , and the orientation of fe0 ; e1 ; e2 g is the standard orientation on E3 . Definition 4.3. Given a line congruence and adapted null frame, the optical scalars are defined by D hre0 eC ; e i; D hre0 eC ; eC i; where r is the Euclidean connection on E3 . These have the following geometric interpretation. Consider a specific ray in the line congruence and a point p along this ray. Now consider the unit circle in the plane orthogonal to the ray at p. As we flow this circle along the line congruence this circle will become distorted. To first order in the affine parameter along the ray the real part of measures the divergence or contraction of the circle, the imaginary part determines the rotation of the circle, while j j measures the shearing [17] (see Figure 3).

p

Divergence

Shear

Twist

Figure 3

4.3 Parametric line congruences. For computational purposes, we must give explicit local parameterizations of the line congruence. In practice, this will be given locally by a map C ! L W 7! .. ; /; N . ; //. N A convenient choice of parameterization will often depend upon the specifics of the situation, but our formalism holds for arbitrary parameterizations. Proposition 4.4 ([7]). For a parameterized line congruence the optical scalars have the following expressions in terms of first derivatives of the parameterization: D C i D

@C @N N @ @N @ @ @C @C

;

D

@C @N @ @N N @ @ @C @C

;

(2)


where @C @ C r@

N 2@ ; 1 C N

N C r @ N @ @

159

N 2N @ ; 1 C N

and @ and @N are differentiation with respect to and , N respectively. 4.4 The optical scalars and the Kähler metric. The twist has the following important interpretation: Proposition 4.5 ([7]). There exists surfaces orthogonal to the rays of a line congruence if and only if the twist of the line congruence is zero. Moreover, in terms of the Kähler structure, the optical scalars have the following significance: Theorem 4.6 ([8]). A line congruence † L is Lagrangian .i.e. j† D 0/ iff the twist of † is zero. A line congruence † L is holomorphic .i.e. J preserves the tangent space T †/ iff the shear of † is zero. The metric induced on † by G is Riemannian, degenerate or Lorentz iff j j2 < 2 , 2 j j D 2 or j j2 > 2 , respectively, where is the imaginary part of . 4.5 Generalized curvature. A further geometric quantity is the curvature of the line congruence, which is defined to be D N N . A line congruence will be said to be flat if D 0. If a line congruence is non-flat, then the direction of the congruence can be used as a parameterization [7]. In other words the line congruence is locally given N The point source line congruence is non-flat, while the set of rays by 7! .; .; //. orthogonal to a given line forms a flat line congruence. In the case where the line congruence is Lagrangian, the curvature is just the Gauss curvature of the orthogonal surfaces to the lines.

5 Constructing surfaces using line congruences 5.1 Surfaces from line congruences. We now describe how to construct surfaces in E3 using line congruences. Given a line congruence † L, a map r W † ! R determines a map † ! E3 by .; / 7! ˆ..; /; r.; // for .; / 2 †. In other words, we pick out one point on each line in the congruence (see Figure 4). With a local parameterization of †, composition with the above map yields a map C ! E3 which comes from substituting r D r. ; / N in equations (1). 5.2 Normal line congruences. Of particular interest are the surfaces in E3 orthogonal to the line congruence – when the line congruence is normal. As mentioned earlier, these exist iff the twist of the congruence vanishes. By the first of equation (2), this is an integrability condition for a real function:

160


r D r. ; / N

0

Figure 4

Theorem 5.1 ([7]). A line congruence .. ; /; N . ; // N is orthogonal to a surface in E3 iff there exists a real function r. ; / N satisfying NN N N D 2@ C 2N @ : @r N 2 .1 C /

(3)

If there exists one solution, there exists a 1-parameter family generated by a real constant of integration. An explicit parameterization of these surfaces in E3 is given by inserting r D r. ; / N in (1). 5.3 Surfaces given by zero of a function. We now show how to construct the normal line congruence of a non-flat oriented surface S E3 . As it is non-flat, we can parameterize S by the direction of the unit normal. Equivalently, we use the real stereographic projection coordinates .; /. Suppose S is defined by the pre-image of zero of a function G W E3 ! R. Then the unit normal to S is, using standard Euclidean coordinates .x 1 ; x 2 ; x 3 /, grad G @ @ @ Ny D D A1 .x 1 ; x 2 ; x 3 / 1 C A2 .x 1 ; x 2 ; x 3 / 2 C A3 .x 1 ; x 2 ; x 3 / 3 ; @x @x @x j grad Gj where Ai D

X n j D1

@G @x j

2 1=2

@G : @x i

161


The link with the coordinates D tan

e i is

2

A1 .x 1 ; x 2 ; x 3 / D cos sin ;

A2 .x 1 ; x 2 ; x 3 / D sin sin ;

(4)

A3 .x 1 ; x 2 ; x 3 / D cos : The tangent plane through a point .; / on S is given by A1 x 1 C A2 x 2 C A3 x 3 D B;

(5)

where Ai are given by (4) and B is a function of .; / determined by the surface. N is the distance of the point on the surface to the point on the The function r.; / normal line which lies closest to the origin. This is given by B rDq D B: A21 C A22 C A23 Finally, the exact functional relationship between and is given by N D .; /

1 N 2 @r : .1 C / 2 @N

N The task then, reduces to finding r as a function of and , or equivalently, and . 1 2 3 In many simple cases it is possible to solve (4) and express .x ; x ; x / as functions of and . These can be directly inserted into (5) to find r.; /. Once we have r and as functions of , equations (1) give the explicit parameterization of S in terms of . 5.4 Example: elliptic and hyperbolic paraboloids. Suppose the surface S is determined by .x 1 /2 .x 2 /2 G D x3 C C D 0; a b for some constants a and b. The elliptic paraboloid with a D 1 b D 1 and the hyperbolic paraboloid with a D b D 1 are graphed below. The unit normal is

4.x 1 /2 4.x 2 /2 C Ny D 1 C 2 a b2

1=2

2x 1 @ 2x 2 @ @ C C 3 : 1 2 a @x b @x @x

We can invert the relations (4) to a x D cos tan ; 2 1

b x D cos tan ; 2 2

a b x D cos2 C sin2 tan2 : 4 4 3

Note that the coordinate domain 0 < =2, 0 < 2 cover all of the paraboloid. These give sin2 : B D .a cos2 C b sin2 / 4 cos

162


Finally converting to holomorphic coordinates to get rD

N 2 b. / N 2 a. C / ; N N 4.1 /.1 C /

D

N C b. /.1 N N N 3 / a. C /.1 C 3 / : N 2 4.1 /

The equations (1) now give the explicit parameterization of the paraboloid. Figure 5 shows the resulting parameterization of the paraboloids with a D b D 1 and a D b D 1.

Figure 5

The lines on these surface are the images of the lines of longitude and latitude around the north pole under the inverse of the Gauss map.

6 Reflection 6.1 Reflection in a surface. Given a C 1 surface S in E3 we consider the reflection of a ray in S. The reflected ray lies in the plane containing the initial ray and the oriented normal at the point of reflection, with the normal bisecting the angle formed by the initial and reflected rays. This is equivalent to a certain action on the space of oriented lines, as described by: Theorem 6.1 ([10]). Consider a parametric line congruence D 1 . 1 ; N 1 /, D 1 . 1 ; N 1 / reflected off an oriented surface with parameterized normal line congruence D 0 . 0 ; N 0 /, D 0 . 0 ; N 0 / and r D r0 . 0 ; N 0 / satisfying (3) with D 0 and D 0 . Then the reflected line congruence .2 ; 2 / is 2 D 2 D

.N0 N1 /2 0 ..10 N0 /N1 2N0 /2

20 N1 C 1 0 N0 ; .1 0 N0 /N1 2N0

.1C0 N1 /2 N 0 ..10 N0 /N1 2N0 /2

C

.N0 N1 /.1C0 N1 /.1C0 N0 / r0 ; ..10 N0 /N1 2N0 /2

(6)

(7)

where the incoming rays are only reflected if they satisfy the intersection equation 1 D

.1CN0 1 /2 0 .1C0 N0 /2

.0 1 /2 N 0 .1C0 N0 /2

C

.0 1 /.1CN0 1 / r0 : 1C0 N0

(8)


163

By virtue of the intersection equation, an alternative way of writing (7) is 2 D

.1C0 N0 /2 N 1 ..10 N0 /N1 2N0 /2

C

2.N0 N1 /.1C0 N1 /.1C0 N0 / r0 : ..10 N0 /N1 2N0 /2

The geometric content of this is: reflection of an oriented line can be decomposed into the sum of a translation and a rotation about the origin. 6.2 Example: plane wave scattered off a paraboloid. Consider the paraboloid, as given earlier in Section 5.4. Thus, it is parameterized by the direction 0 of the normal, with a.0 C N0 /.1 C 03 N0 / C b.0 N0 /.1 03 N0 / 0 D ; 4.1 0 N0 /2 and r0 D

a.0 C N0 /2 b.0 N0 /2 : 4.1 0 N0 /.1 C 0 N0 /

Assume that the incoming plane wave has normal direction along the positive x 1 axis, that is, 1 D 1. By the reflection equation (6), the resulting direction is D

20 C 1 0 N0 ; 1 0 N0 2N0

and substituting the equation of the paraboloid into (7) yields D

a.0 C N0 /2 .3 20 C 2N0 C 20 N0 20 N02 C 202 N0 302 N02 / .1 0 N0 /2 .1 0 N0 2N0 /2 C

b.0 N0 /.2 C 30 3N0 C 202 C 2N02 20 N0 C 20 N02 202 N0 / .1 0 N0 /2 .1 0 N0 2N0 /2

C

b.0 N0 /.202 N02 20 N03 203 N0 302 N03 C 303 N02 203 N03 / : .1 0 N0 /2 .1 0 N0 2N0 /2

Since the incoming ray direction is fixed, we are parameterizing the reflected line congruence by the direction 0 of the normal to the surface at the point of reflection. A direct integration of equation (3) gives the function r as r D

2.0 C N0 /Œa.0 C N0 /2 b.0 N0 /2 C C: .1 C 0 N0 /2 .1 0 N0 /

Finally, the wavefronts form a one-parameter family of parameterized surfaces, which can be obtained by substituting for , and r in equation (1). The result in spherical polar coordinates, after some simplifications, boils down to: x 1 D 8a.1 cos / tan cos3 C C.1 2 sin2 cos2 / x 2 D 2.2a sin2 cos2 C b/ tan sin 2C sin2 sin cos x 3 D a.4 cos2 1/ tan2 cos2 b tan2 sin2 2C sin cos cos ; where 0 < =2 and 0 < 2.

164


Figure 6

7 The focal set of a line congruence 7.1 Focal points of a line congruence. Let and be the optical scalars of a line congruence † as described in Section 4.4. Definition 7.1. A point p on a line in a line congruence is a focal point if and

blow-up at p. The set of focal points of a line congruence † generically form surfaces in E3 , which will be referred to as the focal surfaces of †. Theorem 7.2. The focal set of a parametric line congruence † is fˆ.; r/ j 2 † and 1 . 0 C N0 /r C . 0 N0 0 N 0 /r 2 D 0g; where the coefficients of the quadratic equation are given locally by (2) evaluated at r D 0. Proof. In terms of the affine parameter r along a given line, the Sachs equations, which

and must satisfy, are [17]: @ D 2 C N ; @r

@

D . C / : N @r

These are equivalent to the vanishing of certain components of the Ricci tensor of the Euclidean metric. They have solution: D

0 0 . 0 N0 0 N 0 /r ; D ; 1 . 0 C N0 /r C . 0 N0 0 N 0 /r 2 1 . 0 C N0 /r C . 0 N0 0 N 0 /r 2

where 0 and 0 are the values of the optical scalars at r D 0. The theorem follows. This has the following corollary: Corollary 7.3. Let † be a line congruence, D ‚ C i , the associated optical scalars and 0 , ‚0 , 0 , 0 their values at r D 0. If † is flat with non-zero divergence, then there exists a unique focal surface S given by r D .2‚0 /1 . If it is flat with zero divergence, then the focal set is empty.


165

If † is non-flat, then there exists a unique focal point on each line iff j 0 j2 D 20 , there exist two focal points on each line iff j 0 j2 < 20 and there are no focal points on each line iff j 0 j2 > 20 . The focal set is given by 1

rD

‚0 ˙ .j 0 j2 20 / 2 : 0 N0 0 N 0

Proof. The focal set of a parameterized line congruence are given by r D r. ; / N satisfying the quadratic equation in Theorem 7.2. If D 0, then there is none or one solution depending on whether ‚0 D 0 or not. If ¤ 0 then there are two, one or no solutions iff j 0 j2 20 is greater than, equal to or less than zero (respectively). The solution of the quadratic equation in each case is as stated. 7.2 Alternative definition of focal surfaces. There is also the equivalent definition for focal surfaces: Proposition 7.4. A continuously differentiable surface S in E3 is a focal surface of a line congruence † iff every line in † is tangent to S at some point. Proof. Let † be locally parameterized by 7! .. ; /; N . ; //, N where .; / are the canonical coordinates above. Then a surface in E3 given by r W † ! R is tangent to the line congruence iff 2 3 N N 2

6 1C N Det 4 @z N @z

2 1C N

1 1C N 7

@t 5 D 0; N @t

@zN @N zN

where the partial derivatives are in and . N This determinant equation, which is N @N z@t N @t N @z@t N / C .1 /.@z N N z/ 2.@zN @t N / C 2.@z @N zN @z@ N D 0; is a quadratic equation for r D r. ; / N with coefficients given by the first derivatives of z. ; /, N zN . ; / N and t . ; /. N Carrying out the differentiation we find, for example, that N N 2r 4.C / N 2.1C2 / 4. 2 / N 2r N @ C @z D N 2 N 3 N 2 N 3 @ .1C /

C

.1C /

2 N 2 @ .1C /

2 2 N N 2 @ .1C /

.1C /

C

.1C /

2 @r: 1C N

Similar computations finally yield the quadratic that appears in Theorem 7.2. 7.3 Focal sets and the Kähler metric Theorem 7.5. Let † be an immersed surface in L. If † is flat, there is exactly one focal point on each line of the congruence. If † is not flat then there is none, one or two focal points on each line iff the metric induced on † by G is Riemannian, degenerate or Lorentz .respectively/.

166


Proof. The number of focal points on a given line is determined by the sign of the discriminant of the quadratic equation in Theorem 7.2: j 0 j2 20 . By Theorem 4.6 (cf. Theorem 2 of [8]), this is precisely what determines the sign of the metric induced on † by G: the metric is Riemannian, degenerate or Lorentz iff j 0 j2 20 is less than, equal to or greater than zero. The result follows. 7.4 Further geometric properties. Consider now the case where there are two focal points on each line of †. Thus j 0 j2 20 > 0, and further suppose that these focal points form two continuously differentiable surfaces S1 and S2 in E3 . Let L be the distance between the focal points and ' the angle between the normals to S1 and S2 at corresponding points. Theorem 7.6. The distance L and angle ' defined above are given by 1 j 0 j2 20 2 2 LD2 ; cos2 ' D 0 2 : 0 N0 0 N 0 j 0 j Proof. The first of these follows trivially from the fact that the two focal surfaces are given by 1 1 ‚0 C .j 0 j2 20 / 2 ‚0 .j 0 j2 20 / 2 r1 D ; r2 D : 0 N0 0 N 0 0 N0 0 N 0 The line congruence †, by assumption, is not flat, and so we parameterize it by its direction . To compute the angle ' we note that parametric equations for S1 and S2 N t D t1 .; /; N z D z1 .; /;

N t D t2 .; /; N z D z2 .; /;

are obtained by inserting r D r1 and r D r2 in equations (1). Let 1 ; 2 2 P 1 be the directions of the normals to S1 and S2 , respectively. Thus, for i D 1; 2, N i @ti @z N i @zi @t i D : N i @zi @N zNi 1 i N i @zNi @z If we introduce, for i D 1; 2, N i @ti @z N i ˛i D @zi @t

N i @zi @N zNi ; bi D @Nzi @z

and

a straightforward computation shows that b1 b2 C 2.˛1 ˛N 2 C ˛2 ˛N 1 /

cos ' D ˙

12 :

.b12 C 4˛1 ˛N 1 /.b22 C 4˛2 ˛N 2 /

A lengthy computation involving the explicit expressions for ˛i and bi obtained by differentiation of (1), yields b1 b2 C 2.˛1 ˛N 2 C ˛2 ˛N 1 / D

40 i

N 2 C 20 i @L@L N

N 0 .@L/2 0 .@L/ 2 N C / N ˇ @L/ N 4ˇ 2 N 0 C 4ˇN 2 0 80 iˇ ˇN ; C 2L.ˇ@L

02 .1


167

and .b12 C 4˛1 ˛N 1 /.b22 C 4˛2 ˛N 2 / D

16 0 N 0 N 2 C 20 i @L@L N

N .@L/2 0 .@L/ N 4 0 C /

04 .1

N ˇ @L/ N 4ˇ 2 N 0 C 4ˇN 2 0 80 iˇ ˇN C 2L.ˇ@L where we have introduced N 2 @N ˇ D .1 C /

0

N 2

0 .1 C /

0 C i@

0

2

;

:

The expression for the angle ' follows.

8 Reflection off a cylinder 8.1 The normal congruence of a cylinder. Consider a cylinder of radius a, with axis lying along the x 3 -axis in E3 . Then we have: Proposition 8.1. The inward pointing normal to such a cylinder is given parametrically by: D e iv ; D u e iv ; for .u; v/ 2 R S 1 . The distance of a point p on the surface from the point on the normal through p that lies closest to the origin is r D a. Proof. This can be checked by noting that, with the aid of (1), the mapping .u; v/ 7! ˆ..u; v/; .u; v/; r.u; v//, with , and r as stated, yields a parameterization of the cylinder: .u; v/ 7! .a cos v; a sin v; u/. Moreover, the oriented normal at the point .u; v/ on the cylinder is given by the expression in the proposition. 8.2 The focal set of a plane wave reflected off the inside of a cylinder. The coffeecup caustic is the focal set of a plane wave reflected off the inside of this cylinder. This turns out to be: Proposition 8.2. Consider the reflection off the inside of a cylinder of radius a of a line congruence consisting of parallel rays traveling along the x 1 -axis making an angle ˇ with the x 3 -axis. The focal set of the reflected line congruence is a surface given parametrically by

3 x D a cos v cos v ; 2 1

2

x 3 D u for u 2 R and =2 v 3=2.

x 2 D a sin v.cos2 v 1/;

a cos v cot ˇ; 2

168


Proof. The normal congruence to the plane wave is .1 ; 1 /, where 1 and 1 2 C is free. Reflecting this off the cylinder given above, we have by Theorem 6.1: 2 D N1 e 2iv ; 2 D 12 ae iv 2uN1 ae iv N12 e 2iv ; or if we parameterize by the point of reflection D 0 D ue iv N1 2 D ; N

2 D

12

ae

iv

N2 2 N1 a N 1

12 N

:

We now compute the optical scalars for this line congruence via equations (2) and find that the line congruence is flat (i.e. N N D 0). Thus by Corollary 7.3 there is exactly one focal point on each line, given by, after some computation, r D u cos ˇ

a cos v.2 cos2 ˇ 1/ : 2 sin ˇ

Inserting this, along with the expressions for 2 and 2 in (1) yields the stated result. The domain of v must be restricted to half a circle as the incoming rays reflect on the inside of only one half of the cylinder. 8.3 A point source reflected off a cylinder. We now consider the focal set formed by reflection of a point source off the inside of the cylinder. To this end, the following theorem describes reflection in a cylinder as a mapping .1 ; 1 / 7! .2 ; 2 /: Theorem 8.3. A ray .1 ; 1 / intersects a cylinder of radius a lying along the x 3 -axis if and only if ˇ ˇ ˇ 1 N 1 N1 1 ˇ ˇ ˇ a: (9) ˇ ˇ 1 .1 C 1 N1 / For such a ray, the reflected ray is 1 1 N 1 N1 1 ˙ .a2 1 N1 .1 C 1 N1 /2 C .1 N 1 N1 1 /2 / 2 2 N 2 D 1 ; aN1 .1 C 1 N1 / 1 N 1 1 N1 2 N 2 2 1 N N 2 1 1 ˙ .a 1 1 .1 C 1 1 / C .1 N 1 1 1 / / 2 D 1 1 C 1 N1 1 1 N 1 N1 1 ˙ .a2 1 N1 .1 C 1 N1 /2 C .1 N 1 N1 1 /2 / 2 2 ; aN1 .1 C 1 N1 /

where exterior and interior reflection are given by the plus and minus signs, respectively. Proof. Consider an incoming ray .1 ; 1 /. The reflection equations (6) and (7) tell us again that the reflected ray is 2 D N1 e 2iv ;

2 D

1 iv 2uN1 ae iv N12 e 2iv : ae 2

(10)


169

This incoming ray intersects the cylinder iff (cf. (8)): 1 D

1 iv ae C 2u1 ae iv 12 : 2

We eliminate u from this equation by combining it with its conjugate and solving the resulting equation for v. The solution, which exists iff (9) holds, is e

iv

1 1 N 1 N1 1 ˙ .a2 1 N1 .1 C 1 N1 /2 C .1 N 1 N1 1 /2 / 2 : D aN1 .1 C 1 N1 /

Substituting this back into the intersection equation we get that uD

1 1 N 1 C N1 1 ˙ 21 N1

11 N1 .a2 1 N1 .1 1C1 N1

1 C 1 N1 /2 C .1 N 1 N1 1 /2 / 2 :

Finally, putting these last two equations into the reflected ray equation (10) yields the stated result. 8.4 Multiple reflections of a point source off a cylinder. For multiple reflection we have the following: Theorem 8.4. The kth reflection of a ray .1 ; 1 / off the inside of the cylinder is " k

kC1 D .1/ 1

kC1

1

‰1 i .j1 j2 ‰12 / 2 j1 j

#2k ;

" #2k 1 ‰1 i .j1 j2 ‰12 / 2 .1/k N 2 2 2 1 1 1 ka.1 j1 j /.j1 j ‰1 / 2 D ; j1 j N1

where ‰1 D

1 N 1 N1 1 : ai.1 C j1 j2 /

Proof. This follows from iterations of the above theorem once we realize that jl j and ‰l

l N l Nl l : ai.1 C jl j2 /

are preserved by reflection in a cylinder. 8.5 The focal set of a point source reflected off the inside of a cylinder. For a point source at a finite distance, the following theorem describes the focal set of the kth reflection:

170


Theorem 8.5. Consider the kth reflection off the inside of a cylinder lying along x3 axis with radius a of a point source at .l; 0; 0/. The focal set of the reflected line congruence consists of a surface: 1

kC1

z D .1/

l

1

Œl sin v i C .a2 l 2 sin2 v/ 2 2k Œ2kl cos v sin v e iv C .a2 l 2 sin2 v/ 2 1

a2k Œ2kl cos v C .a2 l 2 sin2 v/ 2 1

x3 D

k.1 u2 /Œa2 l 2 2l 2 sin2 v C 2kl cos v.a2 l 2 sin2 v/ 2 1

uŒ2kl cos v C .a2 l 2 sin2 v/ 2

;

and a curve in the x 1 x 2 -plane: 1

1

z D .1/kC1 ka2k Œl sin v i C .a2 l 2 sin2 v/ 2 2k Œl C 2ke iv .a2 l 2 sin2 v/ 2 ; where z D x 1 C ix 2 , u 2 R and v is in the domain 0 v for l a; and

sin1 .a= l/ v sin1 .a= l/ for l > a:

Proof. Consider a point source lying at .l; 0; 0/ in E3 . This line congruence can be parameterized by its direction 1 and 1 D l.112 /=2. The line congruence obtained from k reflections of this point source off the inside of a cylinder of radius a is given by Theorem 8.4. We then compute the optical scalars of this line congruence parameterized by 1 using (2). We find, for example, that 1

j 0 j k.1 C u2 /.kl cos v C .a2 l 2 sin2 v/ 2 /2 D ; 1

0 2u.2kl cos v C .a2 l 2 sin2 v/ 2 / where 1 D ue iv . The similar expression for 0 shows that the reflected congruence is not flat. Thus each line contains exactly two focal points which can be obtained by inserting the solutions of the quadratic equation of Theorem 7.2 into (1). The results are as stated above. 8.6 Discussion of results. The focal surface obtained from a plane wave reflected off the inside of a cylinder (Proposition 8.2) is symmetric along the x 3 -axis and intersects any plane parallel to the x 1 x 2 -plane in a curve. This curve, called a nephroid, is often observed on the top of a cup of coffee in the presence of a strong, low and distant light source – hence the sobriquet – the coffeecup caustic. Note that this level set is independent of the angle ˇ of incidence of the incoming light. The focal surface of the kth reflection of a point source also has a symmetry: Corollary 8.6. The focal surface generated by the kth reflection of a point source is invariant under translation along the cylindrical axis.


171

Proof. This follows from the fact that x 1 and x 2 of the surface in Theorem 8.5 are independent of u. This symmetry is not shared by the reflected wavefront itself – just its focal surface. In Figure 7 we illustrate the 1st reflected wavefront – the lack of translational symmetry is obvious.

0:5

1

0

0:5

0

0

0:4

0:8

1

Figure 7

In the sequence of pictures in Figure 8 we show the evolution of the level sets of the first focal set as the distance of the source decreases. The cylinder is shown by the heavy circle.

Figure 8

172


The focal curve lies entirely outside of the cylinder and so is not seen in reality. In fact, these properties are shared by the focal sets of point sources reflected off any translation invariant surface: Theorem 8.7 ([12]). The focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source and is not physically visible, while the focal surface is translation invariant. The cross-section of the focal surface inside the cylinder for varying values of l=a is illustrated in Figure 9. The parallel wave limit, given by Proposition 8.2, is also indicated with a broken line.

Figure 9

In Figure 10 we compare the higher reflection caustics for varying values of l=a. These higher reflection caustics are in fact visible physically. At each reflection some of the light intensity is lost and one expects to see a series of overlapping caustics of lessening brightness. In fact, the detailed profile of light intensity near a caustic varies in ways that geometric optics does not model well. Nonetheless, the accompanying plate is a photograph of the caustics formed by a 7 cm diameter brass cylinder and agrees well with the geometric optics approximation. The photograph (Figure 11), which was taken by the first author in collaboration with Grace Weir, shows the first and second reflection caustic formed by a light source at l=a D 1 (compare with the first two curves on the top row of Figure 10).

9 Generalizations There are a number of different ways in which the Kähler structure presented in the preceding can be generalized. Here we mention only three.

173


Number of reflections 2

3

4

5

0.1

0.3

l/a 0.5

0.7

1

1

Figure 10

9.1 Higher dimensions. Let Ln be the space of oriented affine lines in En . This is a 2.n 1/-dimensional manifold which is homeomorphic to T S n1 . The Euclidean group of En sends oriented lines to oriented lines, and hence maps n L to itself. A natural question to ask is whether there is a Kähler metric on Ln which is invariant under this action. In general, the answer is no, the exceptions being dimension 3 and 7. In more detail: Theorem 9.1 ([18]). Suppose that Ln admits a Kähler metric that is invariant under the induced transitive action of a connected closed subgroup of the Euclidean group. Then either n D 3 or n D 7. In these two cases, the metric is neutral and unique up to addition of the round metric on S 2 and S 6 .

174


Figure 11

9.2 Geodesics on 3-manifolds other than E3 . Another natural question to ask is whether there exists a natural Kähler structure on the space of oriented geodesics on an arbitrary Riemannian 3-manifold M . One of the first problems is that this space may not be Hausdorff: the global behaviour of the geodesics can be quite pathological. In addition, while one can identify the tangent space to the space of geodesics with the Jacobi fields, one cannot define a complex structure by rotation through 90ı about the line, as this operation does not, in general, send Jacobi fields to Jacobi fields. In fact, the necessary and sufficient condition that such a rotation preserves the Jacobi equations is that M be of constant curvature [13]. In the Riemannian case we are thus led to consider the space of oriented geodesics on hyperbolic 3-space H3 and the 3-sphere S3 . The space of oriented geodesics of H3 , which we denote by L.H3 /, has the following description: Definition 9.2. Let W S 2 ! S 2 be the antipodal map and define the reflected diagonal by x D f. 1 ; 2 / 2 S 2 S 2 j 1 D . 2 /g: Proposition 9.3. The space of oriented geodesics on hyperbolic 3-space L.H3 / is x homeomorphic to S 2 S 2 . Proof. Consider the unit ball model of H3 . In this model, the geodesics are either


175

diameters, or circles which are asymptotically orthogonal to the boundary 2-sphere. An oriented geodesic can thus be uniquely identified by its beginning and end point on the boundary. Moreover, any ordered pair of points on the boundary 2-sphere define a unique oriented geodesic, as long as the points are distinct. Thus, the space of oriented geodesics is homeomorphic to S 2 S 2 fdiagg. In fact, for geometric reasons which will become clear below, we will identify an oriented geodesic by the direction of its tangent vector at past and future infinity. Since these directions are inward and outward pointing at past and future infinity (respectively), we see that the oriented geodesics can also be identified with S 2 S 2 minus antipodal directions, as claimed. We define the map ˆ W L.H3 / R ! H3 as before: it takes an oriented geodesic and a real number r to the point in H3 which is an affine parameter distance r along the geodesic (the specific choice of affine parameter is immaterial for what follows). The derivative Dˆ then identifies the tangent space T L.H3 / with the Jacobi fields along the oriented geodesic in H3 , which when projected orthogonal to the line gives h W T L.H3 / ! J ? . /. As in the E3 case, we now define a complex structure J on L.H3 / by rotation of the Jacobi fields through 90ı about the oriented geodesics. If we denote S 2 with its standard complex structure by P 1 , then Theorem 9.4 ([5]). The 4-manifold L.H3 / with the complex structure J is biholomorx phic to P 1 P 1 . The gist of this is that the complex structure J comes from the complex structure on the 2-spheres at past and future infinity, the former being opposite to the standard complex structure since the geodesic is pointing inwards there. Proceeding as in the E3 case, we define a symplectic structure by ˝ ˛ ˝ ˛ .X; Y / D h.X/; r.0/ h.Y / h.Y /; r.0/ h.X/ ; where X; Y 2 T L.H3 /, h ; i is the hyperbolic metric on H3 and r.0/ is the covariant derivative in the direction of the oriented geodesic . Finally we get the metric by G. ; / D .J ; /. Theorem 9.5 ([5]). .L.H3 /; J ; ; G/ is a neutral Kähler surface. The metric G is conformally flat and scalar flat, but is not Einstein. The case of oriented geodesics in S3 has not been considered before, although analogous results should hold. If one considers geodesics on 3-dimensional Lorentz space forms, the construction of a neutral Kähler structure is also possible – see below. 9.3 Neutral Kähler structures on TN . It is possible to give a different construction for the neutral Kähler structure on T S 2 that can be easily generalized. Consider S 2 with the round metric g and let j be the standard complex structure.

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The Levi-Civita connection associated with g splits the tangent bundle T T S 2 T S ˚T S 2 and we define a complex structure on T S 2 by J D j ˚j . The integrability of J follows from the integrability of j . To define the symplectic form, consider the metric g as a mapping from T S 2 to 2 T S and pull back the canonical symplectic form on T S 2 to on T S 2 . This symplectic structure is compatible with the complex structure. Finally, the metric is defined as above by G. ; / D .J ; /. The triple .J ; ; G/ determine a Kähler structure on T S 2 . 2

Theorem 9.6 ([8]). The Kähler structure defined above on T S 2 coincides with that on L. Now, given a Riemannian 2-manifold .N; g/ we can carry through the preceding construction and thus get a Kähler structure .J ; ; G/ on the tangent bundle T N . This structure is similar to that on L in that: Theorem 9.7 ([8]). The metric G has neutral signature .C C / and is scalar-flat. Moreover, G is Kähler–Einstein iff g is flat, and G is conformally flat iff g is of constant curvature. Infinitesimal isometries of .N; g/ lift to two infinitesimal isometries of .T N; G): Theorem 9.8 ([9]). Let Iso.T N; G/ be the vector space of Killing vectors of .T N; G/ and Iso.N; g/ be the space of Killing vectors of .N; g/, where the metric g is assumed to be complete. If g is non-flat, then Iso.T N; G/ Š Iso.N; g/ ˚ .Hol.T N; N / \ Lag.T N; N // ; where Hol.T N; N / and Lag.T N; N / are the spaces of holomorphic and Lagrangian sections of the canonical bundle T N ! N , respectively. In addition, dim.Iso.N; g// D dim.Hol.T N; N / \ Lag.T N; N //. If g is flat, then Iso.T N; G/ Š Hty.N; g/ ˚ .Hol.T N; N / \ Lag.T N; N // ˚ V ; where Hty.N; g/ is the space of homotheties of g and V is a certain 3-dimensional vector space. The geodesics of .N; g/ and those of .T N; G) are related by Theorem 9.9 ([9]). The linear subspaces of the fibres of the bundle T N ! N are null geodesics. These are the only geodesics that lie in the fibres. The geodesics that do not lie in the fibres project under the bundle map to geodesics on N. Finally, the signature of the metric on a Lagrangian surface is: Proposition 9.10 ([9]). The metric induced on a Lagrangian surface by the neutral Kähler metric is either Lorentzian or degenerate. The latter occurs when the surface is both Lagrangian and holomorphic.


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This generalization contains a further interesting special case: that of the space of future-pointing time-like geodesics of Lorentz 3-space E31 , which we denote by L31; . This space can be identified with the tangent bundle to the hyperbolic disc H 2 [9], on which we can construct a neutral Kähler metric, as above. This has the following property: Theorem 9.11 ([9]). The identity component of the isometry group of the Kähler metric on TH 2 is isomorphic to the identity component of the Lorentzian isometry group. The geodesics of the neutral Kähler metric are characterized by: Theorem 9.12 ([9]). The geodesics of the Kähler metric on TH 2 are generated by the 1-parameter transvections of the Lorentzian isometry group. Finally, we note the following. A surface S immersed in Euclidean or Lorentz 3space is called Weingarten if there exists a functional relationship between the eigenvalues of the 2nd fundamental form of the immersion. This property can be characterized as follows: Theorem 9.13 ([9]). Let S be a C 2 -smooth .space-like/ surface in E3 .E31 / and † be the oriented normal congruence, considered as a surface in L .L31; /. Then S is Weingarten iff the Lorentzian metric induced by G on † is scalar flat.

References [1] Arnold, V. I., Gusein-Zade, S. M., and Varchenko, A. N., Singularities of differentiable maps. Volume I, Monogr. Math. 82, Birkhäuser, Basel 1986. [2] Bruce, J., Giblin, P., and Gibson, C., On caustics of plane curves. Amer. Math. Monthly 88 (1981), 651–657. 151 [3] Bruce, J., Giblin, P., and Gibson, C., On caustics by reflection. Topology 21 (1982), 179–199. 151 [4] Cayley, A., A memoir upon caustics. Philos. Trans. Roy. Soc. London 147 (1857), 273–312. 151 151 [5] Georgiou, N., and Guilfoyle, B., On the space of oriented geodesics of hyperbolic 3-space. Rocky Mountain J. Math., to appear. 175 [6] Glaeser, G., Reflections on spheres and cylinders of revolution. J. Geometry Graphics 3 (1999), 121–139. 151 [7] Guilfoyle, B., and Klingenberg, W., Generalised surfaces in R3 . Proc. Roy. Irish Acad. Sect. A 104 (2004), 199–209. 150, 153, 158, 159, 160 [8] Guilfoyle, B., and Klingenberg, W., An indefinite Kähler metric on the space of oriented lines. J. London Math. Soc. 72 (2005), 497–509. 154, 157, 159, 166, 176 [9] Guilfoyle, B., and Klingenberg, W., A neutral Kähler metric on the space of time-like lines in Lorentzian 3-space. Preprint 2006, arXiv:math.DG/0608782. 176, 177

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[10] Guilfoyle, B., and Klingenberg, W., Reflection of a wave off a surface. J. Geom. 84 (2006), 55–72. 151, 162 [11] Guilfoyle, B., and Klingenberg, W., On Hamilton’s characteristic functions for reflection. Irish Math. Soc. Bulletin 57 (2006), 29–40. 151 [12] Guilfoyle, B., and Klingenberg, W., Reflection in a translation invariant surface. Math. Phys. Anal. Geom. 9 (2006), 225–231. 150, 151, 172 [13] Hitchin, N. J., Monopoles and geodesics. Comm. Math. Phys. 83 (4) (1982), 579–602. 155, 174 [14] Holditch, H., On the n-th caustic by reflexion from a circle. Quart. J. Math. 2 (1858), 301–322. 151 [15] Izumiya, S., Saji, K., and Takeuchi, N., Singularities of line congruences. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1341–1359. 151 [16] Kobayashi, S., and Nomizu, K., Foundations of Differential Geometry. Volume II, Wiley Classics Lib., Wiley and Sons, New York 1996. 154 [17] Penrose, R., and Rindler, W., Spinors and spacetime. Volume 1 and 2, Cambridge University Press, Cambridge 1986. 158, 164 [18] Salvai, M., On the geometry of the space of oriented lines in Euclidean space. Manuscripta Math. 118 (2) (2005), 181–189. 157, 173

A primer on the .2 C 1/ Einstein universe Thierry Barbot, Virginie Charette, Todd Drumm, William M. Goldman, and Karin Melnick

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

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7 Three kinds of dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Barbot was partially supported by theANR project GEODYCOS during the writing of this paper. Charette is grateful for support from NSERC. Goldman is grateful to a Semester Research Award from the University of Maryland, and to NSF grants DMS-0405605 and DMS-070781. Melnick was partially supported by NSF fellowship DMS-855735 during the writing of this paper.

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Projective singular limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Cartan’s decomposition G D KAK . . . . . . . . . . . . . . . . . . . . . . . . 212 Maximal domains of properness . . . . . . . . . . . . . . . . . . . . . . . . . 214

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1 Introduction We will explore the geometry of the conformal compactification of Minkowski .n C 1/space inside of Rn;2 . We shall call this conformal compactification Einn;1 , or the f n;1 . The Einstein universe Einstein universe, and its universal cover will be denoted Ein is a homogeneous space G=P , where G D PO.n; 2/, and P is a parabolic subgroup. When n D 3, then G is locally isomorphic to Sp.4; R/. The origin of the terminology “Einstein universe” is that A. Einstein himself considered as a paradigmatic universe the product S 3 R endowed with the Lorentz metric ds02 dt 2 , where ds02 is the usual constant curvature Riemannian metric on S 3 . The conformal transformations preserve the class of lightlike geodesics and provide a more flexible geometry than that given by the metric tensor. Our motivation is to understand conformally flat Lorentz manifolds and the Lorentzian analog of Kleinian groups. Such manifolds are locally homogeneous geometric structures modeled on Ein2;1 . The Einstein universe Einn;1 is the conformal compactification of Minkowski space n;1 E in the same sense that the n-sphere S n D En [ f1g conformally compactifies Euclidean space En ; in particular, a Lorentzian analog of the following theorem holds (see [11]): f

Theorem (Liouville’s theorem). Suppose n 3. Then every conformal map U ! En n defined on a nonempty connected subdomain U E extends to a conformal automorphism fN of S n . Furthermore fN lies in the group PO.n C 1; 1/ generated by inversions in hyperspheres and Euclidean isometries. Our viewpoint involves various geometric objects in Einstein space: points are organized into 1-dimensional submanifolds which we call photons, as they are lightlike

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geodesics. Photons in turn form various subvarieties, such as lightcones and hyperspheres. For example, a lightcone is the union of all photons through a given point. Hyperspheres fall into two types, depending on the signature of the induced conformal metric. Einstein hyperspheres are Lorentzian, and are models of Einn1;1 , while spacelike hyperspheres are models of S n with conformal Euclidean geometry. The Einstein universe Einn;1 can be constructed by projectivizing the nullcone in the inner product space RnC1;2 defined by a symmetric bilinear form of type .n C 1; 2/. Thus the points of Einn;1 are null lines in RnC1;2 , and photons correspond to isotropic 2-planes. Linear hyperplanes H in RnC1;2 determine lightcones, Einstein hyperspheres, and spacelike hyperspheres, respectively, depending on whether the restriction of the bilinear form to H is degenerate, type .n; 2/, or Lorentzian, respectively. Section 4 discusses causality in Einstein space. Section 5 is specific to dimension 3, where the conformal Lorentz group is locally isomorphic to the group of linear symplectomorphisms of R4 . This establishes a close relationship between the symplectic geometry of R4 (and hence the contact geometry of RP 3 ) and the conformal Lorentzian geometry of Ein2;1 . Section 6 reinterprets these synthetic geometries in terms of the structure theory of Lie algebras. Section 7 discusses the dynamical theory of discrete subgroups of Ein2;1 due to Frances [13], and begun by Kulkarni [19]. Section 8 discusses the crooked planes, discovered by Drumm [8], in the context of Ein2;1 ; their closures, called crooked surfaces are studied and shown to be Klein bottles invariant under the Cartan subgroup of SO.3; 2/. The paper concludes with a brief description of discrete groups of conformal transformations and some open questions. Acknowledgements. Much of this work was motivated by the thesis of Charles Frances [11], which contains many constructions and examples, his paper [13] on Lorentzian Kleinian groups, and his note [11] on compactifying crooked planes. We are grateful to Charles Frances and Anna Wienhard for many useful discussions. We are also grateful to the many institutions where we have been able to meet to discuss the mathematics in this paper. In particular, we are grateful for the hospitality provided by the Banff International Research Station [5] where all of us were able to meet for a workshop in November 2004, the workshop in Oostende, Belgium in May 2005 on “Discrete groups and geometric structures,” the miniconference in Lorentzian geometry at the E.N.S. Lyon in July 2005, the special semester at the Newton Institute in Cambridge in Fall 2005, the special semester at the Erwin Schrödinger Institute, Vienna in Fall 2005, and a seminar at the University of Maryland in summer 2006, when the writing began. Goldman wishes to thank the Erwin Schrödinger Institute for hospitality during the writing of this paper.

2 Synthetic geometry of Einn;1 In this section we develop the basic synthetic geometry of Einstein space, or the Einstein universe, starting with the geometry of Minkowski space En;1 .

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2.1 Lorentzian vector spaces. We consider real inner product spaces, that is, vector spaces V over R with a nondegenerate symmetric bilinear form h ; i. A nonsingular symmetric n n-matrix B defines a symmetric bilinear form on Rn by the rule: hu; viB WD u Bv: where u denotes the transpose of the vector u. We shall denote by Rp;q a real inner product space whose inner product is of type .p; q/. For example, if 2 3 2 3 u1 v1 6 :: 7 6 :: 7 6 : 7 6 : 7 6 7 6 7 6 up 7 6 vp 7 6 6 7 7 uD6 7 ; v D 6vpC1 7 ; 6upC1 7 6 7 6 :: 7 6 :: 7 4 : 5 4 : 5 upCq vpCq then hu; vi WD u1 v1 C C up vp upC1 vpC1 upCq vpCq defines a type .p; q/ inner product, induced by the matrix Ip ˚ Iq on RpCq , where Ip is the p p identity matrix. The group of linear automorphisms of Rp;q is O.p; q/. If B is positive definite – that is, q D 0 – then we say that the inner product space .V; h ; i/ is Euclidean. If q D 1, then .V; h ; i/ is Lorentzian. We may omit reference to the bilinear form if it is clear from context. If V is Lorentzian, and v 2 V , then v is: • • • •

timelike if hv; vi < 0; lightlike (or null or isotropic) if hv; vi D 0; causal if hv; vi 0; spacelike if hv; vi > 0.

The nullcone N.V / in V consists of all null vectors. If W V , then define its orthogonal complement: W ? WD fv 2 V j hv; wi D 0 for all w 2 W g: The hyperplane v ? is null (respectively, timelike, spacelike) if v is null (respectively spacelike, timelike). In the sequel, according to the object of study, we will consider several symmetric n n-matrices and the associated type .p; q/ symmetric bilinear forms. For different bilinear forms, different subgroups of O.p; q/ are more apparent. For example: • Using the diagonal matrix Ip ˚ Iq ; invariance under the maximal compact subgroup O.p/ O.q/ O.p; q/


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is more apparent. • Under the bilinear form defined by the matrix Ipq ˚

q M

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is more apparent. • Another bilinear form which we use in the last two sections is: 0 1 Ip1 ˚ Iq1 ˚ 1=2 1 0 which is useful in extending subgroups of O.p 1; q 1/ to O.p; q/. 2.2 Minkowski space. Euclidean space En is the model space for Euclidean geometry, and can be characterized up to isometry as a simply connected, geodesically complete, flat Riemannian manifold. For us, it will be simpler to describe it as an affine space whose underlying vector space of translations is a Euclidean inner product space Rn . That means En comes equipped with a simply transitive vector space of translations p 7! p C v; where p 2 En is a point and v 2 Rn is a vector representing a parallel displacement. Under this simply transitive Rn -action, each tangent space Tp .En / naturally identifies with the vector space Rn . The Euclidean inner product on Rn defines a positive definite symmetric bilinear form on each tangent space – that is, a Riemannian metric. Minkowski space En;1 is the Lorentzian analog. It is characterized up to isometry as a simply connected, geodesically complete, flat Lorentzian manifold. Equivalently, it is an affine space whose underlying vector space of translations is Rn;1 . The geodesics in En;1 are paths of the form

R ! En;1 ; t 7! p0 C t v; where p0 2 En;1 is a point and v 2 Rn;1 is a vector. A path as above is timelike, lightlike, or spacelike, if the velocity v is timelike, lightlike, or spacelike, respectively. Let p 2 En;1 . The affine lightcone Laff .p/ at p is defined as the union of all lightlike geodesics through p: Laff .p/ WD fp C v 2 En;1 j hv; vi D 0g:

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Equivalently Laff .p/ D p C N where N Rn;1 denotes the nullcone in Rn;1 . The hypersurface Laff .p/ is ruled by lightlike geodesics; it is singular only at fpg. The Lorentz form on En;1 restricts to a degenerate metric on Laff .p/ n fpg. A lightlike geodesic ` En;1 lies in a unique null affine hyperplane. (We denote this `? , slightly abusing notation.) That is, writing ` D p C Rv, where v 2 Rn;1 is a lightlike vector, the null hyperplane p C v ? is independent of the choices of p and v used to define `. The de Sitter hypersphere of radius r centered at p is defined as Sr .p/ WD fp C v 2 En;1 j hv; vi D r 2 g: The Lorentz metric on En;1 restricts to a Lorentz metric on Sr .p/ having constant sectional curvature 1=r 2 . It is geodesically complete and homeomorphic to S n1 R. It is a model for de Sitter space dSn1;1 . As in Euclidean space, a homothety .centered at x0 / is any map conjugate by a translation to scalar multiplication: En;1 ! En;1 ; x 7! x0 C r.x x0 /: A Minkowski similarity transformation is a composition of an isometry of En;1 with a homothety: f W x 7! rA.x/ C b; where A 2 O.n; 1/, r > 0 and b 2 Rn;1 defines a translation. Denote the group of similarity transformations of En;1 by Sim.En;1 /. 2.3 Einstein space. Einstein space Einn;1 is the projectivized nullcone of RnC1;2 . The nullcone is NnC1;2 WD fv 2 RnC1;2 j hv; vi D 0g and the .n C 1/-dimensional Einstein universe Einn;1 is the image of NnC1;2 f0g under projectivization: P

RnC1;2 f0g ! RP nC2 : In the sequel, for notational convenience, we will denote P as a map from RnC1;2 , implicitly assuming that the origin 0 is removed from any subset of RnC1;2 on which we apply P . c n;1 is defined as the quotient of the nullcone NnC1;2 by the The double covering Ein action by positive scalar multiplications. For many purposes the double covering may f n;1 be more useful than Einn;1 , itself. We will also consider the universal covering Ein in §4. Writing the bilinear form on RnC1;2 as InC1 ˚ I2 , that is, 2 2 2 hv; vi D v12 C C vnC1 vnC2 vnC3 ;


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the nullcone is defined by 2 2 2 D vnC2 C vnC3 : v12 C C vnC1

This common value is always nonnegative, and if it is zero, then v q D 0 and v does not correspond to a point in Einn;1 . Dividing by the positive number may assume that 2 2 2 v12 C C vnC1 D vnC2 C vnC3 D1

2 2 vnC2 C vnC3 we

which describes the product S n S 1 . Thus c n;1 S n S 1 : Ein Scalar multiplication by 1 acts by the antipodal map on both the S n and the S 1 -factor. On the S 1 -factor the antipodal map is a translation of order two, so the quotient c n;1 =f˙1g Einn;1 D Ein is homeomorphic to the mapping torus of the antipodal map on S n . When n is even, c n;1 is an orientable double covering. If n is odd, then Einn;1 is nonorientable and Ein n;1 Ein is orientable. The objects in the synthetic geometry of Einn;1 are the following collections of points in Einn;1 : • Photons are projectivizations of totally isotropic 2-planes. We denote the space of photons by Phon;1 . A photon enjoys the natural structure of a real projective line: each photon 2 Phon;1 admits projective parametrizations, which are diffeomorphisms of with RP 1 such that if g is an automorphism of Einn;1 preserving , then gj corresponds to a projective transformation of RP 1 . The projective parametrizations are unique up to post-composition with transformations in PGL.2; R/. • Lightcones are singular hypersurfaces. Given any point p 2 Einn;1 , the lightcone L.p/ with vertex p is the union of all photons containing p: [ L.p/ WD f 2 Phon;1 j p 2 g: The lightcone L.p/ can be equivalently defined as the projectivization of the orthogonal complement p ? \ NnC1;2 . The only singular point on L.p/ is p, and L.p/ n fpg is homeomorphic to S n1 R. • The Minkowski patch Min.p/ determined by an element p of Einn;1 is the complement of L.p/ and has the natural structure of Minkowski space En;1 , as will be explained in §3 below. In the double cover, a point pO determines two Minkowski patches: c n;1 j hp; qi > 0 for all p; q 2 RnC1;2 representing p; O WD fqO 2 Ein O qg; O MinC .p/ c n;1 j hp; qi < 0 for all p; q 2 RnC1;2 representing p; Min .p/ O WD fqO 2 Ein O qg: O

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• There are two different types of hyperspheres. – Einstein hyperspheres are closures in Einn;1 of de Sitter hyperspheres Sr .p/ in Minkowski patches as defined in §2.2. Equivalently, they are projectivizations of v ? \ NnC1;2 for spacelike vectors v. – Spacelike hyperspheres are one-point compactifications of spacelike hyperplanes like Rn in a Minkowski patch Rn;1 Einn;1 . Equivalently, they are projectivizations of v ? \ NnC1;2 for timelike vectors v. • An anti-de Sitter space AdSn;1 is one component of the complement of an Einstein hypersphere Einn1;1 Einn;1 . It is homeomorphic to S 1 Rn . Its ideal boundary is Einn1;1 . 2.4 2-dimensional case. Because of its special significance, we discuss in detail the geometry of the 2-dimensional Einstein universe Ein1;1 . • Ein1;1 is diffeomorphic to a 2-torus. • Each lightcone L.p/ consists of two photons which intersect at p. • Ein1;1 has two foliations F and FC by photons, and the lightcone L.p/ is the union of the leaves through p of the respective foliations. • The leaf space of each foliation naturally identifies with RP 1 , and the mapping Ein1;1 ! RP 1 RP 1 is equivariant with respect to the isomorphism Š

! PGL.2; R/ PGL.2; R/: O.2; 2/ Here is a useful model (compare Pratoussevitch [25]): The space Mat2 .R/ of 2 2 real matrices with the bilinear form associated to the determinant gives an isomorphism of inner product spaces: Mat2 .R/ ! R2;2 ; 3 2 m11 6m12 7 m11 m12 7 7! 6 4m21 5 ; m21 m22 m22 where R2;2 is given the bilinear form defined by 2 0 0 0 16 0 0 1 6 2 40 1 0 1 0 0

3 1 07 7: 05 0

The group GL.2; R/ GL.2; R/ acts on Mat2 .R/ by .A;B/

X ! AXB 1


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and induces a local isomorphism SL˙ .2; R/ SL˙ .2; R/ ! O.2; 2/; where SL˙ .2; R/ WD fA 2 GL.2; R/ j det.A/ D ˙1g: Here we will briefly introduce stems, which are pieces of crooked planes, as will be discussed in §8 below. Let p0 ; p1 2 Ein1;1 be two points not contained in a common photon. Their lightcones intersect in two points p1 and p2 , and the union L.p0 / [ L.p1 / Ein1;1 comprises four photons intersecting in the four points p0 ; p1 ; p1 ; p2 , such that each point lies on two photons and each photon contains two of these points. This stem configuration of four points and four photons can be represented schematically as in Figure 1 below. 1

4

3

2

Figure 1. Stem configuration.

The complement

Ein1;1 n L.p0 / [ L.p1 /

consists of four quadrilateral regions (see Figure 2). In §8 the union S of two nonadjacent quadrilateral regions will be studied; this is the stem of a crooked surface. Such a set is bounded by the four photons of L.p0 / [ L.p1 /.

Figure 2. Two lightcones in Ein1;1 .

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2.5 3-dimensional case. Here we present several observations particular to the case of Ein2;1 . • We will see that Pho2;1 identifies naturally with a 3-dimensional real projective space (§5.5). • A lightcone in Ein2;1 is homeomorphic to a pinched torus. • Suppose p ¤ q. Define C.p; q/ WD L.p/ \ L.q/: If p and q are incident – that is, they lie on a common photon – then C.p; q/ is the unique photon containing them. Otherwise C.p; q/ is a submanifold that we will call a spacelike circle. Spacelike circles are projectivized nullcones of linear subspaces of R3;2 of type .2; 1/. The closure of a spacelike geodesic in E2;1 is a spacelike circle. • A timelike circle is the projectivized nullcone of a linear subspace of R3;2 of metric type .1; 2/. • Einstein hyperspheres in Ein2;1 are copies of Ein1;1 . In addition to their two rulings by photons, they have a foliation by spacelike circles. • Lightcones may intersect Einstein hyperspheres in two different ways. These correspond to intersections of degenerate linear hyperplanes in R3;2 with linear hyperplanes of type .2; 2/. Let u; v 2 R3;2 be vectors such that u? is degenerate, so u determines a lightcone L, and v ? has type .2; 2/, so v defines the Einstein hypersphere H . In terms of inner products, hu; ui D 0;

hv; vi > 0:

If hu; vi ¤ 0, then u; v span a nondegenerate subspace of signature .1; 1/. In that case L \ H is a spacelike circle. If hu; vi D 0, then u; v span a degenerate subspace and the intersection is a lightcone in H , which is a union of two distinct but incident photons. • Similarly, lightcones intersect spacelike hyperspheres in two different ways. The generic intersection is a spacelike circle, and the non-generic intersection is a single point, such as the intersection of L.0/ with the spacelike plane z D 0 in R2;1 . • A pointed photon is a pair .p; / 2 Ein2;1 Pho2;1 such that p 2 . Such a pair naturally extends to a triple p 2 L.p/ which corresponds to an isotropic flag, that is, a linear filtration of R3;2 0 `p P .`p /? R3;2 ; where `p is the 1-dimensional linear subspace corresponding to p; P is the 2dimensional isotropic subspace corresponding to ; and .`p /? is the orthogonal


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subspace of `p . These objects form a homogeneous space, an incidence variety, denoted Flag2;1 , of O.3; 2/, which fibers both over Ein2;1 and Pho2;1 . The fiber of the fibration Flag2;1 ! Ein2;1 over a point p is the collection of all photons through p. The fiber of the fibration Flag2;1 ! Pho2;1 over a photon identifies with all the points of . Both fibrations are circle bundles.

3 Einn;1 Now we shall describe the geometry of Einn;1 as the compactification of Minkowski space En;1 . We begin with the Euclidean analog. 3.1 The conformal Riemannian sphere. The standard conformal compactification of Euclidean space En is topologically the one-point compactification, the n-dimensional sphere. The conformal Riemannian sphere S n is the projectivization P .NnC1;1 / of the nullcone of RnC1;1 . For U S n an arbitrary open set, any local section

U ! RnC1;1 n f0g of the restriction of the projectivization map to U determines a pullback of the Lorentz metric on EnC1;1 to a Riemannian metric g on U . This metric depends on , but its conformal class is independent of . Every section is 0 D f for some non-vanishing function f W U ! R. Then g 0 D f 2 g ; so the pullbacks are conformally equivalent. Hence the metrics g altogether define a canonical conformal structure on S n . The orthogonal group O.n C 1; 1/ leaves invariant the nullcone NnC1;1 RnC1;1 . The projectivization S n D P .NnC1;1 / is invariant under the projective orthogonal group PO.n C 1; 1/, which is its conformal automorphism group. Let Sn ! NnC1;1 RnC1;1 n f0g be the section taking values in the unit Euclidean sphere. Then the metric g is the usual O.n C 1/-invariant spherical metric. Euclidean space En embeds in S n via a spherical paraboloid in the nullcone NnC1;1 . Namely consider the quadratic form on RnC1;1 defined by 2 3 In 0 1 0 1=25 : In ˚ 1=2 D4 1 0 1=2 0

190


The map En ! NnC1;1 RnC1;1 ; 2 3 x x 7! 4hx; xi5 ; 1

(1)

P

composed with projection NnC1;1 ! S n is an embedding E of En into S n , which is conformal. The Euclidean similarity transformation fr;A;b W x 7! rAx C b; where r 2 RC , A 2 O.n/, and b 2 Rn , is represented by 3 3 2 0 b A 0 0 0 5 2 O.n C 1; 1/: 1 hb; bi5 4 0 r 1 0 0 r 0 1

2

Fr;A;b

In 4 WD 2b 0

That is, for every x 2 En , Fr;A;b E.x/ D E fr;A;b .x/ : Inversion in the unit sphere hv; vi D 1 of En is represented by the element

0 1 In ˚ 1 0 which acts on En n f0g by W x 7!

1 x: hx; xi

The origin is mapped to the point (called 1) having homogeneous coordinates 2 3 0n 415 0 where 0n 2 Rn is the zero vector. The map E 1 is a coordinate chart on the open set En D S n n f1g and E 1 ı is a coordinate chart on the open set .En [ f1g/ n f0g D S n n f0g.

(2)


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3.2 The conformal Lorentzian quadric. Consider now the inner product space RnC1;2 . Here it will be convenient to use the inner product hu; vi WD u1 v1 C : : : C un vn unC1 vnC1 12 unC2 vnC3 12 unC3 vnC2 0 1 D u In ˚ I1 ˚ 1=2 v: 1 0 In analogy with the Riemannian case, consider the embedding E W En;1 ! Einn;1 via a hyperbolic paraboloid defined by (1) as above, where the Lorentzian inner product on En;1 is defined by Q D In ˚ I1 . The procedure used previously in the Riemannian case naturally defines an O.nC1; 2/-invariant conformal Lorentzian structure on Einn;1 , and the embedding we have just defined is conformal. Minkowski similarities fr;A;b map into O.n C 1; 2/ as in the formula (2), where r 2 RC I A 2 O.n C 1; 1/I b 2 Rn;1 ; h ; i is the Lorentzian inner product on Rn;1 ; and 2b is replaced by 2b Q. The conformal compactification of Euclidean space is the one-point compactification; the compactification of Minkowski space, however, is more complicated, requiring the addition of more than a single point. Let p0 2 Einn;1 denote the origin, corresponding to 3 2 0nC1 4 0 5: 1 To see what lies at infinity, consider the Lorentzian inversion in the unit sphere defined by the matrix InC1 ˚ 01 10 , which is given on En;1 by the formula W x 7!

1 x: hx; xi

(3)

Here the whole affine lightcone Laff .p0 / is thrown to infinity. We distinguish the points on .Laff .p0 //: • The improper point p1 is the image .p0 /. It is represented in homogeneous coordinates by 3 2 0nC1 4 1 5: 0 • The generic point on .Laff .p0 // has homogeneous coordinates 2 3 v 415 0 where 0 ¤ v 2 Rn;1 ; it equals .E.v//.

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We have described all the points in En;1 [ .En;1 / which are the points defined by vectors v 2 RnC1;2 with coordinates vnC2 ¤ 0 or vnC3 ¤ 0. It remains to consider points having homogeneous coordinates 2 3 v 405 0 where necessarily hv; vi D 0. This equation describes the nullcone in Rn;1 ; its projectivization is a spacelike sphere S1 , which we call the ideal sphere. When n D 2, we call this the ideal circle and its elements ideal points. Each ideal point is the endpoint of a unique null geodesic from the origin; the union of that null geodesic with the ideal point is a photon through the origin. Every photon through the origin arises in this way. The ideal sphere is fixed by the inversion . The union of the ideal sphere S1 with .Laff .p0 // is the lightcone L.p1 / of the improper point. Photons in L.p1 / are called ideal photons. Minkowski space En;1 is thus the complement of a lightcone L.p1 / in Einn;1 . This fact motivated the earlier definition of a Minkowski patch Min.p/ as the complement in Einn;1 of a lightcone L.p/. Changing a Lorentzian metric by a non-constant scalar factor modifies timelike and spacelike geodesics, but not images of null geodesics (see for example [3], p. 307). Hence the notion of (non-parametrized) null geodesic is well-defined in a conformal Lorentzian manifold. For Einn;1 , the null geodesics are photons. 3.3 Involutions. When n is even, involutions in SO.n C 1; 2/ Š PO.n C 1; 2/ correspond to nondegenerate splittings of RnC1;2 . For any involution in PO.3; 2/, the fixed point set in Ein2;1 must be one of the following: • • • • •

the empty set ;; a spacelike hypersphere; a timelike circle; the union of a spacelike circle with two points; an Einstein hypersphere.

In the case that Fix.f / is disconnected and equals fp1 ; p2 g [ S where p1 ; p2 2 Ein2;1 , and S Ein2;1 is a spacelike circle, then S D L.p1 / \ L.p2 /: Conversely, given any two non-incident points p1 ; p2 , there is a unique involution fixing p1 ; p2 and the spacelike circle L.p1 / \ L.p2 /.


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3.3.1 Inverting photons. Let p1 be the improper point, as above. A photon in Ein2;1 either lies on the ideal lightcone L.p1 /, or it intersects the spacelike plane S0 consisting of all 2 3 x 4 p D y5 z for which z D 0. Suppose is a photon intersecting S0 in the point p0 with polar coordinates 2 3 r0 cos. / p0 D 4 r0 sin. / 5 2 S0 E2;1 : 0 Let v0 be the null vector

2

3 cos. / v0 D 4 sin. / 5 1

and consider the parametrized lightlike geodesic .t / WD p0 C t v0 for t 2 R. Then inversion maps .t / to 2

3 cos. 2 / . ı /.t / D .p0 / C tQ 4 sin. 2 / 5 1 where tQ WD

t r02 C 2r0 cos.

/t

:

Observe that leaves invariant the spacelike plane S0 and acts by Euclidean inversion on that plane. 3.3.2 Extending planes in E2 ;1 to Ein2 ;1 • The closure of a null plane P in E2;1 is a lightcone and its frontier Px n P is an ideal photon. Conversely a lightcone with vertex on the ideal circle S1 is the closure of a null plane containing p0 , while a lightcone with vertex on L.p1 / n .S1 [ fp1 g/ is the closure of a null plane not containing p0 . • The closure of a spacelike plane in E2;1 is a spacelike sphere and its frontier is the improper point p1 . • The closure of a timelike plane in E2;1 is an Einstein hypersphere and its frontier is a union of two ideal photons (which intersect in p1 ).

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• The closure of a timelike (respectively spacelike) geodesic in E2;1 is a timelike (respectively spacelike) circle containing p1 , and p1 is its frontier. Consider the inversion on the lightcone of p0 : 02 31 2 3 t sin t sin B6t cos 7C 6t cos 7 B6 7C 6 7 6 7C 6 7 B B6 t 7C D 6 t 7 : @4 0 5A 4 1 5 1 0 The entire image of the light cone L.p0 / lies outside the Minkowski patch E2;1 . Let us now look at the image of a timelike line in E2;1 under the inversion. For example, 02 3 2 31 2 3 2 3 0 0 0 0 B6 0 7C 6 0 7 6 0 7 6 0 7 B6 7 6 7C 6 7 6 7 6 7 6 7C 6 7 6 7 B B6 t 2 7C D 6 t 7 6 1=t2 7 D 6 s 2 7 @4t 5A 4 1 5 41=t 5 4s 5 1 1 1 t 2 where s D 1=t . That is, the inversion maps the timelike line minus the origin to itself, albeit with a change in the parametrization.

4 Causal geometry In §3.2 we observed that Einn;1 is naturally equipped with a conformal structure. This c n;1 . As in the Riemannian case in §3.1, a global structure lifts to the double cover Ein c n;1 is the pullback by a global secrepresentative of the conformal structure on Ein n;1 nC1;2 c tion W Ein ! R of the ambient quadratic form of RnC1;2 . The section n;1 nC1;2 c W Ein ! R taking values in the set where 2 2 2 D vnC2 C vnC3 D1 v12 C C vnC1

c n;1 Š S n S 1 as in §2.3; it is now apparent that Ein c n;1 exhibits a homeomorphism Ein 2 n 1 is conformally equivalent to S S endowed with the Lorentz metric ds0 d 2 , where ds02 and d 2 are the usual round metrics on the spheres S n and S 1 of radius one. In the following, elements of S n S 1 are denoted by .'; /. In these coordinates, we distinguish the timelike vector field D @ tangent to the fibers fg S 1 . 4.1 Time orientation. First consider Minkowski space En;1 with underlying vector space Rn;1 equipped with the inner product: hu; vi WD u1 v1 C C un vn unC1 vnC1 : A vector u in Rn;1 is causal if u2nC1 u21 C C u2n . It is future-oriented (respectively past-oriented) if the coordinate unC1 is positive (respectively negative); equivalently,


195

u is future-oriented if its inner product with 2 3 0 6 :: 7 6 7 0 D 6 : 7 405 1 is negative. The key point is that the choice of the coordinate unC1 – equivalently, of an everywhere timelike vector field like 0 – defines a decomposition of every affine lightcone Laff .p/ in three parts: • fpg; • the future lightcone Laff C .p/ of elements p C v where v is a future-oriented null vector; • the past lightcone Laff .p/ of elements p Cv where v is a past-oriented null vector. The above choice is equivalent to a continuous choice of one of the connected components of the set of timelike vectors based at each x 2 En;1 ; timelike vectors in these components are designated future-oriented. In other words, 0 defines a time orientation on En;1 . c n;1 , replace 0 by the vector field on Ein c n;1 . Then a To import this notion to Ein n;1 c causal tangent vector v to Ein is future-oriented (respectively past-oriented) if the inner product hv; i is negative (respectively positive). We already observed in §2.3 that the antipodal map is .'; / 7! .'; / on S n S 1 ; in particular, it preserves the timelike vector field , which then descends to a well-defined vector field on Einn;1 , so that Einn;1 is time oriented, for all integers n. Remark 4.1.1. The Einstein universe does not have a preferred Lorentz metric in its conformal class. The definition above is nonetheless valid since it involves only signs of inner products and hence is independent of the choice of metric in the conformal class. The group O.n C 1; 2/ has four connected components. More precisely, let SO.n C 1; 2/ be the subgroup of O.n C 1; 2/ formed by elements with determic n;1 . Let nant 1; these are the orientation-preserving conformal transformations of Ein C O .n C 1; 2/ be the subgroup comprising the elements preserving the time orientation c n;1 . The identity component of O.n C 1; 2/ is the intersection of Ein SOC .n C 1; 2/ D SO.n C 1; 2/ \ OC .n C 1; 2/: Moreover, SO.n C 1; 2/ and OC .n C 1; 2/ each have two connected components. The center of O.n C 1; 2/ has order two and is generated by the antipodal map, which belongs to SO.n C 1; 2/ if and only if n is odd. Hence the center of SO.n C 1; 2/ is trivial if n is even – in particular, when n D 2. On the other hand, the antipodal map always preserves the time orientation.

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The antipodal map is the only element of O.n C 1; 2/ acting trivially on Einn;1 . Hence the group of conformal transformations of Einn;1 is PO.n C 1; 2/, the quotient of O.nC1; 2/ by its center. When n is even, PO.nC1; 2/ is isomorphic to SO.nC1; 2/. 4.2 Future and past. A C 1 -immersion c

Œ0; 1 ! E1;n is a causal curve (respectively a timelike curve) if the tangent vectors c 0 .t / are all causal (respectively timelike). This notion extends to any conformally Lorentzian space – c n;1 , or Ein f n;1 . Furthermore, a causal curve c is futurein particular, to Einn;1 , Ein oriented (respectively past-oriented) if all the tangent vectors c 0 .t / are future-oriented (respectively past-oriented). c n;1 ; or Ein f n;1 . The future IC .A/ (respectively Let A be a subset of En;1 , Einn;1 , Ein the past I .A/) of A is the set comprising endpoints c.1/ of future-oriented (respectively past-oriented) timelike curves with starting point c.0/ in A. The causal future JC .A/ (respectively the causal past J .A/) of A is the set comprising endpoints c.1/ of futureoriented (respectively past-oriented) causal curves with starting point c.0/ in A. Two points p, p 0 are causally related if one belongs to the causal future of the other: p 0 2 J˙ .p/. The notion of future and past in En;1 is quite easy to understand: p 0 belongs to the future IC .p/ of p if and only if p 0 p is a future-oriented timelike element of Rn;1 . Thanks to the conformal model, these notions are also quite easy to understand c n;1 ; or Ein f n;1 : let dn be the spherical distance on the homogeneous in Einn;1 , Ein n f n;1 is conformally Riemannian sphere S of radius 1. The universal covering Ein n isometric to the Riemannian product S R where the real line R is endowed with the negative quadratic form d 2 . Hence, the image of any causal, C 1 , immersed curve f n;1 S n R is the graph of a map f W I ! S n where I is an interval in R and in Ein where f is 1-Lipschitz – that is, for all , 0 in R: dn .f . /; f . 0 // j 0 j: Moreover, the causal curve is timelike if and only if the map f is contracting – that is, satisfies dn .f . /; f . 0 // < j 0 j: f n;1 S n R is: It follows that the future of an element .'0 ; 0 / of Ein IC .'0 ; 0 / D f.'; / j 0 > dn .'; '0 /g f n;1 is the closure of the future IC .p/: and the causal future JC .p/ of an element p of Ein JC .'0 ; 0 / D f.'; / j 0 dn .'; '0 /g: c n;1 is the As a corollary, the future IC .A/ of a nonempty subset A of Einn;1 or Ein f n;1 , but entire spacetime. In other words, the notion of past or future is relevant in Ein n;1 n;1 c not in Ein or Ein .


197

c n;1 that will There is, however, a relative notion of past and future still relevant in Ein 0 be useful later when considering crooked planes and surfaces: let p, O pO be two elements c n;1 such that pO 0 ¤ ˙p. of Ein O First observe that the intersection MinC .p/ O \ MinC .pO 0 / C is never empty. Let p1 be any element of this intersection, so Min .pO1 / contains pO c n;1 induces a time orientation on such a Minkowski and pO 0 . The time orientation on Ein C patch Min .pO1 /. Fact 4.2.1. The points pO 0 and pO are causally related in MinC .pO1 / if and only if, for any lifts p, p 0 of p, O pO 0 , respectively, to RnC1;2 , the inner product hp; p 0 i is positive. Hence, if pO and pO 0 are causally related in some Minkowski patch, then they are causally related in any Minkowski patch containing both of them. Therefore, (slightly c n;1 are abusing language) we use the following convention: two elements p, O pO 0 of Ein 0 nC1;2 causally related if the inner product hp; p i in R is positive for any lifts p, p 0 . 4.3 Geometry of the universal covering. The geometrical understanding of the embedding of Minkowski space in the Einstein universe can be a challenge. In particular, the closure in Einn;1 of a subset of a Minkowski patch may be not obvious, as we will see for crooked planes. This difficulty arises from the nontrivial topology of Einn;1 . f n;1 is easy to visualize; On the other hand, the topology of the universal covering Ein indeed, the map S f n;1 S n R ! RnC1 n f0g Ein S W .'; / 7! exp. /'

f n;1 can be considered as a subset of RnC1 – one that is an embedding. Therefore, Ein is particularly easy to visualize when n D 2. Observe that the map S is O.n C 1/f n;1 and RnC1 . equivariant for the natural actions on Ein The antipodal map .'; / 7! .'; / lifts to the automorphism ˛ of f n;1 S n R; Ein defined by

˛

.'; / 7! .'; C /: f n;1 RnC1 n f0g this lifting ˛ is expressed by x ! x, where In the coordinates Ein

D exp. /. Since null geodesics in Einn;1 are photons, the images by S of null geodesics of n;1 f are curves in RnC1 n f0g characterized by the following properties: Ein • They are contained in 2-dimensional linear subspaces. • Each is a logarithmic spiral in the 2-plane containing it.

198


f 2;1 (that is, the union of Hence, for n D 2, the lightcone of an element p of Ein the null geodesics containing p) is a singular surface of revolution in R3 obtained by rotating a spiral contained in a vertical 2-plane around an axis of the plane. In f n;1 RnC1 n f0g, every null geodesic containing x particular, for every x in Ein contains ˛.x/ D x. The image ˛.x/ D x is uniquely characterized by the following properties, so that it can be called the first future-conjugate point to x: • It belongs to the causal future JC .x/. • For any y 2 JC .x/ such that y belongs to all null geodesics containing x, we have ˛.x/ 2 J .y/: All these considerations allow us to visualize how Minkowski patches embed in f n;1 and pO be its projection to Ein c n;1 . The RnC1 n f0g (see Figure 3): let pQ 2 Ein c n;1 of IC .p/ O is the projection in Ein Q n JC .˛.p//, Q which can Minkowski patch MinC .p/ C 2 n;1 c Q \ I .˛ .p//. Q The projection in Ein of also be defined as I .p/ f n;1 n .JC .p/ Q [ J .p// Q Ein O which is the set of points non-causally related to p. O is the Minkowski patch Min .p/,

O ˛ 2 .p/

MinC .p/ O pO

˛.p/ O

f Figure 3. A Minkowski patch in E in1;1 .

4.4 Improper points of Minkowski patches. We previously defined the improper point p1 associated to a Minkowski patch in Einn;1 : it is the unique point such that the Minkowski patch is Min.p1 /.


199

c n;1 , to every Minkowski patch are attached two improper In the double-covering Ein points: sp

• the spatial improper point, the unique element p1 such that the given Minkowski sp patch is Min .p1 /; ti such that the given Minkowski • the timelike improper point, the unique element p1 C ti patch is Min .p1 /. ti c n;1 . Let Let MinC .p1 / D Min .p1 / be a Minkowski patch in Ein sp

ti R! MinC .p1 / En;1

x the closure in Ein c n;1 of . be a geodesic. Denote by the image of , and by • If is spacelike, then

• If is timelike, then

sp x D [ fp1 g:

ti x D [ fp1 g:

ti x is a photon avoiding p1 and p1 • If is lightlike, then . sp

5 4-dimensional real symplectic vector spaces In spatial dimension n D 2, Einstein space Ein2;1 admits an alternate description as the Lagrangian Grassmannian, the manifold Lag.V / of Lagrangian 2-planes in a real symplectic vector space V of dimension 4. There results a kind of duality between the conformal Lorentzian geometry of Ein2;1 and the symplectic geometry of R4 . Photons correspond to linear pencils of Lagrangian 2-planes (that is, families of Lagrangian subspaces passing through a given line). The corresponding local isomorphism Sp.4; R/ ! O.3; 2/ manifests the isomorphism of root systems of type B2 (the odd-dimensional orthogonal Lie algebras) and C2 (the symplectic Lie algebras) of rank 2. We present this correspondence below. 5.1 The inner product on the second exterior power. Begin with a 4-dimensional vector space V over R and choose a fixed generator vol 2 ƒ4 .V /: The group of automorphisms of .V; vol/ is the special linear group SL.V /.

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The second exterior power ƒ2 .V / has dimension 6. The action of SL.V / on V induces an action on ƒ2 .V / which preserves the bilinear form B

!R ƒ2 .V / ƒ2 .V / defined by: ˛1 ^ ˛2 D B.˛1 ; ˛2 / vol: This bilinear form satisfies the following properties: • B is symmetric; • B is nondegenerate; • B is split – that is, of type .3; 3/. (That B is split follows from the fact that any orientation-reversing linear automorphism of V maps B to its negative.) The resulting homomorphism SL.4; R/ ! SO.3; 3/

(4)

is a local isomorphism of Lie groups, with kernel f˙ I4 g and image the identity component of SO.3; 3/. Consider a symplectic form ! on V – that is, a skew-symmetric nondegenerate bilinear form on V . Since B is nondegenerate, ! defines a dual exterior bivector ! 2 ƒ2 .V / by !.v1 ; v2 / D B.v1 ^ v2 ; ! /: We will assume that

! ^ ! D 2 vol:

(5)

Thus B.! ; ! / D 2 < 0, so that its symplectic complement W0 WD .! /? ƒ2 .V / is an inner product space of type .3; 2/. Now the local isomorphism (4) restricts to a local isomorphism Sp.4; R/ ! SO.3; 2/ (6) with kernel f˙ I4 g and image the identity component of SO.3; 2/. 5.2 Lagrangian subspaces and the Einstein universe. Let V , !, B, ! ; and W0 be as above. The projectivization of the null cone in W0 is equivalent to Ein2;1 . Points in Ein2;1 correspond to Lagrangian planes in V – that is, 2-dimensional linear subspaces P V such that the restriction !jP 0. Explicitly, if v1 ; v2 constitute a basis for P , then the line generated by the bivector w D v1 ^ v2 2 ƒ2 .V /


201

is independent of the choice of basis for P . Furthermore, w is null with respect to B and orthogonal to ! , so w generates a null line in W0 Š R3;2 , and hence defines a point in Ein2;1 . For the reverse correspondence, first note that a point of Ein2;1 Š P .N.W0 // is represented by a vector a 2 W0 such that a â D 0. Elements a 2 ƒ2 V with a â D 0 are exactly the decomposable ones – that is, those that can be written a D v1 ^ v2 for v1 ; v2 2 V . Then the condition a ? ! is equivalent by construction to !.v1 ; v2 / D 0, so a represents a Lagrangian plane, spanfv1 ; v2 g, in V . Thus Lagrangian 2-planes in V correspond to isotropic lines in W0 Š R3;2 . For a point q 2 Ein2;1 , denote by Lq the corresponding Lagrangian plane in V . 5.2.1 Complete flags. A photon in Ein2;1 corresponds to a line ` in V , where \ ` D Lp : p2

A pointed photon .p; /, as defined in §2.5, corresponds to a pair of linear subspaces ` Lp

(7)

where ` V is the line corresponding to and where Lp V is the Lagrangian plane corresponding to p. Recall that the incidence relation p 2 extends to p 2 L.p/; corresponding to the complete linear flag 0 `p P P? .`p /? W0 where P is the null plane projectivizing to . The linear inclusion (7) extends to a complete linear flag 0 ` Lp .` /? V where now .` /? denotes the symplectic orthogonal of ` . Clearly the lightcone L.p/ corresponds to the linear hyperplane .` /? V . 5.2.2 Pairs of Lagrangian planes. Distinct Lagrangian subspaces L1 ; L2 may intersect in either a line or in 0. If L1 \ L2 ¤ 0, the corresponding points p1 ; p2 2 Ein2;1 are incident. Otherwise V D L1 ˚ L2 and the linear involution of V , D IL1 ˚ IL2 ; is anti-symplectic: !..v1 /; .v2 // D !.v1 ; v2 /: The corresponding involution of Ein2;1 fixes the two points p1 ; p2 and the spacelike circle L.p1 / \ L.p2 /. It induces a time-reversing involution of Ein2;1 .

202


5.3 Symplectic planes. Let P V be a symplectic plane, that is, one for which the restriction !jP is nonzero (and hence nondegenerate). Its symplectic complement P ? is also a symplectic plane, and V D P ˚ P? is a symplectic direct sum decomposition. Choose a basis fu1 ; u2 g for P . We may assume that !.u1 ; u2 / D 1. Then B.u1 ^ u2 ; ! / D 1 and

P WD 2u1 ^ u2 C !

lies in .! /? since B.! ; ! / D 2. Furthermore B. P ; P / D B.2u1 ^ u2 ; 2u1 ^ u2 / C 2 B.2u1 ^ u2 ; ! / C B.! ; ! / D0C42 D 2: whence P is a positive vector in W0 Š R3;2 . In particular P . P? \ N.W0 // is an Einstein hypersphere. The two symplectic involutions leaving P (and necessarily also P ? ) invariant ˙ I jP ˚ I jP ? induce maps fixing P , and acting by 1 on . P /? . The corresponding eigenspace decomposition is R1;0 ˚ R2;2 and the corresponding conformal involution in Ein2;1 fixes an Einstein hypersphere. 5.4 Positive complex structures and the Siegel space. Not every involution of Ein2;1 arises from a linear involution of V . Particularly important are those which arise from compatible complex structures, defined as follows. A complex structure on V is an J

V such that J ı J D I. The pair .V; J/ then inherits the structure automorphism V ! of a complex vector space for which V is the underlying real vector space. The complex structure J is compatible with the symplectic vector space .V; !/ when !.J x; J y/ D !.x; y/: (In the language of complex differential geometry, the exterior 2-form ! has Hodge type .1; 1/ on the complex vector space .V; J/.) Moreover V V ! C; .v; w/ 7! !.v; J w/ C i !.v; w/; defines a Hermitian form on .V; J/.


203

A compatible complex structure J on .V; !/ is positive if !.v; J v/ > 0 whenever v ¤ 0. Equivalently, the symmetric bilinear form defined by v w WD !.v; J w/ is positive definite. This is in turn equivalent to the above Hermitian form being positive definite. The positive compatible complex structures on V are parametrized by the symmetric space of Sp.4; R/. A convenient model is the Siegel upper-half space S2 , which can be realized as the domain of 2 2 complex symmetric matrices with positive definite imaginary part (Siegel [27]). A matrix M 2 Sp.4; R/ acts on a complex structure J by J 7! M J M 1 and the stabilizer of any positive compatible J is conjugate to U.2/, the group of unitary transformations of C 2 . Let the symplectic structure ! be defined by the 2 2-block matrix 02 I2 J WD : I2 02 This matrix also defines a complex structure. Write M as a block matrix with A B M D C D where the blocks A; B; C; D are 2 2 real matrices. Because M 2 Sp.4; R/, M JM D J:

(8)

The condition that M preserves the complex structure J means that M commutes with J, which together with (8), means that M M D I4 ; that is, M 2 O.4/. Thus the stabilizer of the pair .!; J/ is Sp.4; R/ \ O.4/, which identifies with the unitary group U.2/ as follows. If M commutes with J, then its block entries satisfy B D C; Relabelling X D A and Y D C , then M D

X Y

D D A:

Y X

corresponds to a complex matrix Z D X C iY . This matrix is symplectic if and only if Z is unitary, x Z D I2 : Z

204


5.5 The contact projective structure on photons. The points of a photon correspond to Lagrangian planes in V intersecting in a common line. Therefore, photons correspond to linear 1-dimensional subspaces in V , and the photon space Pho2;1 identifies with the projective space P .V /. This space has a natural contact geometry defined below. Recall that a contact structure on a manifold M 2nC1 is a vector subbundle E TM of codimension one that is maximally non-integrable: E is locally the kernel of a nonsingular 1-form ˛ such that ˛^.d˛/n is nondegenerate at every point. This condition is independent of the 1-form ˛ defining E, and is equivalent to the condition that any two points in the same path-component can be joined by a smooth curve with velocity field in E. The 1-form ˛ is called a contact 1-form defining E. For more details on contact geometry, see [23], [16], [28]. The restriction of d˛ to E is a nondegenerate exterior 2-form, making E into a symplectic vector bundle. Such a vector bundle always admits a compatible complex structure JE W E ! E (an automorphism such that JE ı JE D I), which gives E the structure of a Hermitian vector bundle. The contact structure we define on photon space P .R4 / Š Pho2;1 will have such Hermitian structures and contact 1-forms arising from compatible complex structures on the symplectic vector space R4 . 5.5.1 Construction of the contact structure. Let v 2 V be nonzero, and denote the corresponding line by Œv 2 P .V /. The tangent space TŒv P .V / naturally identifies with Hom.Œv; V =Œv/ (Œv V denotes the 1-dimensional subspace of V , as well). If V1 V is a hyperplane complementary to Œv, then an affine patch for P .V / containing Œv is given by AV1

Hom.Œv; V1 / ! P .V /; 7! Œv C .v/: That is, AV1 ./ is the graph of the linear map in V D Œv ˚ V1 . This affine patch defines an isomorphism TŒv P .V / ! Hom.Œv; V1 / Š Hom.Œv; V =Œv/ that is independent of the choice of V1 . Now, since ! is skew-symmetric, the symplectic product with v defines a linear functional ˛v

V =Œv ! R; u 7! !.u; v/: The hyperplane field Œv 7! f' W ˛v ı ' D 0g is a well-defined contact plane field on P .V /. It possesses a unique transverse orientation; we denote a contact 1-form for this hyperplane field by ˛.


205

5.5.2 The contact structure and polarity. The contact structure and the projective geometry of P .V / interact with each other in an interesting way. If p 2 P .V /, then the contact structure at p is a hyperplane Ep Tp P .V /. There is a unique projective hyperplane H D H.p/ tangent to Ep at p. Conversely, suppose H P .V / is a projective hyperplane. The contact plane field is transverse to H everywhere but one point, and that point p is the unique point for which H D H.p/. This correspondence results from the correspondence between a line ` V and its symplectic orthogonal `? V . The above correspondence is an instance of a polarity in projective geometry. A polarity of a projective space P .V / is a projective isomorphism between P .V / and its dual P .V / WD P .V /, arising from a nondegenerate bilinear form on V , which can be either symmetric or skew-symmetric. Another correspondence is between the set of photons through a given point p 2 Ein2;1 and the set of 1-dimensional linear subspaces of the Lagrangian plane Lp V . The latter set projects to a projective line in P .V / tangent to the contact plane field, a contact projective line. All contact projective lines arise from points in Ein2;1 in this way. 5.6 The Maslov cycle. Given a 2n-dimensional symplectic vector space V over R, the set Lag.V / of Lagrangian subspaces of V is a compact homogeneous space. It identifies with U.n/= O.n/, given a choice of a positive compatible complex structure on V Š R2n . The fundamental group 1 Lag.V / Š Z: An explicit isomorphism is given by the Maslov index, which associates to a loop in Lag.V / an integer. (See McDuff–Salamon [23], §2.4, or Siegel [27], for a general discussion.) Let W 2 Lag.V / be a Lagrangian subspace. The Maslov cycle MaslovW .V / associated to W is the subset of Lag.V / consisting of W 0 such that W \ W 0 ¤ 0: Although it is not a submanifold, MaslovW .V / carries a natural co-orientation (orientation of its conormal bundle) and defines a cycle whose homology class generates HN 1 .Lag.V /; Z/ where n.n C 1/ D dim Lag.V / : 2 The Maslov index of a loop is the oriented intersection number of with the Maslov cycle (after is homotoped to be transverse to MaslovW .V /). If p 2 Ein2;1 corresponds to a Lagrangian subspace W V , then the Maslov cycle MaslovW .V / corresponds to the lightcone L.p/. (We thank A. Wienhard for this observation.) N D

5.7 Summary. We now have a dictionary between the symplectic geometry of R4! and the orthogonal geometry of R3;2 :

206


Symplectic R4! and contact P .V /

Pseudo-Riemannian R3;2 and Ein2;1

Lagrangian planes L R4!

Points p 2 Ein2;1

Contact projective lines in P .V /

Points p 2 Ein2;1

Lines ` R4!

Photons ?

Hyperplanes `

R4!

Photons

Symplectic planes (splittings) in

R4!

Einstein hyperspheres

Linear symplectic automorphisms

Time-preserving conformal automorphisms

Linear anti-symplectic automorphisms

Time-reversing conformal automorphisms

?

Flags ` L `

in

R4!

Incident pairs p 2 L.p/

Positive compatible complex structures

Free involutions of Ein2;1

Lagrangian splittings V D L1 ˚ L2

Nonincident pairs of points

Lagrangian splittings V D L1 ˚ L2

Spacelike circles

6 Lie theory of Pho2 ;1 and Ein2 ;1 This section treats the structure of the Lie algebra sp.4; R/ and the isomorphism with o.3; 2/. We relate differential-geometric properties of the homogeneous spaces Ein2;1 and Pho2;1 with the Lie algebra representations corresponding to the isotropy. This section develops the structure theory – Cartan subalgebras, roots, parabolic subalgebras – and relates these algebraic notions to the synthetic geometry of the three parabolic homogenous spaces Ein2;1 , Pho2;1 and Flag2;1 . Finally, we discuss the geometric significance of the Weyl group of Sp.4; R/ and SO.2; 3/. 6.1 Structure theory. Let V Š R4 , equipped with the symplectic form !, as above. We consider a symplectic basis e1 ; e2 ; e3 ; e4 in which ! is 2 3 0 1 0 0 61 0 0 0 7 7 JD6 40 0 0 15 0 0 1 0 The Lie algebra g D sp.4; R/ consists of all 4 4 real matrices M satisfying M J C J M D 0; that is,

2

a 6 a21 M D6 4r22 r21

where a; b; aij ; bij ; rij 2 R.

a12 a r12 r11

r11 r21 b b21

3 r12 r22 7 7 b12 5 b

(9)


207

6.1.1 Cartan subalgebras. A Cartan subalgebra a of sp.4; R/ is the subalgebra stabilizing the four coordinate lines Rei for i D 1; 2; 3; 4, and comprises the diagonal matrices 2 3 a 0 0 0 60 a 0 0 7 7 H.a; b/ WD 6 40 0 b 0 5 0 0 0 b for a; b 2 R. The calculation 2 0 6 .2a/a21 ŒH; M D 6 4 .a b/r22 .a C b/r21

.2a/a12 0 .a b/r12 .a C b/r11

.a b/r11 .a b/r21 0 .2b/b21

3 .a C b/r12 .a C b/r22 7 7 .2b/b12 5 0

implies that the eight linear functionals assigning to H.a; b/ the values 2a;

2a; 2b; 2b; a b; a C b; a b; a C b

define the root system

WD f.2; 0/; .2; 0/; .0; 2/; .0; 2/; .1; 1/; .1; 1/; .1; 1/; .1; 1/g a pictured below.

Figure 4. Root diagram of sp.4; R/.

6.1.2 Positive and negative roots. A vector v0 2 a such that .v0 / ¤ 0 for all roots

2 partitions into positive roots C and negative roots depending on whether

.v0 / > 0 or .v0 / < 0 respectively. For example, 1 v0 D 2 partitions into

C D f .2; 0/; .1; 1/; .0; 2/; .1; 1/ g;

D f .2; 0/; .1; 1/; .0; 2/; .1; 1/ g:

208


The positive roots ˛ WD .2; 0/;

ˇ WD .1; 1/

form a pair of simple positive roots in the sense that every 2 C is a positive integral linear combination of ˛ and ˇ. Explicitly:

C D f˛; ˛ C ˇ; ˛ C 2ˇ; ˇg: 6.1.3 Root space decomposition. For any root 2 , define the root space g WD fX 2 g j ŒH; X D .H /X g: In g D sp.4; R/, each root space is 1-dimensional, and the elements X 2 g are called root elements. The Lie algebra decomposes as a direct sum of vector spaces: M g : gDa˚ 2

For more details, see Samelson [26]. 6.2 Symplectic splittings. The basis vectors e1 ; e2 span a symplectic plane P V and e3 ; e4 span its symplectic complement P ? V . These planes define a symplectic direct sum decomposition V D P ˚ P ?: The subalgebra hP sp.4; R/ preserving P also preserves P ? and consists of matrices of the form (9) that are block-diagonal: 2 3 0 a a12 0 6a21 a 0 0 7 6 7: 4 0 0 b b12 5 0 0 b21 b Thus hP Š sp.2; R/ ˚ sp.2; R/ Š sl.2; R/ ˚ sl.2; R/: The Cartan subalgebra a of sp.4; R/ is also a Cartan subalgebra of hP , but only the four long roots

0 D f.˙2; 0/; .0; ˙2/g D f˙˛; ˙.˛ C 2ˇ/g are roots of hP . In particular hP decomposes as M g : hP D a ˚ 2 0


209

6.3 The orthogonal representation of sp.4; R/. Let e1 ; : : : ; e4 be a symplectic basis for V as above and vol WD e1 ^ e2 ^ e3 ^ e4 a volume element for V . A convenient basis for ƒ2 V is: f1 WD e1 ^ e3 f2 WD e2 ^ e3 1 f3 WD p .e1 ^ e2 e3 ^ e4 / 2 f4 WD e4 ^ e1 f5 WD e2 ^ e4 for which the matrix

2

0 60 6 60 6 40 1

0 0 0 1 0

0 0 1 0 0

0 1 0 0 0

(10)

3 1 07 7 07 7 05 0

defines the bilinear form B associated to this volume element. The matrix M defined in (9) above maps to 2 3 aCb a12 r12 b12 0 6 a21 a C b r22 0 b12 7 6 7 6 z r11 0 r22 r12 7 M D 6 r21 7 2 so.3; 2/: 4 b21 0 r11 a b a12 5 0 b21 r21 a21 a b

(11)

For a fixed symplectic plane P V , such as the one spanned by e1 andPe2 , denote by P ^P ? the subspace of ƒ2 V of elements that can be written in the form i vi ^wi , where vi 2 P and wi 2 P ? for all i . The restriction of the bilinear form B to this subspace, which has basis ff1 ; f2 ; f4 ; f5 g, is type .2; 2/. Its stabilizer is the image hQ P of hP in o.3; 2/. Note that this image is isomorphic to o.2; 2/ Š sl.2; R/ ˚ sl.2; R/: 6.4 Parabolic subalgebras. The homogeneous spaces Ein2;1 , Pho2;1 and Flag2;1 identify with quotients G=P of G D Sp.4; R/ where P G is a proper parabolic subgroup. When G is algebraic, then any parabolic subgroup P of G is algebraic, and the quotient G=P is a compact projective variety. See Chapter 7 of [18] for more details. As usual, working with Lie algebras is more convenient. We denote the corresponding parabolic subalgebras by p, and they are indexed by subsets S … of the set … WD f˛; ˇg of simple negative roots, as follows.

210


The Borel subalgebra or minimal parabolic subalgebra corresponds to S D ; and is defined as M g : p; WD pS D a ˚ 2 C

In general, let Sz be the set of positive integral linear combinations of elements of S . The parabolic subalgebra determined by S is M pS WD p; ˚ g : 2Sz

6.4.1 The Borel subalgebra and Flag2 ;1 . Let p; be the Borel subalgebra defined above. The corresponding Lie subgroup P; is the stabilizer of a unique pointed photon, equivalently, an isotropic flag, in Flag2;1 ; thus Flag2;1 identifies with the homogeneous space G=P; . The subalgebra X u; WD g sp.4; R/ 2 C

is the Lie algebra of the unipotent radical of P; and is 3-step nilpotent. A realization of the corresponding group is the group generated by the translations of E2;1 and a unipotent one-parameter subgroup of SO.2; 1/. 6.4.2 The parabolic subgroup corresponding to Pho2 ;1 . Now let S D f˛g; the corresponding parabolic subalgebra p˛ is the stabilizer subalgebra of a line in V , or, equivalently, of a point in P .V /. In o.3; 2/ this parabolic is the stabilizer of a null plane in R3;2 , or, equivalently, of a photon in Ein2;1 . 6.4.3 The parabolic subgroup corresponding to Ein2 ;1 . Now let S D fˇg; the corresponding parabolic subalgebra pˇ is the stabilizer subalgebra of a Lagrangian plane in V , or, equivalently, a contact projective line in P .V /. In o.3; 2/, this parabolic is the stabilizer of a null line in R3;2 , or, equivalently, of a point in Ein2;1 . 6.5 Weyl groups. The Weyl group W of Sp.4; R/ is isomorphic to a dihedral group of order 8 (see Figure 4). It acts by permutations on elements of the quadruples in P .V / corresponding to a basis of V . Let A be the connected subgroup of Sp.4; R/, with Lie algebra a. In the symplectic basis e1 ; : : : ; e4 , it consists of matrices of the form 3 2 a1 7 6 a11 7 ; a1 ; a2 > 0: 6 5 4 a2 1 a2


211

The semigroup AC A with a2 > a1 > 1 corresponds to an open Weyl chamber in a. For i D 1; 2; 3; 4, let Hi be the image in P .V / of the hyperplane spanned by ej for j ¤ i . The point Œe3 2 P .V / is an attracting fixed point for all sequences in AC , and Œe4 is a repelling fixed point: Any unbounded an 2 AC converges uniformly on compact subsets of P .V /nH3 to the constant map Œe3 , while an1 converges to Œe4 uniformly on compact subsets of P .V /nH2 . On H3 n.H3 \ H1 /, an unbounded sequence fan g converges to Œe1 , while on H4 n.H4 \ H2 /, the inverses an1 converge to Œe2 . We will call the point Œe1 a codimension-one attracting fixed point for sequences in AC and Œe2 a codimension-one repelling fixed point. Every Weyl chamber has associated to it a dynamical quadruple like .Œe3 ; Œe4 ; Œe1 ; Œe2 /, consisting of an attracting fixed point, a repelling fixed point, a codimension-one attracting fixed point, and a codimension-one repelling fixed point. Conversely, given a symplectic basis v1 ; : : : ; v4 , the intersection of the stabilizers in Sp.4; R/ of the lines Rvi is a Cartan subgroup A. The elements a 2 A such that .Œv1 ; : : : ; Œv4 / is a dynamical quadruple for the sequence an form a semigroup AC that is an open Weyl chamber in A. The Weyl group acts as a group of permutations of such a quadruple. These permutations must preserve a stem configuration as in Figure 1, where now two points are connected by an edge if the corresponding lines in V are in a common Lagrangian plane, or, equivalently, the two points of P .V / span a line tangent to the contact structure. The permissible permutations are those preserving the partition fv1 ; v2 gjfv3 ; v4 g. In O.3; 2/, the Weyl group consists of permutations of four points p1 ; : : : ; p4 of Ein2;1 in a stem configuration that preserve the configuration. A Weyl chamber again corresponds to a dynamical quadruple .p1 ; : : : ; p4 / of fixed points, where now sequences an 2 AC converge to the constant map p1 on the complement of L.p2 / and to p3 on L.p2 /n.L.p4 / \ L.p2 //; the inverse sequence converges to p2 on the complement of L.p1 / and to p4 on L.p1 /n.L.p1 / \ L.p3 //.

7 Three kinds of dynamics In this section, we present the ways sequences in Sp.4; R/ can diverge to infinity in terms of projective singular limits. In [13], Frances defines a trichotomy for sequences diverging to infinity in O.3; 2/: they have bounded, mixed, or balanced distortion. He introduces limit sets for such sequences and finds maximal domains of proper discontinuity for certain subgroups of O.3; 2/. We translate Frances’ trichotomy to Sp.4; R/, along with the associated limit sets and maximal domains of properness. 7.1 Projective singular limits. Let E be a finite-dimensional vector space, and let .gn /n2N be a sequence of elements of GL.E/. This sequence induces a sequence .gN n /n2N of projective transformations of P .E/. Let k k be an auxiliary Euclidean norm on E and let k k1 be the associated operator norm on the space of endomorphisms End.E/. The division of gn by its norm kgn k1 does not modify the projective

212


transformation gN n . Hence we can assume that gn belongs to the k k1 -unit sphere of End.E/. This sphere is compact, so .gn /n2N admits accumulation points. Up to a subsequence, we can assume that .gn /n2N converges to an element g1 of the k k1 -unit sphere. Let I be the image of g1 , and let L be the kernel of g1 . Let gN 1 W P .E/ n P .L/ ! P .I / P .E/ be the induced map. Proposition 7.1.1. For any compact K P .E/ n P .L/, the restriction of the sequence .gN n /.n2N/ on K converges uniformly to the restriction on K of gN 1 . Corollary 7.1.2. Let be a discrete subgroup of PGL.E/. Let be the open subset of P .E/ formed by points admitting a neighborhood U such that, for any sequence .gn / in with accumulation point g1 having image I and kernel L, U \ P .L/ D U \ P .I / D ;: Then acts properly discontinuously on . In fact, the condition U \ P .L/ D ; is sufficient to define (as is U \ P .I / D ;). To see this, note that if gn ! 1 with gn =kgn k1 ! g1 ; gn1 =kgn1 k1 then Hence

! g1 ;

g1 ı g1 D g1 ı g1 D 0: Im.g1 / Ker.g1 /

and

Im.g1 / Ker.g1 /:

7.2 Cartan’s decomposition G D KAK. When .gn /n2N is a sequence in a semisimple Lie group G GL.E/, a very convenient way to identify the accumulation points gN 1 is to use the KAK-decomposition in G: first select the norm k k on E preserved by the maximal compact subgroup K of G. Decompose every gn in the form kn an kn0 , where kn and kn0 belong to K, and an belongs to a fixed Cartan subgroup. We can furthermore require that an is the image by the exponential of an element of the closure 0 , respecof a Weyl chamber. Up to a subsequence, kn and kn0 admit limits k1 and k1 tively. Composition on the right or on the left by an element of K does not change the operator norm, so gn has k k1 -norm 1 if and only if an has k k1 -norm 1. Let a1 be an accumulation point of .an /n2N . Then 0 g1 D k1 a1 k1 : 0 1 The kernel of g1 is the image by .k1 / of the kernel of a1 , and the image of g1 is the image by k1 of the image of a1 . Hence, in order to find the singular projective limit gN 1 , the main task is to find the limit a1 , and this problem is particularly easy when the rank of G is small.


213

7.2.1 Sequences in Sp.4; R/. The image by the exponential map of a Weyl chamber in sp.4; R/ is the semigroup AC A of matrices (see §6.5): 2 6 A.˛1 ; ˛2 / D 6 4

exp.˛1 /

3 exp.˛1 /

exp.˛2 /

7 7; 5

˛2 > ˛1 > 0:

exp.˛2 /

The operator norm of A.˛1 ; ˛2 / is exp.˛2 /. We therefore can distinguish three kinds of dynamical behaviour for a sequence .A.˛1.n/ ; ˛2.n/ //n2N : • no distortion: when ˛1.n/ and ˛2.n/ remain bounded, • bounded distortion: when ˛1.n/ and ˛2.n/ are unbounded, but the difference ˛2.n/ ˛1.n/ is bounded, • unbounded distortion: when the sequences ˛1.n/ and ˛2.n/ ˛1.n/ are unbounded. This distinction extends to any sequence .gn /n2N in Sp.4; R/. Assume that the sequence .gn =kgn k1 /n2N converges to a limit g1 . Then: • For no distortion, the limit g1 is not singular – the sequence .gn /n2N converges in Sp.4; R/. • For bounded distortion, the kernel L and the image I are 2-dimensional. More precisely, they are Lagrangian subspaces of V . The singular projective transformation gN 1 is defined in the complement of a projective line and takes values in a projective line; these projective lines are both tangent everywhere to the contact structure. • For unbounded distortion, the singular projective transformation gN 1 is defined in the complement of a projective hyperplane and admits only one value. 7.2.2 Sequences in SOC .3; 2/. The Weyl chamber of SOC .3; 2/ is simply the image of the Weyl chamber of sp.4; R/ by the differential of the homomorphism Sp.4; R/ ! SOC .3; 2/ defined in §6.2. More precisely, the image of an element A.˛1 ; ˛2 / of AC is A0 .a1 ; a2 / where a1 D ˛1 C ˛2 ; a2 D ˛2 ˛1 and

2

6 6 A .a1 ; a2 / D 6 6 4 0

exp.a1 /

3 exp.a2 /

7 7 7; 7 5

1 exp.a2 /

exp.a1 /

a1 > a2 > 0:

214


The KAK-decomposition of Sp.4; R/ above corresponds under the homomorphism to a KAK-decomposition of SOC .3; 2/. Reasoning as in the previous section, we distinguish three cases: • no distortion: when a1.n/ and a2.n/ remain bounded, • balanced distortion: when a1.n/ and a2.n/ are unbounded, but the difference a1.n/ a2.n/ is bounded, • unbalanced distortion: when the sequences a1.n/ and a1.n/ a2.n/ are unbounded. The dynamical analysis is similar, but we restrict to the closed subset Ein2;1 of P .R3;2 /: • No distortion corresponds to sequences .gn /n2N converging in SOC .3; 2/. • For balanced distortion, the intersection between P .L/ and Ein2;1 , and the intersection between P .I / and Ein2;1 are both photons. Hence the restriction of the singular projective transformation gN 1 to Ein2;1 is defined in the complement of a photon and takes value in a photon. • For unbalanced distortion, the singular projective transformation gN 1 is defined in the complement of a lightcone and admits only one value. 7.3 Maximal domains of properness. Most of the time, applying directly Proposition 7.1.1 and Corollary 7.1.2 to a discrete subgroup of Sp.4; R/ or SOC .3; 2/ in order to find domains where the action of is proper is far from optimal. Through the homomorphism Sp.4; R/ ! SOC .3; 2/, a sequence in Sp.4; R/ can also be considered as a sequence in SO.3; 2/. Observe that our terminology is coherent: a sequence has no distortion in Sp.4; R/ if and only if it has no distortion in SOC .3; 2/. Observe also that since a1 D ˛1 C ˛2 ;

a2 D ˛2 ˛1 ;

a sequence with bounded distortion in Sp.4; R/ is unbalanced in SOC .3; 2/, and a sequence with balanced distortion in SOC .3; 2/ is unbounded in Sp.4; R/. In summary, we distinguish three different kinds of non-converging dynamics, covering all the possibilities: Definition 7.3.1. A sequence .gn /n2N of elements of Sp.4; R/ escaping from any compact subset in Sp.4; R/ has: • bounded distortion if the coefficient a2.n/ D ˛2.n/ ˛1.n/ is bounded, • balanced distortion if the coefficient ˛2.n/ D .a1.n/ C a2.n/ /=2 is bounded, • mixed distortion if all the coefficients a1.n/ , a2.n/ , ˛1.n/ , ˛2.n/ are unbounded. 7.3.1 Action on Ein2 ;1 . The dynamical analysis can be refined in the mixed distortion case. In [13], C. Frances proved:


215

Proposition 7.3.2. Let .gn /n2N be a sequence of elements of SOC .3; 2/ with mixed distortion, such that the sequence .gn =kgn k1 /n2N converges to an endomorphism g1 . Then there are photons and C in Ein2;1 such that, for any sequence .pn /n2N in Ein2;1 converging to an element of Ein2;1 n , all the accumulation points of .gn .pn //n2N belong to C . As a corollary (§4.1 in [13]): Corollary 7.3.3. Let be a discrete subgroup of SOC .3; 2/. Let 0 be the union of all open domains U in Ein2;1 such that, for any accumulation point g1 , with kernel L and image I , of a sequence .gn =kgn k1 /n2N with gn 2 SOC .3; 2/: • When .gn /n2N has balanced distortion, U is disjoint from the photons P .L/ \ Ein2;1 and P .I / \ Ein2;1 ; • When .gn /n2N has bounded distortion, U is disjoint from the lightcone P .L/ \ Ein2;1 ; • When .gn /n2N has mixed distortion, U is disjoint from the photons and C . Then the action of on 0 is properly discontinuous. Observe that the domain 0 is in general bigger than the domain appearing in Corollary 7.1.2. An interesting case is that in which 0 is obtained by removing only photons: Proposition 7.3.4 (Frances [13]). A discrete subgroup of SOC .3; 2/ does not contain sequences with bounded distortion if and only if its action on P .R3;2 /nEin2;1 is properly discontinuous. Frances calls such a subgroup a of the first kind. The following suggests that the domain 0 is optimal. Proposition 7.3.5 (Frances [13]). Let be a discrete, Zariski dense subgroup of SOC .3; 2/ which does not contain sequences with bounded distortion. Then 0 is the unique maximal open subset of Ein2;1 on which acts properly. 7.3.2 Action on P .V /. A similar analysis should be done when is considered a discrete subgroup of Sp.4; R/ instead of SOC .3; 2/. The following proposition is analogous to Proposition 7.3.2: Proposition 7.3.6. Let .gn /n2N be a sequence of elements of Sp.V / with mixed distortion, such that the sequence .gn =kgn k1 /n2N converges to an endomorphism g1 of V . Then there are contact projective lines and C in P .V / such that, for any sequence .pn /n2N 2 P .V / converging to an element of P .V / n , all the accumulation points of .gn .pn //n2N belong to C .

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We can then define a subset 1 of P .V / as the interior of the subset obtained after removing limit contact projective lines associated to subsequences of with bounded or mixed distortion, and removing projective hyperplanes associated to subsequences with balanced distortion. Then it is easy to prove that the action of on 1 is properly discontinuous. An interesting case is that in which we remove only projective lines, and no hypersurfaces – the case in which has no subsequence with balanced distortion. Frances calls such groups of the second kind. The following questions arise from comparison with Propositions 7.3.5 and 7.3.4: Question. Can groups of the second kind be defined as groups acting properly on some associated space? Question. Is 1 the unique maximal open subset of P .V / on which the action of is proper, at least if is Zariski dense? 7.3.3 Action on the flag manifold. Now consider the action of Sp.4; R/ on the flag manifold Flag2;1 . Let v; w 2 V be such that !.v; w/ D 0, so v and w span a Lagrangian plane. Let

1

Flag2;1 ! Pho2;1 ;

2

Flag2;1 ! Ein2;1 be the natural projections. Let gn be a sequence in Sp.4; R/ diverging to infinity with mixed distortion. We invite the reader to verify the following statements: • There are a flag q C 2 Flag2;1 and points Œv 2 P .V / and z 2 Ein2;1 such that, on the complement of 11 .Œv ? / [ 21 .L.z// the sequence gn converges uniformly to the constant map q C . • There are contact projective lines ˛ C , ˛ in P .V / and photons ˇ C , ˇ in Ein2;1 such that, on the complement of 11 .˛ / [ 21 .ˇ / all accumulation points of gn lie in 11 .˛ C / \ 21 .ˇ C /: This intersection is homeomorphic to a wedge of two circles.

8 Crooked surfaces Crooked planes were introduced by Drumm [8], [9], [10] to investigate discrete groups of Lorentzian transformations which act freely and properly on E2;1 . He used crooked


217

planes to construct fundamental polyhedra for such actions; they play a role analogous to equidistant surfaces bounding Dirichlet fundamental domains in Hadamard manifolds. This section discusses the conformal compactification of a crooked plane and its automorphisms. 8.1 Crooked planes in Minkowski space. For a detailed description of crooked planes, see Drumm-Goldman [10]. We quickly summarize the basic results here. Consider E2;1 with the Lorentz metric from the inner product I2 ˚ I1 on R2;1 . A crooked plane C is a surface in E2;1 that divides E2;1 into two cells, called crooked half-spaces. It is a piecewise linear surface composed of four 2-dimensional faces, joined along four rays, which all meet at a point p, called the vertex. The four rays have endpoint p, and form two lightlike geodesics, which we denote `1 and `2 . Two of the faces are null half-planes W1 and W2 , bounded by `1 and `2 respectively, which we call wings. The two remaining faces consist of the intersection between J˙ .p/ and the timelike plane P containing `1 and `2 ; their union is the stem of C . The timelike plane P is the orthogonal complement of a unique spacelike line P ? .p/ containing p, called the spine of C . To define a crooked plane, we first define the wings, stem, and spine. A lightlike geodesic ` D p C Rv lies in a unique null plane `? (§2.2). The ambient orientation of R2;1 distinguishes a component of `? n ` as follows. Let u 2 R2;1 be a timelike vector such that hu; vi < 0. Then each component of `? n ` defined by ˚

W C .`/ WD p C w 2 `? j det.u; v; w/ > 0 ; ˚

W .`/ WD p C w 2 `? j det.u; v; w/ < 0 is independent of the choices above. In particular, every orientation-preserving isometry f of E2;1 maps W C .`/ ! W C .f .`//; W .`/ ! W .f .`//; and every orientation-reversing isometry f maps W C .`/ ! W .f .`//; W .`/ ! W C .f .`//: Given two lightlike geodesics `1 ; `2 containing p, the stem is defined as S.`1 ; `2 / WD J˙ .p/ \ .p C spanf`1 p; `2 pg/: The spine is D p C .S.`1 ; `2 / p/? : Compare Drumm–Goldman [10].

218


The positively-oriented crooked plane with vertex p and stem S.`1 ; `2 / is the union W C .`1 / [ S.`1 ; `2 / [ W C .`2 /: Similarly, the negatively-oriented crooked plane with vertex p and stem S.`1 ; `2 / is W .`1 / [ S.`1 ; `2 / [ W .`2 /: Given an orientation on E2;1 , a positively-oriented crooked plane is determined by its vertex and its spine. Conversely, every point p and spacelike line containing p determines a unique positively- or negatively-oriented crooked plane. A crooked plane C is homeomorphic to R2 , and the complement E2;1 n C consists of two components, each homeomorphic to R3 . The components of the complement of a crooked plane are called open crooked half-spaces and their closures closed crooked half-spaces. The spine of C is the unique spacelike line contained in C . 8.2 An example. Here is an example of a crooked plane with vertex the origin and spine the x-axis: 2 3 2 3 0 1 p D 405 ; D R 405 : 0 0 The lightlike geodesics are 2

3 0 `1 D R 415 ; 1 the stem is

2 3 0 `2 D R 415 ; 1

82 3 9 < 0 = 4y 5 W y 2 z 2 0 : ; z

and the wings are 82 3 9 < x = W1 D 4 y 5 W x 0; y 2 R : ; y

and

82 3 9 < x = W2 D 4y 5 W x 0; y 2 R : : ; y

The identity component of Isom.E2;1 / acts transitively on the space of pairs of vertices and unit spacelike vectors, so it is transitive on positively-oriented and negativelyoriented crooked planes. An orientation-reversing isometry exchanges positively- and negatively-oriented crooked planes, so Isom.E2;1 / acts transitively on the set of all crooked planes.


219

8.3 Topology of a crooked surface. The closures of crooked planes in Minkowski patches are crooked surfaces. These were studied in Frances [12]. In this section we describe the topology of a crooked surface. Let C E2;1 be a crooked plane. Theorem 8.3.1. The closure Cx Ein2;1 is a topological submanifold homeomorphic c 2;1 is the oriented double to a Klein bottle. The lift of Cx to the double covering Ein covering of Cx and is homeomorphic to a torus. Proof. Since the isometry group of Minkowski space acts transitively on crooked planes, it suffices to consider the single crooked plane C defined in §8.2. Recall the stratification of Ein2;1 from §3.2. Write the nullcone N3;2 of R3;2 as 2 3 X 6Y 7 6 7 6Z 7 where X 2 C Y 2 Z 2 U V D 0: 6 7 4U 5 V The homogeneous coordinates of points in the stem S.C / satisfy X D 0;

Y 2 Z 2 0;

V ¤0

and thus the closure of the stem S.C / is defined by (homogeneous) inequalities X D 0;

Y 2 Z 2 0:

The two lightlike geodesics 2

3 0 `1 D R 415 ; 1

2 3 0 `2 D R 415 1

defining S.C / extend to photons 1 ; 2 with ideal points represented in homogeneous coordinates 2 3 2 3 0 0 617 617 6 7 6 7 7 ; p2 D 617 : 1 p1 D 6 6 7 6 7 405 405 0 0 The closures of the corresponding wings W1 ; W2 are described in homogeneous coor-

220


dinates by

9 82 3 X > ˆ > ˆ > ˆ6 7 > ˆ = ˆ > ˆ4 U 5 > ˆ > ˆ ; : V 82 3 9 X ˆ > ˆ > ˆ > ˆ > 7 ˆ > 4U 5 ˆ > ˆ > : ; V

The closure of each wing intersects the ideal lightcone L.p1 / (described by V D 0) in the photons: 82 3 82 3 9 9 0 0 ˆ > ˆ > ˆ ˆ > > ˆ ˆ > > ˆ ˆ 6 7 > 6 7 > Y Y ˆ ˆ > > 4 5 4 5 ˆ > ˆ > ˆ U ˆ U > > : : ; ; 0 0 Thus the crooked surface Cx decomposes into the following strata: • four points in a stem configuration: the vertex p0 , the improper point p1 , and the two ideal points p1 and p2 ; • eight line segments, the components of 1 n fp0 ; p1 g;

2 n fp0 ; p2 g;

1

n fp1 ; p1 g and

2

n fp1 ; p2 gI

• two null-half planes, the interiors of the wings W1 ; W2 ; • the two components of the interior of the stem S. Recall that the inversion in the unit sphere D I3 ˚ 01 10 fixes p1 and p2 , and interchanges p0 and p1 . Moreover interchanges i with i , i D 1; 2. Finally leaves invariant the interior of each Wi and interchanges the two components of the interior of S. The original crooked plane equals fp0 g [ 1 n fp1 g [ 2 n fp2 g [ int.W1 / [ int.W2 / [ int.S/ and is homeomorphic to R2 . The homeomorphism is depicted schematically in Figure 5. The interiors of W1 ; W2 , and S correspond to the four quadrants in R2 . The wing Wi is bounded by the two segments of i , whereas each component of S is bounded by one segment of 1 and one segment of 2 . These four segments correspond to the four coordinate rays in R2 .


221

Now we can see that C is a topological manifold: points in int.W1 /; int.W2 /; or int.S/ have coordinate neighborhoods in these faces. Interior points of the segments have two half-disc neighborhoods, one from a wing and one from the stem. The vertex p0 has four quarter-disc neighborhoods, one from each wing, and one from each component of the stem. (See Figure 5.) W1

1

1 S

p0

S

2

2 W2

Figure 5. Flattening a crooked plane around its vertex.

Coordinate charts for the improper point p1 and points in i nfp1 ; pi g are obtained by composing the above charts with the inversion . It remains to find coordinate charts near the ideal points p1 ; p2 . Consider first the case of p1 . The linear functionals on R3;2 defined by T D Y Z;

W DY CZ

are null since the defining quadratic form factors: X 2 C Y 2 Z 2 U V D X 2 C T W U V: Working in the affine patch defined by T ¤ 0 with inhomogeneous coordinates WD

X ; T

the nullcone is defined by whence

WD

Y ; T

! WD

W ; T

WD

U ; T

WD

V ; T

2 C ! D 0; ! D 2 C

and .; ; / 2 R3 is a coordinate chart for this patch on Ein2;1 . In these coordinates, p1 is the origin .0; 0; 0/, 1 is the line D D 0, and 1 is the line D D 0. The wing W2 misses this patch, but both S and W1 intersect it. In these coordinates S is defined by

and W1 is defined by

D 0;

!0

0;

! D 0:

222


Since on W1

D 2 0

this portion of W1 in this patch has two components: ; < 0; ; > 0; and the projection . ; / defines a coordinate chart for a neighborhood of p1 . (Compare Figure 6.) W1 1 1

S

p1

S 1

1

W1

Figure 6. Flattening a crooked surface around an ideal point p1 .

The case of p2 is completely analogous. It follows that Cx is a closed surface with cell decomposition with four 0-cells, eight 1-cells and four 2-cells. Therefore .Cx / D 4 8 C 4 D 0 and Cx is homeomorphic to either a torus or a Klein bottle. To see that Cx is nonorientable, consider a photon, for example 1 . Parallel translate the null geodesic 1 n fp1 g to a null geodesic ` lying on the wing W1 and disjoint S1 Cx which intersects 1 from 1 n fp1 g. Its closure `N D ` [ fp1 g is a photon on W transversely with intersection number 1. Thus the self-intersection number 1 1 D 1 so 1 Cx is an orientation-reversing loop. Thus Cx is nonorientable, and homeomorphic to a Klein bottle. Next we describe the stratification of a crooked surface in the double covering c 2;1 has both a spatial and a c 2;1 . Recall from §4.4 that a Minkowski patch in Ein Ein timelike improper point. Let C be a crooked plane of E2;1 , embedded in a Minkowski ti patch MinC .p1 /, so p1 D p1 , the timelike improper point of this patch. Denote by sp p1 the spatial improper point. c 2;1 decomposes into the following strata: The closure Cx of C in Ein ti • seven points: p0 ; p1 ; p1 ; p1˙ ; p2˙ ; sp


223

• twelve photon segments: i˙ ; connecting p0 to pi˙ ; ti ˛i˙ ; connecting p1 to pi˙ ;

ˇi˙ ; connecting p1 to pi˙ I sp

• two null half-planes, the interiors of W1 and W2 . The wing Wi is bounded by the curves i˙ and ˇi˙ ; • the two components of the interior of the stem S. The stem is bounded by the curves i˙ and ˛i˙ , for i D 1; 2. c 2;1 is the lift of a crooked surface from The saturation of Cx by the antipodal map on Ein 2;1 Ein . The interested reader can verify that it is homeomorphic to a torus. 8.4 Automorphisms of a crooked surface. Let C be the positively-oriented crooked plane of Section 8.2, and Cx the associated crooked surface in Ein2;1 . First, C is invariant by all positive homotheties centered at the origin, because each of the wings and the stem are. Second, it is invariant by the 1-dimensional group of linear hyperbolic isometries of Minkowski space preserving the lightlike lines bounding the stem. The subgroup A, which can be viewed as the subgroup of SO.3; 2/ acting by positive homotheties and positive linear hyperbolic isometries of Minkowski space, then preserves C , and hence Cx . The element 0 1 1 B 1 C B C B C 1 s0 D B C @ 1 A 1 is a reflection in the spine, and also preserves Cx . Note that s0 is time-reversing. Then we have Z2 Ë A Š Z2 Ë .R>0 /2 Aut.Cx /: Next let `1 ; `2 be the two lightlike geodesics bounding the stem (alternatively bounding the wings) of C . As above, the inversion leaves invariant C n .`1 [ `2 /. In fact, the element 0 B B s1 D B B @

1

1 1

C C C C 1A

1 1

224


is an automorphism of Cx . The involution 0 1 B 1 B B 1 s2 D B @ 1

1 C C C C A 1

also preserves Cx and exchanges the ideal points p1 and p2 . The involutions s0 , s1 ; and s2 pairwise commute, and each product is also an involution, so we have G WD Z32 Ë .R>0 /2 Aut.Cx / To any crooked surface can be associated a quadruple of points in a stem configuration. The stabilizer of a stem configuration in SO.3; 2/ Š PO.3; 2/ is N.A/, the normalizer of a Cartan subgroup A. Suppose that the points .p0 ; p1 ; p2 ; p1 / are associated to Cx . As above, a neighborhood of p0 in Cx is not diffeomorphic to a neighborhood of p1 in Cx , so any automorphism must in fact belong to the subgroup N 0 .A/ preserving each pair fp0 ; p1 g and fp1 ; p2 g. Each g 2 N 0 .A/ either preserves Cx or carries it to its opposite, the closure of the negatively-oriented crooked plane having the same vertex and spine as C . Now it is not hard to verify that the full automorphism group of Cx in SO.3; 2/ is G.

9 Construction of discrete groups A complete flat Lorentzian manifold is a quotient En;1 = , where acts freely and properly discontinuously on En;1 . When n D 2, Fried and Goldman [15] showed that unless is solvable, projection on O.2; 1/ is necessarily injective and, furthermore, this linear part is a discrete subgroup 0 O.2; 1/ [1], [6], [24]. In this section we identify E2;1 with its usual embedding in Ein2;1 , so that we consider such as discrete subgroups of SO.3; 2/. We will look at the resulting actions on Einstein space, as well as on photon space. At the end of the section, we list some open questions. 9.1 Spine reflections. In §8.4, we described the automorphism group of a crooked surface. We recall some of the basic facts about the reflection in the spine of a crooked surface, which is discussed in §3.3 and §5.2.2, and which is denoted s0 in the example above. Take the inner product on R3;2 defined by the matrix 1 0 1 I2 ˚ I1 ˚ 1 0 2 and identify E2;1 with its usual embedding in the Minkowski patch determined by the improper point p1 . Let C be the crooked plane determined by the stem configuration


.p0 ; p1 ; p2 ; p1 / as in §8.2, with 2

3 0 617 6 7 7 p1 D 6 617 405 0

and

225

2 3 0 617 6 7 7 p2 D 6 617 : 405 0

Then s0 is an orientation-preserving, time-reversing involution having fixed set Fix.s0 / D fp1 ; p2 g [ L.p1 / \ L.p2 / : In the Minkowski patch, hs0 i interchanges the two components of the complement of C . If a set of crooked planes in E2;1 is pairwise disjoint, then the group generated by reflections in their spines acts properly discontinuously on the entire space [7], [8], [10]. Thus spine reflections associated to disjoint crooked planes give rise to discrete subgroups of SO.3; 2/. We will outline a way to construct such groups; see [4], for details. Let S1 ; S2 Ein2;1 be a pair of spacelike circles that intersect in a point; conjugating if necessary, we may assume that this point is p1 . Each circle Si , i D 1; 2, is the projectivized nullcone of a subspace Vi R3;2 of type (2,1); V1 C V2 can be written as the direct sum Rv1 ˚ Rv2 ˚ W; where v1 ; v2 are spacelike vectors and W D V1 \ V2 is of type (1,1). We call fS1 ; S2 g an ultraparallel pair if v1? \ v2? is spacelike. Alternatively, we can define the pair to ? be ultraparallel if they are parallel to vectors u1 ; u2 2 R2;1 such that u? 1 \ u2 is a 2;1 spacelike line in E . Let S1 , S2 be an ultraparallel pair of spacelike circles in Ein2;1 . Denote by 1 and 2 the spine reflections fixing the respective circles. (Note that 1 and 2 are conjugate to s0 , since SO.3; 2/ acts transitively on crooked surfaces.) Identifying the subgroup of SO.3; 2/ fixing p0 and p1 with the group of Lorentzian linear similarities Sim.E2;1 / D RC O.2; 1/; then D 2 ı1 has hyperbolic linear part – that is, it has three, distinct real eigenvalues. The proof of this fact and the following proposition may be found, for instance, in [4]. Proposition 9.1.1. Let S1 and S2 be an ultraparallel pair of spacelike circles as above. Then S1 and S2 are the spines of a pair of disjoint crooked planes, bounding a fundamental domain for h i in E2;1 . Note that while hi acts freely and properly discontinuously on E2;1 , it fixes p1 as well as two points on the ideal circle. Next, let Si , i D 1; 2; 3 be a triple of pairwise ultraparallel spacelike circles, all intersecting in p1 , and let D h1 ; 2 ; 3 i be the associated group of spine reflections. Then contains an index-two free group generated by hyperbolic isometries of E2;1 (see [4]). Conversely, we have the following generalization of a well-known theorem in hyperbolic geometry.

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Theorem 9.1.2 ([4]). Let D h1 ; 2 ; 3 j 1 2 3 D Id i be a subgroup of isometries of E2;1 , where each has hyperbolic linear part and such that their invariant lines are pairwise ultraparallel. Then there exist spine reflections i , i D 1; 2; 3, such that 1 D 1 2 , 2 D 2 3 and 3 D 3 1 . Note that as above is discrete. Indeed, viewed as a group of affine isometries of E2;1 , its linear part G O.2; 1/ acts on the hyperbolic plane and is generated by reflections in three ultraparallel lines. As mentioned before, if the spacelike circles are spines of pairwise disjoint crooked planes, then acts properly discontinuously on the Minkowski patch. Applying this strategy, we obtain that the set of all properly discontinuous groups , with linear part generated by three ultraparallel reflections, is non-empty and open [4]. Here is an example. For i D 1; 2; 3, let Vi R3;2 be the .2; 1/-subspace 9 82 3 aui C cpi = < Vi D 4ahui ; pi i C b C chpi ; pi i5 W a; b; c 2 R ; ; : c where u1 D

p 2 0 1 ;

p p1 D 0 2 1 ;

h p u2 D 22 h p p2 D 26

p 6 2 p 22

i

; i

1

1

h p u3 D 22 hp ; p3 D 26

p 6 2 p 22

i

; i 1 :

1

Then the projectivized nullcone of Vi is a spacelike circle – in fact, it corresponds to the spacelike geodesic in E2;1 passing through pi and parallel to ui . The crooked planes with vertex pi and spine p C Rui , respectively, are pairwise disjoint (one shows this using inequalities found in [10]). 9.2 Actions on photon space. Still in the same Minkowski patch as above, let G be a finitely generated discrete subgroup of O.2; 1/ that is free and purely hyperbolic – that is, every nontrivial element is hyperbolic. Considered as a group of isometries of the hyperbolic plane, G is a convex cocompact free group. By the theorem of Barbot [2], Theorem 9.2.1. Let be a subgroup of isometries of E2;1 with convex cocompact linear part. Then there is a pair of non-empty, -invariant, open, convex sets ˙ E2;1 such that • the action of on ˙ is free and proper; • the quotient spaces ˙ = are globally hyperbolic; • each ˙ is maximal among connected open domains satisfying these two properties; • the only open domains satisfying all three properties above are ˙ . The notion of global hyperbolicity is central in General Relativity, see for example [3]. The global hyperbolicity of ˙ = implies that it is homeomorphic to the


227

product .H2 =G/ R, where H2 is the hyperbolic plane, and G is the convex cocompact linear part of . It also implies that no element of preserves a null ray in ˙ . Let be as in Theorem 9.2.1 and consider its action, for instance, on C . Since C = is globally hyperbolic, it admits a Cauchy hypersurface, a spacelike surface S0 which meets every complete causal curve and with complement consisting of two connected components. The universal covering Sz0 is -invariant. The subset Pho2;1 0 Pho2;1 comprising photons which intersect Sz0 is open. 2;1 We claim that acts freely and properly on Pho2;1 0 . Indeed, let K Pho0 be a compact set. Then K is contained in a product of compact subsets K1 K2 , where K1 Sz0 and K2 S 1 , the set of photon directions. The action of restricts to a Riemannian action on Sz0 . Thus the set f 2 j .K1 / \ K1 ¤ ;g is finite. As Sz0 is spacelike, it follows that f 2 j .K/ \ K ¤ ;g is finite too. Finally, global hyperbolicity of C = implies that no photon intersecting C is invariant under the action of any element of . Corollary 9.2.2. There exists a non-empty open subset of Pho2;1 on which acts freely and properly discontinuously. 9.3 Some questions. So far we have considered groups of transformations of Ein2;1 and Pho2;1 arising from discrete groups of Minkowski isometries. Specifically, we have focused on groups generated by spine reflections associated to spacelike circles intersecting in a point. Question. Describe the action on Ein2;1 of a group generated by spine reflections corresponding to non-intersecting spacelike circles. In particular, determine the possible dynamics of such an action. A related question is: Question. What does a crooked surface look like when its spine does not pass through p1 , or the lightcone at infinity altogether? Describe the action of the associated group of spine reflections. More generally, we may wish to consider other involutions in the automorphism group of a crooked surface. Question. Describe the action on Ein2;1 of a group generated by involutions, in terms of their associated crooked surfaces. As for the action on photon space, here is a companion question to those asked in §7: Question. Given a group generated by involutions, what is the maximal open subset of Pho2;1 on which the group acts properly discontinuously?

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References [1] Abels, H., Properly discontinuous groups of affine transformations: a survey. Geom. Dedicata 87 (2001) 309–333. 224 [2] Barbot, T., Global hyperbolic flat space-times. J. Geom. Phys. 53 (2) (2005), 123–165. 226 [3] Beem, J. K., Ehrlich, P. E., Easley, K. L., Global Lorentzian geometry. 2n ed., Monogr. Textbooks Pure Appl. Math. 202, Marcel Dekker, Inc., New York 1996. 192, 226 [4] Charette, V., Affine deformations of ultraideal triangle groups. Geom. Dedicata 97 (2003), 17–31. 225, 226 [5] Charette, V., Drumm, T., and Goldman, W. (eds.), New techniques in Lorentz manifolds (BIRS Workshop, 2004), Geom. Dedicata 126 (2007), 1–291. 181 [6] Charette, V., Drumm, T., Goldman, W., and Morrill, M., Complete flat affine and Lorentzian manifolds. Geom. Dedicata 97 (2003), 187–198. 224 [7] Charette, V., and Goldman, W., Affine Schottky groups and crooked tilings. In Crystallographic Groups and their Generalizations, Contemp. Math. 262, Amer. Math. Soc., Providence, RI, 2000, 69–97, 225 [8] Drumm, T., Fundamental polyhedra for Margulis space-times. Topology 31 (4) (1992), 677–683. 181, 216, 225 [9] Drumm, T., Linear holonomy of Margulis space-times. J. Differential Geom. 38 (1993), 679–691. 216 [10] Drumm, T., and Goldman, W., The geometry of crooked planes. Topology 38 (2) (1999), 323–351. 216, 217, 225, 226 [11] Frances, C., Géometrie et dynamique lorentziennes conformes. Thèse, E.N.S. Lyon, 2002. 180, 181 [12] Frances, C., The conformal boundary of Margulis space-times. C. R. Acad. Sci. Paris Sér. I 332 (2003), 751–756. 219 [13] Frances, C., Lorentzian Kleinian groups. Comment Math. Helv. 80 (4) (2005), 883–910. 181, 211, 214, 215 [14] Frances, C., Une démonstration du théorème de Liouville en géométrie conforme. Enseign. Math. (2) 49 (1–2) (2003), 95–100. [15] Fried, D., and Goldman, W., Three-dimensional affine crystallographic groups. Adv. Math. 47 (1983), 1–49. 224 [16] Goldman, W., Complex Hyperbolic Geometry. Oxford Math. Monogr., Oxford University Press, New York 1999. 204 [17] Goldman, W., Labourie, F., and Margulis, G., Proper affine actions and geodesic flows of hyperbolic surfaces. Submitted. [18] Knapp, A. W., Lie groups beyond an introduction. Progr. Math. 140, Birkhäuser, Boston, MA, 1996. 209 [19] Kulkarni, R., Proper actions and pseudo-Riemannian space forms. Adv. Math. 40 (1) (1981), 10–51. 181 [20] Kulkarni, R., and Raymond, F., 3-dimensional Lorentz space forms and Seifert fiber spaces. J. Differential Geom. 21 (1985), 231–268.


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[21] Margulis, G., Free properly discontinuous groups of affine transformations. Dokl. Akad. Nauk SSSR 272 (1983), 937–940. [22] Margulis, G., Complete affine locally flat manifolds with a free fundamental group. J. Soviet Math. 134 (1987), 129–134. [23] McDuff, D., and Salamon, D., Introduction to symplectic topology. 2nd ed., Oxford Math. Monogr., Oxford University Press, New York 1998. 204, 205 [24] Milnor, J., On fundamental groups of complete affinely flat manifolds. Adv. Math. 25 (1977), 178–187. 224 [25] Pratoussevitch, A., Fundamental domains in Lorentzian geometry. Geom. Dedicata 126 (2007), 155–175. 186 [26] Samelson, H., Notes on Lie Algebras. 2n ed., Universitext, Springer-Verlag, NewYork 1990. 208 [27] Siegel, C. L., Symplectic geometry. Amer. J. Math. 65 (1943), 1–86 203, 205 [28] Thurston, W., Three-dimensional geometry and topology. Vol. 1 (Silvio Levy ed.), Princeton Math. Ser. 35, Princeton University Press, Princeton, NJ, 1997. 204

Essential conformal structures in Riemannian and Lorentzian geometry Charles Frances

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

2

Generalizations of Lichnerowicz’s conjecture . . . . . . . . . . . . . . . . . . . . . 234 2.1 Conjecture in the pseudo-Riemannian framework . . . . . . . . . . . . . . . . 234 2.2 Lichnerowicz’s conjecture for parabolic geometries . . . . . . . . . . . . . . . 236

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Some words about the proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . 240

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Conformal dynamics on compact manifolds . . . . . . . . . . . . . . . . . . . . . . 242

5 The conformal model space in Lorentzian geometry . . . . . 5.1 Geometry of Einstein’s universe . . . . . . . . . . . . 5.2 Examples of essential dynamics on Einstein’s universe 5.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . .

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More complicated examples of compact essential Lorentzian manifolds . . . . . . . 248 6.1 Schottky groups on Einstein’s universe . . . . . . . . . . . . . . . . . . . . . . 248 6.2 More complicated essential dynamics . . . . . . . . . . . . . . . . . . . . . . 249

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Essential versus isometric dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 252 7.1 Stable conformal dynamics and its consequences on the geometry . . . . . . . 253 7.2 Examples where stability imposes conformal flatness . . . . . . . . . . . . . . 255

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Essential actions of simple groups on compact manifolds . . . . . . . . . . . . . . . 257

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

1 Introduction The starting point of what has been called Lichnerowicz’s conjecture, is the very simple and naive question: “Given a Riemannian manifold, is the group of angle-preserving transformations bigger than the group of distance-preserving ones?” At a first glance, the natural answer should be almost always affirmative. Indeed, the data of a Riemannian metric g on a manifold M seems to be stronger than a simple “angle-structure”, most commonly called conformal structure, i.e., the data of a whole family of metrics Œg D fe g j 2 C 1 .M /g. As an illustration, one thinks at once of a similarity x 7! Ax C T of Rn , with jj 6D 1, A 2 O.n/. Such a transformation is conformal for the flat metric on Rn . It fixes a unique point x0 2 R at which its differential is Id. Since u 7! u cannot

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preserve any scalar product, we get that x 7! Ax C T cannot preserve any smooth Riemannian metric in the conformal class of the flat metric. This example motivates the following: Definition 1.1. Let .M; g/ be a smooth Riemannan manifold. Let f be a conformal transformation of .M; g/, i.e., there exists a smooth function f W M ! R such that f g D e 2f g. The transformation is said to be essential if f does not preserve any metric in the conformal class Œg of the metric g. More generally, a subgroup G of conformal transformations of .M; g/ is said to be essential if it does not preserve any metric in the conformal class. One also says that the structure .M; g/ itself is essential when its group of conformal transformations is essential. By the stereographic projection, the similarities x 7! x C T can be extended smoothly to transformations of the sphere Sn fixing the “point at infinity”. It turns out that these extensions act as conformal transformations for the round metric gcan of constant curvature C1 on Sn . Exactly by the same argument as above, such transformations are essential. Another way to see quickly that the conformal group of .Sn ; gcan / is essential, is to notice that this conformal group, the Möbius group PO.1; n C 1/, is not compact. On the other hand, by Ascoli’s theorem, the group of isometries of a compact Riemannian manifold has to be compact. Let us now try to determine the conformal group of a flat torus T n D Rn = , where D Z1 ˚ ˚Zn is a lattice. We endow this torus with the metric gN flat induced by the flat metric on Rn . Any conformal transformation fN lifts to a conformal transformation f of .Rn ; geucl /. Thus f is of the form x 7! Ax C T , with 2 R and A 2 O.n/. Let us suppose that jj 6D 1. We can then assume jj > 1. If U is a small open subset such that the covering map W Rn ! T n is injective on U , then has to be injective on every f k .U /, k 2 N. On the other hand, limk!C1 Vol.f k .U // D C1, where Vol is the euclidean volume form on Rn . But cannot be injective on an open subset with a volume strictly greater than Vol.1 ; : : : ; n /, yielding a contradiction. We infer that jj D 1, and that fN is an isometry of .T n ; gN flat /. Thus, the conformal group of .T n ; gN flat / is exactly the group of isometries. One says in this case that the conformal group is inessential. Looking for more examples we could determine the conformal group of other Riemannian manifolds. For example that of the hyperbolic space Hn , or of RP n . Every time we get that this group is inessential, and reduces to the group of isometries. So, starting with the feeling that essential Riemannian manifolds should be quite numerous, we still have only two examples of such essential structures! This lack of examples led to the: Lichnerowicz’s Conjecture. The only Riemannian manifolds of dimension at least two having an essential conformal group are, up to conformal diffeomorphism, the standard sphere .Sn ; gcan / and the Euclidean space .Rn ; geucl /. Several partial results toward the conjecture were made during the sixties by [4], [30], [28], [31], among others.

Essential conformal structures in Riemannian and Lorentzian geometry 233

Almost simultaneously, but with different approaches, J. Ferrand and M. Obata proved in [12] and [32] that the conjecture was true in the compact case. Finally, in 1996, J. Ferrand answered definitively to the original question of Lichnerowicz by proving: Theorem 1.2 ([13]). Let .M; g/ be a Riemannian manifold of dimension n 2. If the group of conformal transformations of .M; g/ is essential, then .M; g/ is conformally diffeomorphic to (i) .Sn ; gcan / if M is compact; (ii) .Rn ; gcan / if M is not compact. This theorem has been proved independently by R. Schoen in [34] (see also [18]). Ferrand’s result is often presented as a nice example of the following general principle. Generically, rigid geometric structures have a trivial group of automorphisms (even if we just consider the local group of automorphism). So, when the group of automorphism is nontrivial, and even “big”, then the geometric structure has to be very peculiar. Of course, we must precise what we mean by “big”. When we are looking at compact manifolds, a big group of automorphisms is for example a non compact one. To understand why Ferrand’s theorem illustrates this principle, we have to precise that Riemannian conformal structures (and more generally pseudo-Riemannian ones) are rigid geometric structures. Indeed, such a structure defines naturally a parallelism on a subbundle B 2 .M / R2 .M / of the bundle R2 .M / of 2-frames of M (details can be found in [23]). Any local conformal transformation acts on an open subset of B 2 .M / preserving this parallelism. We thus see that a conformal transformation whose 2-jet at a point is the 2-jet of the identical transformation will fix a point of B 2 .M /. Since the parallelism is preserved, this means that the transformation induces the identical transformation on B 2 .M /, hence the transformation is itself the identical transformation of M . We thus see that any conformal transformation is completely determined by its 2-jet at a point of M , so that the dimension of the Lie algebra of infinitesimal conformal transformations is finite. This is a manifestation of the rigidity of conformal structures. Now, how can we interpret the condition of essentiality as a criteria for the conformal group to be big? Let us recall that the action of a group G by homeomorphisms of a manifold M is said to be proper if for every compact subset K M the set GK D fg 2 G j g.K/ \ K 6D ;g has compact closure in Homeo.M / (where Homeo.M /, the group of homeomorphisms of M , is endowed with the compact-open topology). In particular, when the manifold M is compact, the action of G is proper if and only if G is compact. Nonproperness can be thought as the weakest condition of non triviality for the dynamics of a group action on a manifold. A key point is the following theorem of Alekseevski, which, in the Riemannian framework, makes the link between essentiality and dynamics of the conformal group:

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Theorem 1.3 ([2]). Let .M; g/ be a Riemannian manifold. The conformal group of .M; g/ is essential if and only if its action on M is not proper. One part of this theorem is clear: by Ascoli’s theorem, the isometry group of a Riemannian manifold acts properly. When the manifold .M; g/ is compact the converse is quite easy to prove. Assume that the conformal group H acts properly, R i.e. is compact. Then, if d is the bi-invariant Haar measure on H , the metric gN D h2H h gd.h/ is a smooth Riemannian metric of Œg, left invariant by the group H . Thus, the conformal group is inessential. For noncompact manifolds the proof is more technical. In the light of Alekseevski’s result, Theorem 1.2 is a remarkable example of geometrico-dynamical rigidity. Here, nonproperness, a very weak assumption on the dynamics of the conformal group, implies very strong consequences on the geometry of the manifold: the only possible geometries turn out to be, up to conformal diffeomorphism, .Sn ; gcan / and .Rn ; geucl /. Our aim in this article is to study to what extent Theorem 1.2 generalizes (or does not) to more general frameworks. In the next section we will discuss what such generalizations could be. In Section 3 we recall the main arguments to prove Theorem 1.2. Then, in the next sections, we focus on Lorentzian conformal geometry, trying to understand the meaning of essentiality in this case. Acknowledgement. The author is grateful to Professors DimitriAlekseevsky and Helga Baum for the invitation to participate in the Special Research Semester Geometry of pseudo-Riemannian manifolds with applications in Physics.

2 Generalizations of Lichnerowicz’s conjecture 2.1 Conjecture in the pseudo-Riemannian framework. The definition 1.1 of an essential structure carries in an obvious way to general pseudo-Riemannian manifolds (recall that a (smooth) pseudo-Riemannian metric g on a manifold M is a (smooth) field of nondegenerate quadratic forms of constant signature .p; q/ on TM ). It is thus natural to ask wether Ferrand’s theorem also generalizes, in some way, to any signature. Let us point out the first difficulty occurring when we pass from Riemannian conformal geometry to general signature .p; q/. While in Riemannian signature we saw, thanks to Theorem 1.3, that essentiality is equivalent to nonproperness of the action of the conformal group, this equivalence is no longer true in higher signature. It is still true that the properness of the action of the conformal group implies inessentiality, but the converse is false as the following example shows. Endow the space Rn with the Lorentzian metric gmink D .dx1 /2 C.dx2 /2 C C.dxn /2 . If O.1; n1/ is the group of linear transformations preserving the quadratic form x12 C x22 C C xn2 , then the conformal group of .Rn ; gmink / is the group generated by homotheties, translations, and elements of O.1; n 1/. We look at the torus Rn =Zn , endowed with the induced metric gN mink . By a proof analogous to that made in the introduction for a Riemannian flat torus, it is not hard to check that the conformal group of .T n ; gmink / is exactly the group of


isometries of .T n ; gN mink /, so that the structure is inessential. Nevertheless, by a Theorem of Borel and Harisch-Chandra, the subgroup OZ .1; n1/ D O.1; n1/\SL.n; Z/ is a lattice in O.1; n 1/. In particular it is noncompact, and since it normalizes Zn , it induces an isometric action on .T n ; gN mink /. Thus the action of Isom.T n ; gN mink /, and hence of Conf.T n ; gN mink /, is not proper, while the structure is inessential. Hoping for a direct generalization of Theorem 1.2, we could ask: Question 2.1. Let .p; q/ 2 N 2 be two integers. Are there, up to conformal diffeomorphism, only a finite number of pseudo-Riemannian manifolds .M; g/ of signature .p; q/ for which the conformal group is essential? It turns out that with this degree of generality, there is no hope to get a positive answer to this question. Given a basis .e1 ; : : : ; en / of Rn , we look at the one-parameter group of transformations: t W .x1 ; : : : ; xn / 7! .e 2t x1 ; e t x2 ; : : : ; e t xn1 ; xn /. This group acts as conformal transformations for a lot of Lorentzian metrics on Rn . In fact, Alekseevski proved in [3] that t acts as a homothetic flow for every metric of the form g D 2dx1 dxn C P Pn1 n1 2 2 iD2 .dxi / C i;j D2 ij yi dyj dxn C .Q.xn /.x2 ; : : : ; xn1 / C b.xn /x1 /.dxn / , for ij , b smooth functions, and Q.xn / a smooth family of quadratic forms in the variables x2 ; : : : ; xn1 . This means that for such a metric g we have . t / g D e t g. This holds in particular for the fixed points of t , so that t cannot act by isometries of a Lorentz metric. All the structures .Rn ; g/ with g as above are thus essential. Other examples of essential conformal flows of pseudo-Riemannian manifolds preserving an infinite dimensional space of conformal structures were constructed in [25], [26]. The basic idea here is to consider a flow t acting conformally on a Lorentz manifold .M; g/, and having singularities at which the differential is the identity. These flows are then essential, because one knows that a Lorentzian isometry fixing a point, and whose differential is the identity at this point, has to be the identical transformation (the manifold M under consideration is connected). Now, “far from the singularity”, the flow t considered in [25], [26] acts properly. There is a piece of transversal † M such that for every x 2 †, the orbit t x leaves every compact set of M in a finite time. Thus, choosing an open subset U †, perturbing g at the points of U , without changing anything on † n U , and pushing the modified metric along the flow, will yield another conformal structure Œg 0 on M which is preserved by t . Since the perturbation on U is arbitrary, we get a huge class of different Lorentzian conformal structures on M , which are preserved by t , and for which t is an essential subgroup of conformal transformations. Similar constructions by “perturbations” are done in [15]. The process described above uses the noncompactness of the manifold to get pieces of the manifold where the action of the flow is proper. Such constructions break down on compact manifolds. It is thus quite likely that compact essential structures are more unusual than noncompact ones. So, we could reformulate the previous question: Question 2.2. Let .p; q/ 2 N 2 be two integers. Are there, up to conformal diffeomorphism, only a finite number of compact pseudo-Riemannian manifolds .M; g/ of signature .p; q/ for which the conformal group is essential?

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Even with the extra compactness assumption, we will see that no positive answer to Question 2.2 can be expected. In Section 6, following [15], we will construct quite a wide class of different Lorentzian conformal structures on compact manifolds which are all essential. Even if these constructions are achieved in the Lorentzian framework, it is likely that they generalize to any signature (except the Riemannian one, of course). The structures constructed in Section 6 will all be globally distinct, but locally they are all conformally modelled on open subsets of Minkowski’s space (the space Rn endowed with the Lorentzian metric .dx1 /2 C .dx2 /2 C C .dxn /2 ). This still leaves open the following: Generalized Lichnerowicz’s Conjecture. Let .M; g/ be a compact pseudo-Riemannian manifold with an essential group of conformal transformations. Then .M; g/ is conformally flat, i.e., every x 2 M has a neighbourhood Ux which is conformally equivalent to an open subset of Rn , with the conformal structure induced by x12 2 xp2 C xpC1 C C xn2 . This conjecture is stated in [10], p. 96. Notice that the compactness assumption cannot be removed if we want the conjecture to be true. Examples of [25], [26] exhibit conformal flows on noncompact manifolds, which are essential for nonconformally flat pseudo-Riemannian structures. Also, in [33], M. N. Podoksenov gives examples of Lorentzian metrics on Rn which are homogeneous, and for which the flow t above is conformal (and automatically essential). We see that even under the strong assumption of homogeneity, essentiality does not imply local rigidity if we remove the compactness assumption. 2.1.1 Strong essentiality. Until now, we were not very precise on the regularity required for the conformal structures we consider. Ferrand’s theorem 1.2 requires a regularity C 2 of the metric. For a C k pseudo-Riemannian metric g one usually defines the conformal class as Œg D fe g j 2 C k .M /g. But in fact, keeping g of class C k , we could enlarge the conformal class, considering ŒgC 0 D fe g j 2 C 0 .M /g (and in the same way ŒgL1 , etc.). This leads to the following: Definition 2.3. A pseudo-Riemannian manifold .M; g/ is said to be strongly essential, if its group of conformal transformations does not preserve any metric in ŒgC 0 . Notice that, for a conformal structure, the assumption of being strongly essential is weaker than the simple notion of essentiality. These two notions could be distinct, even if we do not have examples of pseudo-Riemannian structures which are essential without being strongly essential. At least, since smooth invariant objects can be difficult to build for dynamical systems, the assumption of strong essentiality in generalized Lichnerowicz’s conjecture could make it a little bit simpler to handle. 2.2 Lichnerowicz’s conjecture for parabolic geometries. We address now another question which puts the generalized Lichnerowicz conjecture in a wider framework.


2.2.1 Cartan geometry. A way to look at pseudo-Riemannian and conformal pseudoRiemannian structures as rigid geometric structures is to present them as what is called Cartan geometries. We will not give a lot of details about Cartan geometries in this section, but we refer to [35], which is a very good reference. Let us consider a Lie group G and a closed subgroup P of G. A Cartan geometry on a manifold M is, roughly speaking, a geometric structure on M which is infinitesimally modelled on the homogeneous space G=P . So, Cartan geometries are curved generalizations of Klein’s geometries modelled on G=P , namely manifolds which are locally modelled on G=P . Formally, a Cartan geometry on a manifold M , modelled on the homogeneous space X D G=P , is the data of: (i) a P -principal bundle B ! M over M ; (ii) a 1-form ! on B, with values in the Lie algebra g, called Cartan connection, and satisfying the following conditions: – at every point p 2 B, !p is an isomorphism between Tp B and g; – if X is a vector field of B, coming from the action by right multiplication of some one-parameter subgroup t 7! ExpG .tX / of P , then !.X / D X ; – for every a 2 P , Ra ! D Ad.a1 /! (Ra standing for the right action of a on B). A Cartan geometry on a manifold M will be denoted by .M; B; !/. To each Cartan geometry is associated a 2-form D ! C 12 Œ!; !, the curvature of !, whose vanishing characterizes the manifolds M which are locally modelled on the homogeneous space X D G=P . Such Cartan geometries are called flat. Pseudo-Riemannian manifolds .M; g/ are examples of Cartan geometries. The model space is X D SO.p; q/ Ë Rn = SO.p; q/. We can choose for the fiber bundle B the bundle of orthonormal frames on M . The metric g defines in a unique way a LeviCivita connection, which can be reinterpreted as a Cartan connection !g on B, with values in so.p; q/ ˚ Rn . Each isometry of .M; g/ acts on B and leaves the connection !g invariant. For conformal pseudo-Riemannian structures .M; Œg/ of signature .p; q/, the model space is X D SO.p C 1; q C 1/=P , where P is the stabilizer, in SO.p C 1; q C 1/, of 2 2 2 an isotropic line for the quadratic form x12 xpC1 C xpC2 C C xnC2 . The n group P is isomorphic to the semi-direct product .R SO.p; q// Ë R . In particular, in the Riemannian case p D 0, X is just the sphere Sn considered as the homogeneous space SO.1; n C 1/=P , i.e., Sn endowed with the conformal class of the round metric. The fact that a conformal structure .M; Œg/ determines in a canonical way a Cartan geometry .M; B; !/ is not at all an obvious fact (see for example [23]). Nevertheless, when the dimension of the manifold M is at least three, the data of the conformal class Œg defines on a subbundle B of the bundle of 2-jets of frames a Cartan connection ! with values in so.p C 1; q C 1/. This connection is flat if and only if the manifold is locally conformally equivalent to open subsets of the space Rn , endowed with .dx1 /2 .dxp /2 C .dxpC1 /2 C C .dxn /2 .

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2.2.2 Parabolic geometries versus reductive geometries. Among all types of Cartan geometries we can isolate two large and interesting families. The first family is constituted by reductive geometries. These are the Cartan geometries .M; B; !/ modelled on X D G=P such that g D p ˚ n, where p is the Lie subalgebra of the subgroup P , and n is Ad.P /-invariant. For example, pseudoRiemannian metrics give rise to reductive Cartan geometries, since g D so.p; q/ ˚ Rn in this case, and Rn is Ad.SO.p; q//-invariant. For general reductive geometries it is possible to define a notion of covariant derivative (see [35], Chapter 5), so that these geometries behave quite closely to pseudo-Riemannian ones. Other examples of reductive geometries are, for instance, affine structures. The second family is constituted by parabolic geometries. A good exposition of these geometries is given in [8]. They correspond to Cartan geometries .M; B; !/ modelled on X D G=P , where G is a simple Lie group, whose Lie algebra g is endowed with a k-grading, namely g D gk ˚ ˚ g1 ˚ g0 ˚ gC1 ˚ ˚ gCk (Œgi ; gj giCj ), and the Lie algebra of P is p D g0 ˚ gC1 ˚ ˚ gCk . In this case there is no natural Ad.P /-invariant complement to p, what makes this kind of geometries more difficult to handle. Pseudo-Riemannian conformal structures .M; Œg/ (in dimension 3) are examples of parabolic geometries. Indeed, in this case we saw that g D so.p C 1; q C 1/, and there is a 1-grading of so.p C 1; q C 1/, namely so.p C 1; q C 1/ D n ˚ g0 ˚ nC , with n and nC n-dimensional abelian subalgebras, and g0 D R ˚ so.p; q/. The Lie subalgebra corresponding to the group P is g0 ˚ nC . Apart from pseudo-Riemannanian conformal structures, there are a lot of interesting geometric structures, which define canonically a parabolic Cartan geometry: projective structures, CR and quaternionic CR-structures, some models of path geometries (see [8] and references therein for a lot of examples). Now let us remark that a parabolic geometry .M; B; !/ modelled on X D G=P defines a family of reductive geometries that we will call subordinated to .M; B; !/. Indeed, in the case of a parabolic geometry, the Lie algebra g0 turns out to be reductive (in the sense of Lie algebras). So, g0 D Œg0 ; g0 ˚ z, where z is the center of g0 , and s D Œg0 ; g0 is semi-simple (see [22]). At the Lie group level P can be written as a semi-direct product .Z S/ Ë N , where Z is abelian and centralizes S , which is semi-simple. The group N is nilpotent with Lie algebra gC1 ˚ ˚ gCk . The action of Z ËN is proper on B, so that B0 D B=.Z ËN / is a smooth manifold. In fact, since S normalizes Z Ë N , we still have a right action of S on B0 , which makes B0 an S -principal bundle over M . Definition 2.4. We call a unimodular reductive geometry on M , subordinated to .M; B; !/, the data of an S -equivariant section W B0 ! B. Let us make this definition more explicit. Pick an S -equivariant section W B0 ! B and write † D s.B0 /. The bundle map W B ! M , when restricted to †, makes † into an S -principal bundle over M . Now, let us denote by g0 the Lie algebra gk ˚ ˚ g1 ˚ s (where s D Œg0 ; g0 is the Lie algebra of S), and by W g ! g0 , the


projection onto g0 relatively to z ˚ gC1 ˚ ˚ gCk . On † we define ! x D ı !. We claim that .M; †; !/ x is a Cartan geometry over M , modelled on X 0 D S Ë N =S . By N we denote the connected Lie subgroup of G, whose lie algebra is gk ˚ ˚ g1 . Notice that the Lie algebra of S Ë N is g0 , so that the geometry .M; †; !/ x is reductive. This reductive geometry is unimodular because, since S is semi-simple, Ad.S / acts on n D gk ˚ ˚ g1 by elements of SL.n /. Let us illustrate this construction for conformal pseudo-Riemannian structures. In this case, g D so.p C 1; q C 1/ D n ˚ g0 ˚ nC , as we already said. The Lie algebra g0 equals R ˚ so.p; q/ in this case, and thus s D so.p; q/. For each point p 2 B, denote Vp D ! 1 .p/ (recall that p D g0 ˚ nC ). Then, Dp defines an isomorphism from Tp B=Vp on Tx M , where x.p/. Also, !p induces an isomorphism from Tp B=Vp onto g=p. Hence !p ı .Dp /1 defines an isomorphism ip W Tx M ! g=p. It is not difficult to check that ipb D Ad.b 1 / ı ip , for every b 2 B. The conformal structure at Tx M is just the pullback through ip of the unique Ad.P /-invariant conformal class of scalar products of signature .p; q/ on g=p. We choose now h ; i0 , an Ad.S /-invariant scalar product of signature .p; q/ on g=p. Let W B0 ! B be an SO.p; q/-equivariant section. This data defines a metric g on M , in the conformal class Œg. Indeed, for each x 2 M , choose p 2 † over x and define g .x/ D .ip / h ; i0 . This does not depend on the choice of p 2 † above x, since ip s D Ad.s 1 /ip , for s 2 SO.p; q/, and h ; i0 is SO.p; q/invariant. Hence, a unimodular reductive geometry .M; †; !/ x subordinated to the Cartan geometry .M; B; !/ associated to a pseudo-Riemannian conformal structure .M; Œg/ is just the data of a metric in the conformal class. Given a Cartan geometry .M; B; !/ we define Aut.M; B; !/ as the set of diffeomorphisms fO of B such that fO ! D !. Such a diffeomorphism has to respect the fibers of B, so that it induces a diffeomorphism f of M . The set of such induced diffeomorphisms is called Aut.M; !/. Let now .M; B; !/ be a parabolic geometry modelled on X D G=P . We will say that Aut.M; B; !/ leaves invariant a subordinated unimodular reductive geometry, if there exists an S -invariant section W B0 ! B such that fO.†/ D †. Notice that in x D !. x In the case of a pseudo-Riemannian conformal structure, saying this case, fO ! that the group Aut.M; B; !/ preserves a subordinated unimodular reductive geometry means that the conformal group preserves a metric in the conformal class, i.e., the conformal group is inessential. We can now formulate: Lichnerowicz’s Conjecture for parabolic Cartan geometries. Let .M; B; !/ be a compact parabolic geometry modelled on X D G=P . Then either the geometry is flat, or there is a C 0 unimodular reductive geometry, subordinated to .M; B; !/, which is preserved by Aut.M; B; !/. Schoen’s theorem [34] on CR-structures, and more generally the results of [18], are evidences that the conjecture must be true when G is simple of rank one (here, regularity of the connection, a mild restriction on its curvature, is made).

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3 Some words about the proof of Theorem 1.2 Let us now explain the main ideas of the proof of Theorem 1.2. In dimension 2, the theorem is a consequence of the uniformization theorem of Riemann surfaces. We will explain the proof when dim.M / 3. ByAlekseevski’s theorem, the assumption on the essentiality of the conformal group can be replaced, by the assumption that this group does not act properly. The first, and most difficult step in the proof, is to show that under this hypothesis the manifold has to be conformally flat, i.e., every point x of M has a neighbourhood Ux conformally equivalent to an open subset V of the Euclidean space Rn . If we know that .M; g/ is conformally flat, then we also know since Kuiper that z stands for the universal cover of z ! Sn (where M there is a conformal immersion ı W M z ; g/ M ), called developping map, as well as a morphism W Conf.M Q ! PO.1; n C 1/ z ; g/ satisfying the equivariance relation ı ı D . / ı ı. When Conf.M Q does not act z properly, the dynamics of .Conf.M ; g// Q PO.1; n C 1/ allows to understand the map ı (see for example [27], [19]), and we get: Proposition 3.1. Let .M; g/ be a Riemannian manifold which is conformally flat. If the conformal group of .M; g/ does not act properly on M , then the developping map z ! Sn is a diffeomorphism on Sn , or on Sn n fpg for some point p 2 Sn . ıW M Notice that at the beginning of the seventies, notions like .G; X /-structures, and tools like developping maps were not yet very “popular”, so that this part of the proof in [32], for example, is not really correct. On a general pseudo-Riemannian manifold .M; g/ of dimension n 3, the conformal flatness is detected by tensorial conditions. The Weyl tensor is the .1; 3/ tensor given by the formula

1 S S gg Ric g g: W DR 2n.n 1/ n2 n Here R, Ric and S stand for the Riemann, Ricci and Scalar curvature associated to g, and h q stands for the Kulkarni–Nomizu product of two symmetric 2-tensors (see [6], p. 47). In dimension n 4, the vanishing of the Weyl tensor is equivalent to the manifold .M; g/ being conformally flat. In dimension 3, the Weyl tensor always vanishes, but another tensor substitutes it. The Schouten tensor on .M; g/ is given by

SD

S 1 Ric g : n2 2.n 1/

Then one defines the Cotton tensor by C.X; Y; Z/ D .rX S /.Y; Z/ .rY S /.X; Z/. In dimension 3, the tensor C vanishes if and only if .M; g/ is conformally flat. Thanks to Proposition 3.1, Theorem 1.2 is proved if one can show that essentiality implies conformal flatness. If we suppose that the manifold M is compact, there is a


trick to do that. Indeed, on a manifold which is not conformally flat one can build the 2 following nontrivial singular metric hg D kW kg g (resp. kC k 3 g when dim.M / D 3). One checks that the conformal group acts by isometries for this singularR metric. In fact, this singular metric defines a singular distance dh .x; y/ D inf hg . 0 ; 0 /, the infimum being taken over all ’s joining x to y. If K denotes the closed subset on which the Weyl tensor (resp. the Cotton tensor in dimension 3) vanishes, and if K D fx 2 M; dh .x; K/ g, then .K ; dh / is, for sufficiently small, a nonempty, compact, non-singular metric space, left invariant by the conformal group. One then infers that the conformal group of .M; g/ is compact, and thus inessential (see [19] for details). For a noncompact Riemannian manifold .M; g/ the previous demonstration breaks down, and far more involved tools must be used. It is J. Ferrand who first gave a correct proof in the noncompact case. For this she introduced conformal invariants, which allowed her to understand the global dynamical behaviour of sequences of conformal transformations which do not act properly on .M; g/. Let H.M / (resp. H0 .M /) denote the space of continuous functions on M (resp. continuous with compact support) with an Ln -integrable differential distribution. Thanks to the metric g, the latter can be considered as a gradient vector field. Now, if C1 and C2 are closed connected sets of M , let A.C0 ; C1 / denote the set of functions u 2 H.M / such that R u Dn0 on C0 and u D 1 on C1 . Then one defines Cap.C0 ; C1 / D inf u2A.C0 ;C1 / M jruj d Volg . Notice that Cap.C1 ; C2 / is invariant by conformal change of metric g ! e 2 g. If .x; y; z/ 2 M 3 with z 6D x and z 6D y, J. Ferrand defines .x; y; z/ D inf C0 ;C1 Cap.C0 ; C1 /, for C0 a noncompact closed connected set containing z, and C1 a compact connected set containing x and y. Ferrand then proves that noncompact Riemannian manifolds split into two classes. On the first class she can define a conformally-invariant distance. So the conformal group of manifolds of the first class acts properly, and these manifolds are inessential. For manifolds which are not in this class, the function can be extended to W .M M MO / n ! RC [ fC1g (where MO is the Alexandroff compactification of M , and is the diagonal) and satisfies: – .x; y; z/ D 0 if and only if y D x or z D 1, – when x 6D y, .x; y; z/ D C1 if and only if z D x or z D y. Now let us consider a noncompact essential Riemannian manifold .M; g/. By Alekseevski’s theorem, the action of the conformal group of .M; g/ is nonproper, and thus we can find a sequence .xk / of M converging to x1 , and a sequence .fk / of conformal transformations, leaving every compact subset of Homeo.M /, and such that yk D fk .xk / converges to y1 2 M . Now three cases have to be considered: (i) There is a subsequence of .fk /, also noted .fk /, and a converging sequence zk ! z1 , z1 6D x1 , such that fk .zk / ! w1 , w1 6D y1 . (ii) There is a subsequence of .fk /, also noted .fk /, and a converging sequence zk ! z1 , z1 6D x1 , such that fk .zk / ! y1 .

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(iii) For any converging sequence zk ! z1 , z1 6D x1 , fk .zk / ! 1. In the first case, Ferrand proves that .fk / has a subsequence converging in Conf.M; g/, which contradicts the hypothesis on .fk /. Now, if we are in the case (ii), we look at a converging sequence ak ! a1 , a1 2 M , and, looking at subsequence if necessary, we suppose that fk .ak / tends to b1 2 MO . Then .x1 ; z1 ; a1 / > 0, and .x1 ; z1 ; a1 / D limk!C1 .xk ; zk ; ak /. On the other hand, if b1 6D y1 , then .x1 ; z1 ; a1 / D limk!C1 .fk .xk /; fk .zk /, fk .ak // D .y1 ; y1 ; b1 / D 0 by the conformal invariance of , yielding a contradiction. We thus must have b1 D y1 . This implies the following dynamical property for .fk /: For any open subset U M with compact closure in M , and for any > 0, fk .U / lies in the g-ball Bg .y1 ; / with center y1 and radius , for k sufficiently big. But such a dynamical behaviour implies that .M; g/ is conformally flat. Indeed, let n R us suppose that dim.M / 4, and look at the integral U kW kg2 d Volg . This inten R gral is conformally invariant, so that by the previous assertion, U kW kg2 d Volg n R 2 Bg .y1 ;/ kW kg d Volg , and this holds for every > 0 arbitrary small. This implies n R that U kW kg2 d Volg D 0, and finally W D 0 on U . In dimension 3 one considers n

kC k instead of kW kg2 to conclude the proof. The invariant allows also to conclude in case (iii). Let us consider any sequence ak ! a1 in M . Since fk .zk / tends to 1, we see that .yk ; ak ; fk .zk // tends to .y1 ; a1 ; 1/, namely to 0. But by conformal invariance .yk ; ak ; fk .zk // D .xk ; fk1 .ak /; zk /. Since any cluster value of .xk ; fk1 .ak /; zk / has to be 0, we infer that the only possible cluster values for .fk1 .ak // are 1 and x1 . But the set of cluster values of .fk1 .ak // over all the convergent sequences .ak / has to be connected, because M is, and since fk1 .yk / ! x1 , we infer that for any convergent sequence ak ! a1 , fk1 .ak / ! x1 . The end of the proof of case (ii), when applied to .fk1 /, yields the conclusion W D 0 on M .

4 Conformal dynamics on compact manifolds Given a pseudo-Riemannian compact manifold .M; g/ and a conformal transformation f , can we describe all the possible dynamical patterns for the dynamics of .f k / on M ? In the Riemannian case, Theorem 1.2 allows a complete description. Proposition 4.1. Let .M; g/ be a compact Riemannian manifold, and let f be a conformal transformation. Then two cases can occur: (i) The sequence .f k / is contained in a group Isom.M; e 2 g/ for some 2 C 1 .M /. In this case, for every x 2 M , the closure of .f k .x// in M is a torus on which f acts by translation. (ii) The manifold .M; g/ is conformally diffeomorphic to .Sn ; gcan /, and under this


identification, .f k / is a non relatively compact sequence of Möbius transformations. To complete the dynamical description, let us recall that Möbius transformations generating a non relatively compact group have a so called North–South dynamic: Lemma 4.2. Let .fk / be a sequence of PO.1; n C 1/ tending to infinity .i.e. leaving every compact subset of PO.1; n C 1//. Then there exist two points p C and p on Sn such that, considering a subsequence of .fk / if necessary, (i) for every x 2 Sn n fp g, limk!C1 fk .x/ D p C , the convergence is uniform on every compact subset of Sn n fp g; (ii) for every x 2 Sn n fp C g, limk!C1 .fk /1 .x/ D p , the convergence is uniform on every compact subset of Sn n fp C g. We see that the dynamics of essential sequences .f k / (case (ii) of the proposition, described in the lemma) is a North–South dynamics, qualitatively very different from that of isometric sequences (case (i) of the proposition). More interesting, the proof of Theorem 1.2 done by J. Ferrand (and in fact all existing proofs) consists roughly in showing that an essential conformal transformation on a Riemannian manifold, must have a North–South dynamics. And the key point, at the end of the proof, is to observe that a North–South dynamics forces the Weyl tensor to vanish, and the geometry to be conformally flat (see the end of the previous section). So, an answer to Lichnerowicz’s conjecture for general pseudo-Riemannian signatures should begin by a good understanding of the dynamics of essential transformations. A central question is: Question 4.3. Let .M; g/ be a pseudo-Riemannian manifold. What are the qualitative differences between the dynamics of isometries on M and the dynamics of essential conformal transformations? We are far from having an answer to this question, but at least for the Lorentzian signature we have some hints of what the answer could be. To better understand the question, our first task is to exhibit and study quite a lot of compact Lorentzian manifolds having essential conformal transformations. That is what we are going to do in the two next sections.

5 The conformal model space in Lorentzian geometry 5.1 Geometry of Einstein’s universe. Just as the standard sphere is a central geometrical object in conformal Riemannian geometry, there is in the Lorentzian framework a distinguished compact conformal space. This space is called Einstein’s universe, and it is so important from the geometrical and dynamical point of view that we must, at least briefly, describe it. A more detailed description can be found in [16], [15]. For the physical point of view, see [21].

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We write R2;n for the space RnC2 endowed with the quadratic form q 2;n .x/ D 2 . The isotropic cone of q 2;n is the subset of 2x0 xnC1 2x1 xn C x22 C C xn1 2;n 2;n R on which q vanishes. We denote this isotropic cone, with the origin removed, by C 2;n , and denote by the projection from R2;n minus the origin on RP nC1 . The set .C 2;n / is a smooth hypersurface † of RP nC1 . This hypersurface turns out to be endowed with a natural Lorentzian conformal structure. Indeed, for any x 2 C 2;n , the restriction of q 2;n to the tangent space Tx C 2;n , denoted by qO x2;n , is degenerated. Its kernel is just the kernel of the tangent map dx . Thus, pushing qO x2;n by dx , we get a well-defined Lorentzian metric on T.x/ †. If .x/ D .y/, then the two Lorentzian metrics on T.x/ † obtained by pushing qO x2;n and qOy2;n are in the same conformal class. Thus, the form q 2;n determines naturally a well-defined conformal class of Lorentzian metrics on †. In fact, one can check that the manifold † is the quotient of S1 Sn1 by the product of the antipodal maps, and the natural conformal structure is induced on this quotient by the conformal class of the metric dt 2 C gcan . The manifold †, together with its canonical conformal structure, will be called Einstein’s universe and is denoted by Einn . Notice also that the metric induced on † by dt 2 C gcan , gives rise to a smooth n-form, called Vol on †. If .X1 ; : : : ; xn / is a (local) smooth field of orthonormal frames on †, then Vol.X1 ; : : : ; Xn / D 1. 5.1.1 Conformal group and Liouville’s theorem. From the very construction of Einn it is clear that the group PO.2; n/ acts naturally by conformal transformations on Einn . It turns out that PO.2; n/ is the full conformal group of Einn . Moreover, there is a Liouville theorem, asserting that any conformal transformation between connected open subsets of Einn is the restriction of a unique transformation of PO.2; n/ (this theorem is proved in [7]). 5.1.2 Lightlike geodesics and lightcones. The projection on Einn of the intersection of C 2;n with linear subspaces of R2;n will yield various interesting geometrical objects of Einn . For example, the projection on Einn of the intersection of C 2;n with null 2-planes of R2;n (resp. degenerate hyperplanes of R2;n ) is called a lightlike geodesic (resp. lightcones) of Einn . The lightlike geodesics are smooth circles. The lightcones are the sets of lightlike geodesics passing through the same point p, the vertex of the lightcone. If p 2 Einn , we will call C.p/ the lightcone with vertex p. A lightcone C.p/ has always a singularity at its vertex p (this singularity is locally that of a lightcone in Minkowski’s space, as we will see later), but C.p/ n fpg is smooth, diffeomorphic to R Sn2 . 5.1.3 Complementary of a lightlike geodesic. Let Einn be a lightlike geodesic. We call the complementary of in Einn . Since this kind of open subsets will play an important role in the following, we recall some of their main geometrical properties. Open sets like admit a natural foliation by degenerate hypersurfaces, and this


foliation H is preserved by the whole conformal group of . This foliation can be described as follows. Given a point p 2 , we consider the lightcone C.p/ with vertex p. Since is a lightlike geodesic, we have C.p/. Now, the intersection of C.p/ with is a degenerate hypersurface of , diffeomorphic to Rn1 . We denote it by H .p/. If p 6D p 0 , C.p/ and C.p 0 / only intersect along , so that H .p/ \ H .p 0 / D ;. We thus get a foliation H whose leaves are the H .p/’s, for p 2 . We also get a smooth fibration W ! defined as follows: for every x 2 , .x/ is the unique p 2 such that x 2 H .p/. 5.1.4 Complement of a lightcone: stereographic projections. Let us identify Minkowski’s space R1;n1 with the subspace of R2;n spanned by e1 ; : : : ; en , and let us denote by h ; i the restriction of q 2;n to Span.e1 ; : : : ; en /. We define sN W R1;n1 ! C 2;n ;

x 7! hx; xie0 C 2x C enC1 :

The map s D ı sN is a conformal embedding of R1;n1 into Einn , and is called stereographic projection. The image s.R1;n1 / is the complement in Einn of the lightlike cone with vertex p1 D .e0 /. This cone is called cone at infinity and denoted by C1 . To better understand the way R1;n1 compactifies, the following lemma is useful (a proof is given in [14], p. 53): Lemma 5.1. After identifying R1;n1 Einn n C1 thanks to the stereographic projection s, one has the following. (i) Let u be a timelike or a spacelike vector, and a C R u an affine straightline of R1;n1 . Then lim t!˙1 .a C t u/ D p1 . (ii) To each lightlike direction u of R1;n1 is associated a unique lightlike geodesic u C1 , such that the leaves H .p/, p 2 n fp1 g, are the image through the stereographic projection of the affine hyperplanes a C u? . • For any a 2 R1;n1 , lim t!˙1 .a C t u/ D u .a/. • Two straightlines aCRu and b CRu “hit” C1 at the same point of nfp1 g if and only if they belong to a same degenerate affine hyperplane of R1;n1 . To summarize the last part of the lemma, let us say that any lightlike affine straightline of Minkowski’s space (still identified with Einn n C1 ) compactifies in Einn as a lightlike geodesics. Two lightlike affine straightlines are parallel if and only if they hit C1 n fp1 g at points which are on a same lightlike geodesic of C1 . In particular, two lightlike straightlines meet at infinity if and only if they are parallel and in a same lightlike hyperplane. Given a lightlike geodesic Einn , the lemma helps in understanding the foliation H . Indeed, choose a point p1 2 . The “Minkowski component” Einn n C.p1 / is included in . Then the lemma says that, after identifying Einn n C.p1 / with R1;n1 thanks to a stereographic projection, the restriction of H to Einn n C.p1 / is just a foliation by parallel lightlike hyperplanes.

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Remark 5.2. The previous construction generalizes to any signature. If RpC1;qC1 is the 2 2 C CxnC2 , space RnC2 endowed with 2x0 xnC1 2x1 xn 2xp xnpC1 CxpC1 nC1 the projection of the isotropic cone on RP is a smooth manifold, endowed with a conformal structure of signature .p; q/, and with conformal group PO.p C 1; q C 1/. This space is called Einp;q (in Lorentzian signature, we write Einn instead of Ein1;n1 ). Notice that Einp;q is finitely covered by the product Sp Sq , endowed with the product p;q can metric gScan is the conformal compactification of Rp;q . In p gSq . The space Ein particular, it is conformally flat and turns out to be the universal model for conformally flat manifolds (i.e., the universal cover of every conformally flat manifold of signature .p; q/ admits a conformal immersion in Einp;q ). 5.2 Examples of essential dynamics on Einstein’s universe. We are going to show that Einstein’s universe has a lot of essential (and in fact strongly essential) conformal transformations. In the following we fix p1 2 Einn , C1 D C.p1 / and a stereographic projection identifying Einn n C1 with R1;n1 . In this way, any conformal transformation of R1;n1 can be regarded as a conformal transformation of Einn n C1 , and by Liouville’s theorem, extends in a unique way to a conformal transformation of Einn , i.e., to an element of PO.2; n/. So we will always, without further precision, look at conformal transformations of R1;n1 as elements of PO.2; n/ fixing p1 . 5.2.1 Dynamics of translations. Using Lemma 5.1, it is not difficult to understand the dynamics of translations of R1;n1 , when extended to the whole Einn . Lemma 5.3. Let T be a translation of R1;n1 by the vector u D .u1 ; : : : ; un /. Then, as an element of O.2; n/, 0 1 1 hT; e1 i hT; e2 i : : : hT; en i hT; T i B 1 0 ::: 0 2u1 C B C B : :: C : :: :: B : C C T DB B :: C : :: B : : C 0 B C @ 1 2un A 1 For any translation T , the differential of T at p1 is the identity. Let T be a timelike translation. Then T has p1 as unique fixed point in Einn . For every x 2 Einn n fp1 g, limn!˙1 T n x D p1 . Moreover, for any open subset U Einn n C1 with compact closure in Einn n C1 , limn!˙1 T n U D p1 .the convergence is to be understood with respect to the Hausdorff topology/, and limn!˙1 Vol.T n U / D 0. Let T be a spacelike translation. Then the fixed points of T are the points of a lightcone of codimension one in C1 . For any x 2 Einn which is not fixed by T , limn!˙1 T n x D p1 . Moreover, for any open subset U Einn n C1 with compact closure in Einn n C1 , limn!˙1 T n U D p1 , and limn!˙1 Vol.T n U / D 0.


Let T be a lightlike translation by the vector u. Then the fixed points of T are exactly the points of u . For any compact subset K Einn nu , limn!˙1 T n K D u .K/. Moreover, if U is an open subset of Einn n u with compact closure in Einn n u , then limn!˙1 Vol.T n U / D 0. The fact that the differential of a translation at p1 is the identity, proves that translations are essential conformal transformations of Einn . Indeed, an isometry of a C 1 Lorentzian connected manifold (or more generally pseudo-Riemannian manifold) fixing a point and with differential being the identity at this point must be the identical transformation. This is just because at a fixed point, an isometry is conjugate to its differential by the exponential map. Moreover, the fact that translations are volume-collapsing on open subsets of † for the volume form Vol proves that they cannot preserve any L1 metric in the conformal class of a smooth Lorentzian metric on †. Therefore, translations are strongly essential conformal transformations of Einn . 5.2.2 Dynamics of a homothety. Let h be a homothetic transformation of R1;n1 of ratio , h W x 7! x. We suppose jj < 1. We denote by p C the point of Einn corresponding to the origin in Minkowski’s space and set p D p1 . We denote C C and C the lightcones associated to p C and p respectively. Then: Lemma 5.4. The fixed points of h are p C , p and the points of C C \ C .a codimension 2 Riemannian sphere in Einn /. If x 2 C D Einn n C , then limn!C1 .h /n x D p C . If x 2 D Einn n C C , then limn!1 .h /n x D p . If U is an open subset of C .resp. / with compact closure in C .resp. in /, then limn!C1 Vol.hn .U // D 0 .resp. limn!1 Vol.hn .U // D 0/. Since the differential of h at p C is Id, it is clear that h cannot preserve any Lorentzian metric on † (of any regularity). This shows that h is a strongly essential transformation. 5.2.3 A last example of essential dynamics. We consider now the transformations introduced in Section 2, namely W .x1 ; : : : ; xn / 7! .e 2 x1 ; e x2 ; : : : ; e xn1 ; xn /. It is quite simple to check that, as an element of O.2; n/, can be written as 1 0 e C B e C: In2 DB A @ e e We suppose that < 0 and denote by C the lightlike geodesic, compactification of the straightline R en in Einn . We also write D e1 . Since en is not colinear to e1 , C and are two disjoint lightlike geodesics of Einn . Let us denote H C .p/, p 2 C (resp. H .p/, p 2 ), the leaves of the natural foliation on C (resp. of ) introduced in Section 5.1. Now, from basic linear algebra on R2;n , we see

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that if a lightlike geodesic of Einn is not contained in a lightcone, it intersects this lightcone at exactly one point. It follows that each leaf H C .p/ meats exactly once (resp. each leaf H .p/ meats C exactly once). So, there is a natural projection W C ! (resp. C W ! C ), which at each point of a leaf H C .p/ associates the intersection of with H C .p/. We then have: Lemma 5.5. The fixed points of the transformation are the points of C [ . For any open subset U with compact closure in it holds that limn!C1 n U D C .U / and limn!C1 Vol. n U / D 0. For any open subset U C with compact closure in C it holds that limn!1 n U D .U / and limn!1 Vol. n U / D 0 Once again, the volume-collapsing properties of . n /n2N show that cannot preserve any L1 metric in the conformal class of any smooth Lorentzian metric on †. It follows that the transformations are strongly essential. 5.3 Remarks. If we look at the dynamical patterns described above, we see that the dynamics of elements of PO.2; n/ is a little bit more complicated than dynamics of Möbius elements on Sn . Nevertheless, all these dynamics have a rough common pattern. There are attracting sets (these sets are points, or lightlike geodesics), which attract points of a dense open subset. Moreover, the volume form Vol is collapses on this dense open subset under the iterates of the essential transformation. We will focus on this point later on.

6 More complicated examples of compact essential Lorentzian manifolds 6.1 Schottky groups on Einstein’s universe. A subgroup O SL.2; R/, generated by g elements O1 ; : : : ; Og (g 2), is called a Schottky group, if there exist 2g pairwise disjoint half-discs of H2 , denoted by D1C ,…,DgC , D1 ,…,Dg , such that i .H2 nDi / D

DiC for every i 2 f1; : : : ; gg. By a half-disc of H2 , we mean a connected component of the complementary of a geodesic. The interested reader will find more details on Schottky groups in [29], for instance. For what follows, we will just precise that a Schottky group is always a free discrete subgroup of SL.2; R/. When it acts on @H2 ' S1 , a Schottky group has a closed invariant subset ƒ O , homeomorphic to a Cantor set, on which its action is minimal. O The action of O on O D S1 n ƒ O is proper discontinuous, and the quotient n

is O a finite union of circles. 2 We consider R2;n , endowed with q 2;n .x/ D 2x0 xnC1 2x1 xn C x22 C C xn1 , and denote by T0 the projection on Einn of the subspace spanned by .e0 ; e1 ; en ; enC1 /. Since this subspace has signature .2; 2/, T0 is a sub-Einstein’s universe of dimension 2. Conformally, it is just the product S1 S1 with the conformal class of the metric dxdy. On T0 there are two foliations by lightlike geodesics. The first, F1 , has the leaves fxg S1 , and the leaves of the second, F2 , are the S1 fyg.


We now introduce two representations R and L of SL.2; R/ in O.2; n/ defined in the following way: For every A D ac db in SL.2; R/, ! A In2 L .A/ D A and ! bI2 aI2 : R .A/ D In2 cI2 dI2 Notice that L .A/ (resp. R .A/) preserves T0 D S1 S1 and acts projectively by A on the left factor (resp. the right factor) and trivially on the other. In particular, L .A/ (resp. R .A/) leaves every lightlike geodesic of F2 (resp. of F1 ) invariant. Let us now consider a Schottky group O in SL.2; R/, generated by sO1 ; : : : ; sOg . We O and si D L .Osi /. The set ƒ D ƒ O S1 T0 is a closed invariant set D L ./ subset for the action of on Einn . Moreover, we proved in [16], [15]: Theorem 6.1. The action of is proper on D Einn n ƒ . The quotient manifold n is compact, diffeomorphic to S1 .S1 Sn2 /.g1/] , and inherits from Einn a Lorentzian conformal structure. Here .S1 Sn2 /.g1/] stands for the connected sum of .g 1/ copies of S1 Sn2 . The conformal group of this structure is induced by O.n 2/ R .SL.2; R// and is strongly essential. We will explain in the following section why the structures constructed in this way are essential. For the moment, let us just make some remarks on Theorem 6.1. This theorem tells us that the answer to Question 2.2 is negative, and there is no hope to have, in the Lorentzian framework, such a strong statement as the Ferrand–Obata theorem for conformal Riemannian geometry. Indeed, assumption of essentiality is no more sufficient to fix the topology of the manifold, even in the compact case. Moreover, starting from two Schottky groups O 1 and O 2 in SL.2; R/, with the same number g of generators, but which are not conjugate in SL.2; R/, we get two groups 1 D L .O 1 / and 2 D L .O 2 / which are not conjugate in O.2; n/. This gives non conformally equivalent Lorentzian structures on S1 .S1 Sn2 /.g1/] , which are both essential. So, even when one fixes the topology (here, for example, S1 .S1 Sn2 /.g1/] ), there still can be a non trivial moduli space of conformal structures which are essential. 6.2 More complicated essential dynamics. We keep the notations of the previous O We section: O is a Schottky group of SL.2; R/ with g generators, and D L ./. denote M D n . t O t / and t D R . O t /, where O t D 1 t Let us consider the two flows D . R 01 t 0 . Since t and t both centralize and leave invariant, they and O t D e t 0 e

induce two conformal flows N t and N t on M . We are going to show that N t and N t are two strongly essential flows on M .

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6.2.1 Dynamics of the flow N t on M . As a flow of O.2; n/, t can be written as 0 1 1 t B0 1 C B C t B C: In2 DB C @ 1 tA 0 1 By the matrix expression given in Lemma 5.3, we recognize here a “lightlike translation flow”, as already studied in Section 5.2. Let us recall the dynamical properties of t on Einn . The flow t fixes all the points of a lightlike geodesic 0 T0 in F2 (and 0 is exactly the set of fixed points of t ). Any lightlike geodesic of Einn , passing through a point p 2 0 , is preserved by t . t If such a is different from 0 , then as a parabolic transformation (i.e., 1 t acts on the action is conjugate to that of 0 1 on RP 1 ). Now, let us consider the projection W ! M . Since 0 is in F2 , it is transverse to ƒ , and thus 0 \ is a nonempty -invariant closed subset of . We get that .0 \ / is a finite union x 1; : : : ; x s . To see that N t is strongly essential, let us of closed lightlike geodesics pick a point x0 2 0 \ , and an open neighbourhood U of x0 , on which is injective. Let V be an open subset of U n 0 with compact closure in U n 0 . Then lim t!˙1 t V D 0 .V / and lim t!˙1 Vol. t V / D 0. In particular, there is a T0 such that for t > T0 , t V U . Write Ux D .U / (resp. Vx D .V /), and let us define a smooth volume form on Ux by Vol D . 1 / Vol. Then for t > T0 , N t Vx Ux and lim t!C1 Vol. N t Vx / D 0. This proves that N t is strongly essential. To have more intuition on how an essential flow of Lorentzian transformations behaves, we now describe more precisely the dynamics of N t on M . O the action of on 0 is conjugate to that of O on Since we defined to be L ./, S1 D @H2 . Thus ƒ \0 is homeomorphic to the Cantor set ƒ O . The complementary of this Cantor set in 0 is a family I of connected components. Since we supposed that n.0 \ / is a union of s closed lightlike geodesics, this means that S the action of on the family I has exactly s orbits. For each I 2 I, we define I D x2I .C.x/n0 /. This is an open subset of . In fact each I is the set of those S x 2 . n I / such that lim t!C1 t x D lim t!1 t x 2 I . The quotient n. I 2I I / is a finite union x s of open subsets of M . For each j 2 f1; : : : ; sg, one has the following x 1; : : : ;

x j as

x j D fx 2 M n x j j lim t!C1 N t x D dynamical characterization of

t x j g. lim t!1 N x 2 x x s is a dense open subset of M where the dynamics of N t is easy Thus 1 [ [

to understand. x x It remains to understand how the complement S S of 1 [ [ s in M looks like. The complement of I 2I I in is K D x2.ƒ \0 / .C.x/ \ /. This is a closed subset of and K \ . n 0 / is a lamination by lightlike hypersurfaces, transversally modelled on a Cantor set. xs x 1 [ [

Looking at the quotient Kx D nK, we get that Kx is the complement of

x x x x x x in M . The set K contains 1 [ [ s , and K n 1 [ [ s is a lamination L


by lightlike hypersurfaces, transversally modelled on a Cantor set. By the minimality of the action of O on ƒ O we get that each leaf of L is dense in K. 6.2.2 Dynamics of the flow N t on M . In O.2; n/ the flow 0 t

B DB @

et

t

can be written as

1 e

t

In2

et

C C: A e t

This is the third flow that we studied in Section 5.2. We already studied its dynamics on Einn in Lemma 5.5, and we keep the notations of this lemma. The lightlike geodesics C and are both fixed individually by the group . Thus, C \ and \ x C; : : : ; xC both project in M on a finite union of closed lightlike geodesics, s , and 1 x x 1 ; : : : ; s . The C \ (resp. \ ) can be written as the union of infinitely many connected components UI 2I I (resp. UJ 2J J ). For every J 2 J), S S every I 2 I (resp. we define I D x2I .C.x/nC / (resp. C D .C.x/n /). We observe that x2J J

I C for every I 2 I, and that . I / is a connected component of \ , just obtained from I by “sliding along the leaves of F1 ”. In particular, if I and I 0 are in the same -orbit, the same will be true for . I / and . I0 /. Reindexing x ’s, we will now suppose that if I projects on x C , then . / if necessary the j j I x . projects on j S x [ Now, we get that the quotient n I 2I I is a finite union of open subsets

1 x x x x ! [

xj W

s . Each j contains j , and induces a smooth fibration j x , whose fiber are smooth lightlike hypersurfaces, and such that for any x 2

x , j j lim t!1 N t x D xj .x/ . S xC xC Looking at n J 2J C J we get a finite union 1 [ [ s of open subsets. Each C C C C x contains x , and there is a smooth fibration x ! x C , whose fibers are

xj W

j j j j x C. xjC .x/ for every x 2

smooth lightlike hypersurfaces such that lim t!C1 N t x D j x ˙ . As in the previous example, It remains to describe what are the boundaries of the

j S S let us introduce K C D x2.ƒ \ / .C.x/ \ / and K D x2.ƒ \C / .C.x/ \

/. Then KxC D .K C / and Kx D .K / are two closed subsets of M , C x [ [ x xC xC x x containing s and 1 [ [ s respectively. The sets K nf1 [ [ s g 1 C C C x [ [ x s g are two laminations L and L by smooth lightlike and K n f 1 hypersurfaces, transversally modelled on a Cantor set. Each leaf of LC (resp. of L ) x C in M is K C and the is dense in K C (resp. in K ). Finally, the closure of each

j x in M is K . closure of each

j 6.2.3 Interpretation of the previous examples in dimension 3. Let us now say a little bit more about the previous examples when the construction is performed on Ein3 .

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In this case, Ein3 n T0 carries an action of SL.2; R/ SL.2; R/ by L .SL.2; R// L .SL.2; R//. This allows to identify Ein3 nT0 with SL.2; R/, endowed with a Lorentzian conformal structure invariant by the action of SL.2; R/ SL.2; R/ by left and right multiplications. Now Ein3 n T0 is a dense open subset of , and projects to a dense open subset N M . The manifold N is n SL.2; R/ and can thus be identified with a two-fold cover of T 1 .nH2 /, the unit tangent bundle of the noncompact hyperbolic surface nH2 . The Killing form of SL.2; R/ induces on N D n SL.2; R/ a Lorentzian metric of constant curvature 1, which is preserved by the right action of SL.2; R/. Our manifold M can thus be understood as a conformal compactification of N . This compactification is made thanks to a finite union n.T0 \

/ D T1 [ [ Ts of Lorentzian tori. Nt Nt How can weinterpret the flows and ? On n SL.2; R/, the right multiplication 1 t t 0 ) can be considered as the action of the horocyclic flow (resp. by 0 1 (resp. e0 et 1 geodesic flow) on T .nH2 / (once again, up to a two-fold cover). So the flows N t and N t can be regarded as the extension to M of the horocyclic and geodesic flows on N . Notice that the action of N t and N t is inessential on N (as we saw, those flows preserve a Lorentzian metric with constant curvature in the conformal class), but become essential when extended to the conformal compactification M .

7 Essential versus isometric dynamics Now that we have some examples of essential conformal transformations on compact Lorentz manifolds, we can try to guess what could be the answer to Question 4.3, and isolate what are the dynamical properties which distinguish essential actions from inessential ones. A useful notion will be the following. Definition 7.1. Let M be a compact manifold, x0 in M , and .fk / a sequence of homeomorphisms of M . Let us denote by ƒfk .x0 / the set of cluster points of fk .x0 /. Then the sequence .fk / is said to be equicontinuous at x0 , if for every sequence xk tending to x0 the set of cluster points of fk .xk / is also ƒfk .x0 /. An homeomorphism f of M is said to be equicontinuous at x0 if the sequence .f k / is equicontinuous at x0 . Stated briefly, a transformation f is equicontinuous at x0 if the following phe0 0 (x1 6D x1 ) nomenon does not occur: f k .x0 / tends to x1 whereas f k .xk / tends to x1 for a sequence xk tending to x1 . The dynamical study of Lorentzian isometries on a compact manifold gave rise to a great amount of works: [1], [36], [38], [10], [9], [24] among others. One of the basic properties of Lorentzian isometric dynamics is: Theorem 7.2. Let .M; g/ be a compact Lorentz manifold. Let f be an isometry of .M; g/, such that .f k /k2Z does not have compact closure in Isom.M; g/. Then f is nowhere equicontinuous on M .


On the contrary, the dynamical behaviour of the essential transformations we met until now offered a quite different picture. For such an essential conformal transformation f on M , there always existed: – two finite families of closed subsets F1C ; : : : ; FsC and F1 ; : : : ; Fs (this two families being sometimes the same), playing the role of attracting and repelling sets. In the example we had, these sets were finite union of points, or finite union of closed lightlike geodesics. – a family of open subsets . ij /i;j 2f1;:::;sg , endowed with S continuous projections ijC W ij ! FiC and ij W ij ! Fj , and such that i;j 2f1;:::;sg ij is a dense open set of M . The dynamical behaviour of f k on ij was described by the fact that for any compact subset K ij , limk!C1 f k .K/ D ijC .K/ and limk!1 fSk .K/ D ij .K/. In particular, in all the examples we met, f was equicontinuous on i;j 2f1;:::;sg ij . This dynamical pattern could be a general picture for essential transformations, and it would distinguish them from inessential ones. Let us formulate the following dynamical conjecture: Conjecture 7.3. Let .M; g/ be a compact Lorentz manifold. Let f be an essential conformal transformation of .M; g/. Then f is equicontinuous on a dense open subset of M . 7.1 Stable conformal dynamics and its consequences on the geometry. We would like now to explain why Conjecture 7.3 is linked to the generalized Lichnerowicz conjecture (at least for the Lorentzian signature). We are going to see that when an essential conformal transformation f on a compact Lorentzian manifold .M; g/ is equicontinuous on a dense open subset, then it imposes some constraints on the geometry of this open subset. We will see in the following section that sometimes these constraints force the dense open subset (and hence the whole M ) to be conformally flat. Most of the ideas presented here are based on [17]. 7.1.1 Stable conformal transformations. Equicontinuity implies properties on the differential maps Df k which are more tractable technically. We consider a sequence .fk / of conformal diffeomorphisms of a Lorentz manifold .M; g/. We suppose that there is x0 2 M such that xk D fk .x0 / has a limit point x1 2 M . We then choose smooth frame fields x 7! .E1 .x/; E2 .x/; : : : ; En .x// and y 7! .F1 .y/; F2 .y/; : : : ; Fn .y// in neighbourhoods of x0 and x1 respectively. We suppose further that .E1 .x/; : : : ; En .x// and .F1 .y/; : : : ; Fn .y// satisfy gx .E1 .x/; E2 .x// D 1 (respectively gy .F1 .y/; F2 .y// D 1 ) and gx .Ei .x/; Ei .x// D 1, i 3 (respectively gy .Fi .y/; Fi .y// D 1, i 3 ), all the other products being zero. The differential Dx0 fk , read in the frames .E1 .x0 /; : : : ; En .x0 // and .F1 .fk .x0 //; : : : ; Fn .fk .x0 /// yields a matrix Mk .x0 / in R O.1; n 1/. The projection on the R-factor is just the square root of the conformal distorsion, namely e k .x0 / , if f g D e 2k g.

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We now use the Cartan decomposition O.1; n1/ D KAK, where K is the maximal compact subgroup of O.1; n 1/, namely K D O.1/ O.n 1/, and A is a maximal abelian subgroup in O.1; n 1/. We perform a Cartan decomposition of the sequence .k/ Mk .x0 /, so that Mk .x0 / can be written as a product L.k/ 1 .x0 /Dk .x0 /L2 .x0 /. The .k/ two matrices L.k/ 1 .x0 / and L2 .x0 / are in K and Dk .x0 / is a diagonal matrix of the form 0 k .x0 / 1 e k .x0 / e B C B C 1 e k .x0 / B C @ A :: : 1 with k .x0 / 0. In what follows we will use the notation ıkC .x0 / D k .x0 / C k .x0 / and ık .x0 / D k .x0 / k .x0 /. Remark that one always has ık k ıkC . Now, given a sequence .fk / such that fk .x0 / ! x1 , we will say that this sequence is simple if C

(i) e k .x0 / , e k .x0 / , e ık .x0 / and e ık .x0 / all have a limit in R [ fC1g when k ! C1; .k/ (ii) the two sequences L.k/ 1 .x0 / and L2 .x0 / converge in K. Every sequence .fk / admits a simple subsequence. In [36], A. Zeghib performed the dynamical study of sequences of isometries of a compact manifold and introduced the following notion of stability. Definition 7.4. Let .fk / be a simple sequence of conformal transformations of .M; g/ such that fk .x0 / ! x1 . The stable space at x0 for the sequence .fk / is defined as the subspace Hxk dxi2 / where x1 ; : : : ; xn are cartesian coordinates and ' 6D 0 is a scalar function. Let hx; xi denote the pseudoEuclidean scalar product of the point x D .x1 ; : : : ; xn /. We define the following mapping: x 7! y D .y0 ; : : : ; ynC1 / WD '2 hx; xi C 1 ; 'x1 ; : : : ; 'xn ; '2 hx; xi 1 : Then the following conditions are easily checked: 1. .y0 ; : : : ; ynC1 / 6D .0; : : : ; 0/, 2. y lies in the null cone fy j hy; yi D 0g, 3. the induced metric of this immersion is X X X X dyi2 C dyi2 D ' 2 dxi2 C dxi2 : ik

i>k

ik

i>k

The sphere inversion appears essentially as the mapping ynC1 7! ynC1 . Note that the Riemannian case k D 0 is included; in this case the image does not meet the hyperplane y0 D 0. However, the ambient space is EnC2 . 1 With respect to the pseudo-Euclidean metric, the mapping x 7! y is conformal in any case, independent of '. This motivates the following definition of a conformal development map into the real projective space RP nC1 . Definition 3.2 ([72], see also [4]). The conformal development map on a conformally flat pseudo-Riemannian manifold .Mkn ; g/ is defined locally by x 7! y 7! Œy0 ; : : : ; ynC1 2 RP nC1 . If M is simply connected this induces a conformal immersion ı W M ! Qkn RP nC1 , the so-called conformal development. Here Qkn denotes the projective quadric fy j hy; yi D 0g. Qkn can also be regarded as the conformal compactification of Enk . One observes that ı.Enk / D fŒy0 ; y; ynC1 2 Q j ynC1 6D y0 g where the equation fynC1 D y0 g describes the ‘points at infinity’. The quadric Qkn is diffeomorphic with fhy; yi D 0g \ S nC1 Š S k S nk modulo the identification of antipodal pairs of points. Topologically, Qkn can also be regarded as a sphere bundle over RP k if k nk cf. [24], a Euclidean model is the tensor product S k ˝ S nk R.kC1/.nkC1/ . In the special case of Minkowski 4-space R41 the Lie group U.2/ can be considered as the conformal compactification of u.2/ via the Cayley map ı W u.2/ ! U.2/, ı.x/ D .1 C x/.1 x/1 , see [23]. Lemma 3.3. The conformal transformations of the projective quadric Qkn are in 1-1correspondence with those projective transformations of RP nC1 preserving Qkn .

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This lemma is essentially due to Möbius for the classical case k D 0; n D 2 (compare the Möbius geometry). For arbitrary dimensions it is stated in [72]. Theorem 3.4 (Kuiper [72]). If M is simply connected and conformally flat then ı W M ! Qkn is globally defined. If moreover M is compact then ı is either a diffeomorphism between M and S n .if k D 0/ or a two-fold covering .if 2 k n 2/. For k D 1 or k D n 1 the universal covering is non-compact. A conformally flat manifold M is called developable if the conformal development map ı is globally defined. Any simply connected conformally flat manifold is developable. Examples. The key examples of conformal vector fields on pseudo-Euclidean space are again the vector fields V1 and V2 above, extended to Qkn by taking limits of the flow. The fixed points of the flow ˆ.1/ t are the two isolated points Œ1; 0; 1 (the origin) and Œ1; 0; 1 (its image under the inversion at the unit sphere) and the null cone at infinity n2 f0g Qk1 f0g. The fixed point set of ˆ.2/ depends on the type of the translation vector x0 : If t hx0 ; x0 i 6D 0 then the only fixed point is Œ1; 0; 1 D 1. This is a perfect analogue of the conformal flow on the standard sphere with one fixed point. If hx0 ; x0 i D 0 this is different. In this case let I denote the conformal inversion I.x/ D jxjx 2 . Then the

.2/ conjugation I ı ˆ.2/ t ı I of the 1-parameter group ˆ t leads to the vector field V3 .x/ D d I ı ˆ.2/ j t ı I D 2hx; x0 ix C hx; xix0 . This is a third type where the gradient dt tD0 of the conformal factor satisfying LV3 g D 2g is a parallel and isotropic vector. The fixed point set is one isotropic line in the null cone. The flat metric of Minkowski 4-space in coordinates .u; v; x; y/ can be written as g D 2dudv C dx 2 C dy 2 . If we choose especially the isotropic translational vector x0 D 12 @v then this vector field V3 takes the form V3 .u; v; x; y/ D u2 ; 12 .x 2 C y 2 /; ux; uy . This is the standard type of a so-called special conformal vector field, see Section 8.

Corollary 3.5. On the conformal compactification Qkn there exists a conformal vector n2 field Vx2 with one zero, and on Qkn n Qk1 there exists a conformal vector field Vx1 with two zeros. These vector fields are essential and complete. Vx1 is a local gradient field, Vx2 is not a gradient field near the zero. n2 Qkn n Qk1 is nothing but the union of ı.En / and its image under inversion at the n2 ‘unit sphere’. This inversion transforms Vx1 into Vx1 . This space Qkn n Qk1 is not simply connected. In fact, its fundamental group is isomorphic to the integers Z if 2 k n 2, leading to a Z-sheeted universal covering which carries a conformal vector field with infinitely many zeros. n2 Corollary 3.6 ([63]). For 2 k n 2 the universal covering of Qkn n Qk1 defines a manifold M.Z/ together with a conformal structure such that ı W M.Z/ ! n2 becomes a conformal covering. The conformal vector field Vx1 can be lifted Qkn n Qk1 to a vector field V1.Z/ with infinitely many zeros. These zeros are in natural bijection

Conformal transformations of pseudo-Riemannian manifolds

269

to .2Z/ [ .2Z C 1/ Š Z. Similarly, there are intermediate coverings with any even number of zeros of the vector field. For 2 k n 2 the universal covering of the quadric Qkn itself is diffeomorphic to S k S nk . The metric can be chosen as the product of two metrics of constant curvature with opposite signs. This space carries a conformal vector field V2.2/ with two zeros as the lift of Vx2 via the conformal covering ı W S k S nk ! Qkn . Even if we remove one of the zeros, the vector field is still complete. The punctured S k S nk carries a complete conformal vector field with one zero. In a neighborhood of a zero of a conformal gradient field the metric is conformally flat, see Section 7.

4 The Riemannian case In the case of a Riemannian manifold any conformal vector field without zeros can be made into an isometric vector field by a conformal change, see Lemma 2.1. Such a field is called inessential, otherwise it is essential. Since much is known about the isometry groups and Killing fields, it is here more interesting to study essential conformal vector fields, that is, conformal vector fields which never become isometric under a global conformal change of the metric. Theorem 4.1 (Essential conformal vector fields). 1. (Alekseevskii [1], Ferrand [34], [35], Yoshimatsu [108]) Assume that .M; g/ is a Riemannian manifold of dimension n admitting a complete and essential conformal vector field. Then .M; g/ is conformally diffeomorphic with either the standard sphere S n or with the Euclidean space En . 2. (Obata [90], Lelong-Ferrand [77], Lafontaine [76]) Assume that .M; g/ is a compact Riemannian manifold of dimension n admitting an essential conformal vector field. Then .M; g/ is conformally diffeomorphic with the standard sphere S n . Three key steps in the proof are the following: 1. The zeros of the vector field are isolated. 2. In a neighborhood of a zero the manifold is conformally flat. 3. The conformal development map ı W M ! S n is injective. Several steps in the proof were made more precise in various papers, so the result cannot really be attributed to a single person, compare [45]. The case of a complete manifold carrying a complete and closed essential conformal vector field was solved by Bourguignon [19]. No analogous result seems to be known yet in the case of a pseudo-Riemannian manifold with an indefinite metric. It is a conjecture that a compact and pseudo-Riemannian manifold carrying an essential conformal vector field is conformally flat, it is named Lichnerowicz’ conjecture in [38]. Essential conformal vector fields in Finsler geometry are investigated in [85].

270


The situation with respect to inessential conformal vector fields is totally different, even in the compact case and even under additional curvature restrictions. Example 4.2 ([33]). For any n there is a compact Riemannian n-manifold of constant scalar curvature admitting a conformal vector field without zeros. The simplest example of this kind for n D 4 is the product S 1 S 3 with the warped product metric g D dt 2 C .2 C cos t /g1pwhere g1 is the standard metric on the unit sphere. In this case the vector field V D 2 C cos t @ t is conformal (and inessential), see [26, p. 277]. There are similar examples g D dt 2 C .f .t //2 g in any dimension, with a periodic warping function f which can be explicitly given. It has to satisfy the ODE nf 2 C.n2/f 02 C2ff 00 D .n2/ where , are the constant (normalized) scalar curvatures of g, g , respectively. These examples can be extended to the case of a pseudo-Riemannian metric, see [65].

5 Conformal transformations of Einstein spaces Definition 5.1. A pseudo-Riemannian manifold of dimension n 3 is called an Einstein space if the Ricci tensor is a (necessarily constant) multiple of the metric tensor. In this case the metric is called an Einstein metric, and in the equation Ric D g the factor is called the Einstein constant. Hence D S=n where S is the scalar curvature. In General Relativity the case D 0 is precisely the case where the Einstein field equations hold for the vacuum. 4-dimensional Einstein spacetimes with nonvanishing Einstein constants are the de Sitter space and the anti-de Sitter space. Kasner started in the early 20s [53], [54] an investigation about conformal changes of Ricci flat metrics. In more generality, one can ask what happens to Einstein metrics under conformal change. Lemma 5.2. The following formula holds for any conformal change g 7! gN D ' 2 g: RicgN Ricg D ' 2 .n 2/ ' r 2 ' C ' ' .n 1/ jjr'jj2 g : (8) Moreover, if V is a conformal vector field with LV g D 2g then the formula LV Ric D .n 2/r 2 g holds and the following conditions are equivalent: 1. LV Ric D g for a certain function . 2. grad.div V / is conformal. 3. .r 2 /ı D 0.

(9)


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The first equation follows from the relationship between the two Levi-Civita conS associated with g and g: nections r; r N SX Y rX Y D X.log '/Y Y .log '/X C g.X; Y / grad.log '/: r Corollary 5.3. The Einstein property of a metric is in general not preserved under conformal changes. If g is an Einstein metric then the conformally transformed metric gN D ' 2 g is Einstein if and only if .r 2 '/ı D 0; that is, if the Hessian of ' is a scalar multiple of the metric tensor. This equation was already analyzed by Brinkmann [21], [22] in the 1920s. He was the first who proved that in the case g.grad '; grad '/ 6D 0 the metric g is a warped product. Furthermore, he proved that in the case g.grad '; grad '/ D 0 the metric has a specific form carrying a parallel isotropic vector field (now called a Brinkmann space) which in dimension four became later important in physics as a pp-wave, compare [95]. For solutions of the equation .r 2 '/ı D 0 see Section 7 below. Corollary 5.4. Assume that an Einstein space carries a conformal vector field V which is not homothetic or isometric. Then it carries also a conformal gradient field, namely, the gradient of div V . This gradient field does not vanish identically but it can happen that it is a parallel isotropic vector field, hence isometric. Theorem 5.5 (Brinkmann [22]). Assume that .M; g/ is an Einstein space of dimension n 3 admitting a non-constant solution f of the equation .r 2 f /ı D 0. Then the following hold: (a) Then around any point p with g.grad f .p/; grad f .p// 6D 0 the metric tensor is a warped product g D dt 2 C .f 0 .t //2 g where grad f D f 0 @t@ , D ˙1 and where the .n 1/-dimensional Einstein metric g does not depend on t . Moreover, f satisfies the ODE f 00 C f D 0 where denotes the normalized Einstein constant .such that D 1 on the unit sphere in any dimension/. (b) Furthermore, if g.grad f; grad f / D 0 on an open subset then grad f is a parallel isotropic vector field on that subset, and the metric tensor can be brought into @ the form g D dudv C g .u/ where grad f D @u D grad v and where the .n 2/-dimensional metric g .u/ is Ricci flat for any fixed u and does not depend on v. Consequently g itself must be Ricci flat. These coordinates u, v, xi , i D 1; 2; : : : ; n 2 are sometimes called Rosen coordinates. Corollary 5.6. In dimension n D 4 any Einstein space is of constant sectional curvature if it admits either a non-trivial conformal mapping onto some other Einstein space or if it admits a non-homothetic conformal vector field V such that grad.div V / is not parallel and isotropic. For a Riemannian 4-manifold the latter case cannot occur, and for a Lorentzian 4-manifold the latter case is the case of a Ricci flat pp-wave .or vacuum pp-wave/.

272


This result is due to Brinkmann [22], compare also [42]. See Section 8 below for a further discussion of the 4-dimensional Lorentzian case. Theorem 5.7 (Yano and Nagano [107]). Assume that a compact Riemannian Einstein space admits a non-homothetic conformal vector field. Then it is conformally diffeomorphic with the standard sphere. This follows essentially from Theorem 7.7 since on a compact space the gradient of the divergence must have a critical point. Then by Theorem 3.4 the manifold is a quotient of the standard sphere. The case of a covering can be easily excluded, so only the sphere itself is possible. In a similar way one obtains the following. Theorem 5.8 ((Kanai [52]). Assume that a complete Riemannian Einstein space admits a non-homothetic conformal vector field V with a critical point of div V . Then it is of constant sectional curvature. Without any assumption on critical points or zeros of the vector fields, there are counterexamples in form of warped products dt 2 C .cosh t /2 g where g is Einstein with D 1 but not of constant sectional curvature, a fact implicitly contained in [22], compare [6]. The case of a single conformal mapping into some Einstein space which is defined on a complete Einstein space is classified in [61, Theorem 27]. Here the case of a warped product g D dt 2 C e 2t g comes in with a complete and Ricci flat metric g . Conformal vector fields on Riemannian Einstein spaces were classified by Kanai [52]. A pseudo-Riemannian analogue is more complicated since it has to include the case of pp-waves and generalizations, compare [58] and [65]1 . The case of pseudo-Riemannian spaces of constant scalar curvature carrying nonisometric local gradient fields can also be classified, see [65, Theorem 4.3]. Here we obtain generalizations of Ejiri’s example at the end of Section 4, all as warped product metrics. The possible warping functions can be explicitly determined.

6 Spaces which are conformally Einstein We call a pseudo-Riemannian manifold .M; g/ conformally Einstein if every point p has an open neighborhood U such that the conformally equivalent metric .U; gN D 2 g/ for some function W U ! R is an Einstein metric. As a direct consequence of Lemma 5.2 we obtain the following: Proposition 6.1. An n-dimensional pseudo-Riemannian manifold .M; g/ admits a conformal mapping onto an Einstein space .M; g/ N with gN D ' 2 g if and only if the factor ' satisfies the equation ' Ricı C.n 2/.r 2 '/ı D 0: (10) 1

In Theorem 3.2 of this paper the proof has a gap, as kindly pointed out to the authors by Helga Baum. So possibly one case in the classification there was missing.


273

It seems that Brinkmann [21] was the first who discussed this equation, which is also called conformal Einstein equation. Its integration is surprisingly difficult. In dimension 2 the equation is trivial. In higher dimensions the equation implies that the eigenspaces of r 2 ' must coincide with the given eigenspaces of Ric. Furthermore the eigenvalues of r 2 ' are determined by the eigenvalues of Ric and by ' itself. We rewrite this equation for the function D log ' using the following Definition 6.2. For a vector field V on a pseudo-Riemannian manifold we define the Schwarzian tensor as the following traceless .0; 2/ tensor:

1˚ div V C kV k2 g.X; Y /: (11) n Then one can rewrite the conformal Einstein equations with the help of the Schwarzian tensor as follows, see [82]: FV .X; Y / WD g.rX V; Y / C g.X; V /g.Y; V /

Proposition 6.3. An n-dimensional pseudo-Riemannian manifold .M; g/ admits a conformal mapping onto an Einstein space .M; g/ N with gN D exp.2/g if and only if the factor satisfies the following equation: Ricı D .n 2/Fgrad :

(12)

It is an important question treated by many authors to characterize conformally Einstein manifolds by tensorial equations. In dimension four the Bach equation [5] is only a necessary condition, there are Bach flat spaces which are not conformally Einstein, cf. [86]. Tensorial conditions in dimension 4 for certain classes of metrics are discussed for example in Szekeres [102], Kozameh, Newman and Tod [60]. Extensions to arbitrary dimensions are due to Listing [82], [83], Gover and Nurowski [43]. Under a suitable non-degeneracy assumption for the Weyl curvature we present tensorial equations providing necessary and sufficient conditions for the metric to be conformally Einstein following [82] and [43]. A pseudo-Riemannian manifold .M; g/ of dimension n 4 has a harmonic Weyl tensor (or is called a C-space) if the divergence of the Weyl tensor vanishes, i.e. div W D 0. Proposition 6.4. A pseudo-Riemannian Einstein manifold of dimension n 4 has a harmonic Weyl tensor. Proof. Since Ric D .S=n/g and since the scalar curvature is constant the Schouten tensor P is parallel, therefore the Cotton tensor C D dP as the skew-symmetrization of rP vanishes. We conclude from Equation (7) that div W D 0. Let gN D ' 2 g be a conformally equivalent metric and let ' D exp./ and denote S the Weyl tensor resp. its divergence with respect to the metric g. S , divW N Then by W the conformal behaviour of the divergence of the Weyl tensor is given by the following equation, cf. [82, Lemma 1]: S .X; Y; Z/ D .div W / .X; Y; Z/ C .3 n/W .X; Y; Z; grad /: div W (13)

274


If now gN D exp.2/ g is an Einstein metric we conclude from Proposition 6.4 that div4 W . ; ; / C .3 n/W . ; ; ; grad / D 0: (14) Let .E1 ; E2 ; : : : ; En / be a pseudo-orthonormal frame with g.Ei ; Ej / D i ıij and i 2 f˙1g. For a .0; 2/-tensor h we denote by W Œh the following .0; 2/ tensor: X W Œh.X; Y / D i j W .Ei ; X; Y; Ej /h.Ej ; Ei /: i;j

If we take the divergence div1 with respect to the first argument of Equation (14) we obtain: div1 div4 W C .3 n/W Œr 2 .n 3/2 W Œd ˝ d D 0: Since for any function f we have W Œf g D 0 we conclude from the conformal Einstein equation (12) n3 W ŒRic .n 3/.n 4/W Œd ˝ d D 0: (15) n2 Note that W ŒRic D W ŒRicı D W Œ.n 2/P ı . Equation (15) motivates the following div1 div4 W C

Definition 6.5. For a pseudo-Riemannian manifold .M; g/ of dimension n 4 one defines the Bach tensor n3 B D div1 div4 W C W ŒRic: n2 Hence we obtain part (a) of the following theorem presenting a necessary condition for a metric to be conformally Einstein. In the 4-dimensional case this result can be found in [60, Theorem 2], for arbitrary dimension in [82]. Following [43] in part (b) a sufficient condition is formulated. The metric is called weakly generic in [43] if W .V; ; N ; / D 0 holds if and only if V D 0, i.e., the Weyl tensor viewed as map TM ! 3 TM is injective. Theorem 6.6. Let .M; g/ be a pseudo-Riemannian manifold. (a) (n D 4: [60, Theorem 2]; n 4: [82, Remark 2]) If .M; g/ is conformally Einstein such that gN D exp.2/ g is Einstein then the Cotton tensor C and the Bach tensor B satisfy the following equations: C W . ; ; ; grad / D 0; B C .3 n/.n 4/W Œd ˝ d D 0:

(16) (17)

(b) (n D 4 W [60, Theorem 2] n 4 W [43, Theorem 2.2]) Let the metric be weakly generic and let for some vector field V with dual one form V # the following equations be satisfied: C W . ; ; ; V # / D 0; B C .3 n/.n 4/W V # ˝ V # D 0:

(18) (19)

Then the metric is conformally Einstein. The vector field V D grad is locally a gradient field and gN D exp.2/ g is an Einstein metric.


275

If we consider the Weyl tensor as endomorphism W W ƒ2 .T M / ! ƒ2 .T M / then Listing [82] calls the Weyl tensor non-degenerate if W has maximal rank. Then he defines the vector field T WD

n 1 X i k W 1 Œdiv W . ; ; Ei / .Ei ; Ek /Ek : n3

(20)

i;kD1

Using the conformal Einstein equations written in Proposition 6.3 with the help of the Schwarzian tensor one obtains the following tensorial characterization: Theorem 6.7 ([82, Theorem 2], [43, Proposition 2.7]). A pseudo-Riemannian manifold .M; g/ with non-degenerate Weyl tensor is locally conformally Einstein if and only if the vector field T defined in Equation (20) satisfies: Ricı C.n 2/FT D 0: This result can be extended to weakly generic metrics, cf. [83] and it is shown in [43, Proposition 2.7] that the tensor field G WD Ricı C.n2/FT is conformally invariant. It is used in [43, Theorem 2.10] to define a natural polynomial in the Riemannian curvature tensor and its covariant derivatives of conformal weight 2n.n 1/ whose vanishing for a weakly generic metric characterizes conformally Einstein metrics. Conformally Einstein metrics can be characterized as conformal structures for which the standard tractor bundle admits a parallel section, cf. for example [43, Section 3] One can also use the vector field T and Theorem 6.6 (b) to define a generalized Bach tensor which is conformally covariant not only in dimension 4, cf. [83]. For the rest of this section we consider the particular case of dimension four: For a pseudo-Riemannian manifold of dimension 4 the definition of the Bach tensor B given in Definition 6.5 reads: B D div1 div4 W C .W ŒRic/=2 and in index notation Bij D

4 X

r k r l Wkij l C

k;lD1

4 1 X kl R Wkij l : 2 k;lD1

The Bach tensor in dimension 4 is a symmetric, trace free and divergence free tensor and it is conformally covariant, i.e. if gN D f 2 g then BgN D f 2 Bg . For a compact manifold it is the gradient of the functional Z jWg j2 d Vg : (21) W .g/ D M

We call a metric Bach-flat, if the Bach equation B D 0 is satisfied. Hence this equation is the Euler-Lagrange equation of the functional W . In particular metrics which are locally conformal to an Einstein metric are Bach flat. In the Riemannian case half conformally flat metrics are also Bach flat, but they are not weakly generic. As a consequence of Theorem 6.6 we obtain:

276


Corollary 6.8 ([60, Theorem 2]). A pseudo-Riemannian and weakly generic manifold .M; g/ of dimension 4 with non-degenerate Weyl tensor is conformally Einstein if and only if it is Bach-flat and conformally equivalent to a space with a harmonic Weyl tensor, i.e. if B D 0 and for some vector field V : C D W . ; ; ; V # /. As an interesting class of conformally Einstein metrics in dimension 4 one can discuss products of surfaces, cf. [30, Chapter 18]. Here extremal metrics on surfaces play a particular role, we call a metric h on a surface S with Gaussian curvature extremal, if r 2 D h for some function , i.e. if grad is a conformal vector field on the surface. Then one can show that 2 D and the so-called first .classifying/ parameter c and the second .classifying/ parameter p with c WD C 2 ;

p WD c h.grad ; grad /

3 3

(22)

are constants. Let F be the cubic polynomial 3 ; (23) 3 then we conclude from Theorem 5.5 and the formula for the Gauss curvature, cf. [30, Lemma 18.9]: If F . 1 / 6D 0 for some 1 , then we can introduce coordinates ; in a neigborhood of 1 such that the metric is of the following form: F . / D c p

F . /d 2 C

1 d 2; F . /

(24)

with D ˙1. These metrics (in arbitrary dimension) were first introduced by Calabi in the Riemannian setting as critical Kähler metrics for certain curvature functionals, see [15, Chapter 11E]. As an equivalent characterization one can use that the gradient grad S of the scalar curvature is a holomorphic vector field. Then one obtains the following Proposition 6.9. Let .M 4 ; g/ D .M12 ; g1 / .M22 ; g2 / be a product of two surfaces with a pseudo-Riemannian product metric g D g1 ˚ g2 whose scalar curvature S is nowhere vanishing. Then the pseudo-Riemannian manifold .M; g/ is locally conformally Einstein if and only if both surface metrics are extremal and have the same first classifying parameter. The conformally equivalent Einstein metric gN is uniquely determined up to a constant. Hence we can introduce coordinates i , i , i D 1; 2, on .Mi ; gi / and classifying parameters c; p1 ; p2 with the corresponding cubic polynomials Fi . i / D c i pi i3 =3, i D 1; 2, g D 1 F1 . 1 /d 12 C

1 1 d 2 C 1 F2 . 2 /d 22 C d 2: F1 . 1 / 1 F2 . 2 / 2

(25)

Then the conformally equivalent Einstein metric is given by gN D

1 4 gD g S2 . 1 C 2 /2

(26)


277

and has scalar curvature Sx D 3.p1 C p2 /. These examples can be found in several papers, cf. for example [105]. They also occur in the context of conformal gravity where the field equation B D 0 is considered, cf. [37], [30]. Note that the Schwarzschild metric is a Ricci flat metric which is conformally equivalent to the product of two surface metrics one of which has constant Gaussian curvature.

7 Conformal gradient fields This section deals with conformal gradient vector fields V D grad f and simultaneously with vector fields which are integrable (or closed), so that they are locally gradient fields.This means that for every point p 2 M there is a neighborhood U and a function f 2 C 1 .U / such that V D grad f . It follows that grad f is conformal if and only if the Hessian r 2 f .X; Y / WD hrX grad f; Y i satisfies the equation r 2 f D g

(27)

since Lgrad f g.X; Y / D 2r 2 f .X; Y /. The Laplacian f is the divergence of the gradient of f , so in the equation above the factor is nothing but the Laplacian, divided by the dimension: f g: (28) r 2f D n From Equation (27) we obtain the following Ricci identity for the curvature tensor introduced in Equation (4) R.X; Y / grad f D X. /Y Y . /X:

(29)

By contraction we obtain for the Ricci tensor: Ric.X; grad f / D .1 n/X. /:

(30)

It turns out that one can integrate Equation (27) by reducing it to an ODE whenever the gradient of f is not isotropic. This can be done along the lines of Brinkmann’s results [22]. The following lemma was stated by Fialkow [36, p. 471]. Lemma 7.1. Let .M; g/ be a pseudo-Riemannian manifold. Then the following conditions are equivalent: (1) There is a non-constant solution f of the equation r 2 f D f g in a neighborn hood of a point p 2 M with hgrad f .p/; grad f .p/i 6D 0. (2) There is a neighborhood U of p , a C 1 -function f W .; / ! R with f 0 .t / 6D 0 for all t 2 .; / and a pseudo-Riemannian manifold .M ; g / such that .U; g/ is isometric to the warped product .; / ; dt 2 f 0 .M ; g // D .; / M ; dt 2 C f 0 .t /2 g where WD signhgrad f .p/; grad f .p/i 2 f˙1g.

278


Proof. .2/ ) .1/: Define the function f W .; / M ! R by f .t; x/ D f .t /. Then grad f .t; x/ D f 0 .t / @ t and r@ t grad f D f 00 .t / @ t . Let X be a lift of a vector field on M , then by Equation (27) we have rX grad f D f 00 X . .1/ ) .2/ W Let U be a neigborhood of p 2 M with compact closure and with hgrad f .q/; grad f .q/i 6D 0 for all q 2 U . Hence c D f .p/ is a regular value, let M be the connected component of f 1 .c/ containing p. Then there is an > 0 such that the normal exponential map exp? W .; / M ! M defines a diffeomorphism onto the image. Let q 2 U; g.X; grad f .q// D 0, then it follows immediately that Xg.grad f; grad f / D 2

f g.grad f; X / D 0: n

(31)

Hence hgrad f; grad f i is constant along the level hypersurfaces f 1 .c 0 / and the level hypersurfaces f 1 .f .exp.t; x0 ///; t 2 .; / are parallel. Therefore they coincide with the t-levels and f can be regarded as a function only of t , written as f .t; x/ D f .t / by slight abuse of notation and grad f .t; x/ D f 0 .t / @ t as well as r 2 f D 2f 00 g D

f g: n

(32)

The equation g.@ t ; @ t / D D signhgrad f .p/; grad f .p/i follows since the curve t 7! exp.t grad f .x// is a geodesic. Let X be a lift of a vector field on M , then g.@ t ; X/ D 0 by the Gauss Lemma. If X1 ; X2 are vectors tangential to M at x0 and Xi .t/ D d exp.t; x0 /.Xi /, i D 1; 2, then d j tDs g.X1 ; X2 /.t / D L@ t g.X1 ; X2 /.s/ D 0 Lgrad f g.X1 ; X2 /.s/ dt f .s/ f 00 .s/ 2 2 D 0 rX1 .s/;X2 .s/ f D 2 0 g.X1 ; X2 /.s/: f .s/ f .s/ The claim follows from the uniqueness of the solution of the ODE ..f 0 /2 g.X1 ; X2 //0 .t / D 0: The metric g is non-degenerate since it is orthogonal to the time-like or space-like t-direction. Proposition 7.2 ([56, Proposition 2] [63]). Let V be a non-trivial closed conformal vector field on the n-dimensional pseudo-Riemannian manifold .M; g/. 1. If V .p/ D 0, then div V .p/ D n .p/ 6D 0, in particular all zeros of V are isolated. 2. Denote by C D C .M; g/ the vector space of closed conformal vector fields, then dim C n C 1. If the dimension of the space of closed conformal vector fields is maximal, i.e., if dim C.M; g/ D n C 1, then the manifold is of constant sectional curvature.


279

Lemma 7.3. Let @ t be the unit tangent vector in direction of the first factor of the product I M and let X , Y , Z be lifts of vector fields on M . Here I denotes an open interval in R. Denote by r , R , Ric , the Levi-Civita covariant derivative, the Riemannian curvature tensor and the normalized scalar curvature of .M ; g /. .The normalized scalar curvature of the standard sphere with sectional curvature 1 is also 1/. Then we have the following formulae for the corresponding geometric quantities r, R, Ric, of the warped product metric g D dt 2 C f 2 .t /: 1. r@ t @ t D 0, r@ t X D rX @ t D rX Y D

f0 X, f g.X;Y / 0 f f @ t C

rX Y .

2. R.X; Y /Z D R .X; Y /Z R.X; Y /@ t D 0, 00 R.X; @ t /@ t D ff X .

f 02 fg.Y; Z/X; g.X; Z/Y g f2

,

3. Ric.Y; Z/ D Ric .Y; Z/ f2 f.n 2/f 02 C f 00 f gg.Y; Z/, Ric.Y; @ t / D 0, 00 Ric.@ t ; @ t / D .n 1/ ff , 4. f 2 D

n2 n

n2 02 f n

n2 f 00 f .

This follows from the formulae for warped products in general, cf. [91, Chapter 7] since rf D f 0 @ t ; r 2 @ t ;@ t f D g.r@ t rf; @ t / D f 00 : Note, however, that the curvature tensor in [91] has the opposite sign. The formulae in the Riemannian case and the pseudo-Riemannian case coincide if we consider in the case D 1 the warped product gQ D dt 2 C f 2 .t /gQ , gQ D g which is anti-isometric to g (then Q D , Q D ; : : : ). In particular we obtain as in the Riemannian case the Corollary 7.4. The warped product .I; dt 2 / f .M ; g / is an Einstein metric .a metric of constant sectional curvature/ if and only if g is an Einstein metric .a metric of constant sectional curvature/ and f 02 C f 2 D . Near a regular point of a function f satisfying r 2 f D g the metric has the structure of a warped product, cf. Lemma 7.1. Around a critical point we can use geodesic polar coordinates and obtain the following. Proposition 7.5 ([56], [63]). Let .M; g/ be a pseudo-Riemannian manifold with a nonconstant solution f of the equation r 2 f D g for a function and with a critical point p 2 M . 1. (cf. [103], [61, Lemma 18] in the Riemannian case) There are functions f˙ such that the metric in geodesic polar coordinates .r; x/ R † in a neighborhood

280


U of p has the form g.r; x/ D dr 2 C

f0 .r/2 f00 .0/2

g1 .x/;

D hx; xi;

(33)

and f .r; x/ D f .r/, .r; x/ D .r/ with .r/ D f 00 .r/, in particular the metric is conformally flat in a neighborhood of the critical point. 2. If all geodesics through p are defined on the whole real line R then the metric g is of the form above for all .r; x/, as long as f0 .r/ does not vanish. Definition 7.6. We call a pseudo-Riemannian manifold with a conformal vector field C -complete if every point can be joined by a geodesic with some zero of the vector field. In the case of a gradient field grad f this means that every point can be joined by a geodesic with some critical point of the function f . Theorem 7.7. Let .M; g/ be a pseudo-Riemannian manifold carrying a non-constant solution f of the equation r 2 f D g having critical points. We assume either that all geodesics through critical points are defined on R and that .M; g/ is null complete or that .M; g/ is C -complete. Then the manifold .M; g/ is .locally/ conformally flat. One can define neighborhoods Mj for every critical point pj on which the metric has the form as in (33). These neighborhoods Mj cover M . Theorem 7.7 follows from Proposition 7.5 since around each critical point the metric is conformally flat by the equation (33) in polar coordinates. On the other hand, the level .M ; g / in the warped product metric according to Lemma 7.1 cannot change along the geodesic t-lines. Therefore the completeness assumption implies the assertion. With regard to the global geometry of complete manifolds, the main results of [63] and [66] are the following: Theorem 7.8 ([63]). For any signature .k; n k/ with 1 k n 1 there exists a smooth pseudo-Riemannian manifold of dimension n carrying a complete conformal gradient field V D grad f with an arbitrary prescribed number N 1 of isolated zeros .including the case of infinitely many zeros in two different ways corresponding to N or Z/. These manifolds are C -complete. Theorem 7.9 ([63]). Let Mkn be a geodesically complete pseudo-Riemannian manifold of signature .k; n/ with 2 k n 2 carrying a non-trivial conformal gradient field with at least one zero. 1. The diffeomorphism type of Mkn is uniquely determined by the number N of zeros. Here in the case of infinitely many zeros we have to distinguish between N and Z. 2. Every manifold is conformally equivalent to a standard manifold M.J /.˛; ˇ/ defined in [63, p. 468]. 3. If in addition the vector field is complete then the conformal type is uniquely determined by the number N of zeros.


281

In the Lorentzian case k D 1 the disconnectedness of the geodesic distance spheres opens up more possibilities for the global conformal types which can be described by the gluing graph. In the Riemannian case part 3 of Theorem 7.9 is given in [19]. Definition 7.10. Let M be a Lorentzian manifold with a conformal gradient field grad f . The associated graph G.M; f / is defined as follows: 1. The vertices of G.M; f / are the critical points of f in M . 2. Every vertex is contained in three edges, one space-like and two time-like ones. These correspond to the space-like cone and the two components of the time-like cone in M at that point. 3. Two vertices are joined by an edge if and only if there is a trajectory of the vector field from one to the other, in such a way that the trajectory passes through the corresponding space-like or time-like cones. Obviously each edge has a unique character .C/ if it is space-like or ./ if it is timelike. We can use this as a label for each edge. It is possible that an edge is incident with only one vertex. This is called a free edge. It is also possible that two vertices are joined by more than one edge. If M is C -complete then every point can be joined by a geodesic with some critical point of f . Consequently, every point is somehow represented by an edge in the associated graph. In the case of signature 2 k n2 the analogous associated graph has to be a linear graph since the time-like cone is always connected. In the Lorentzian case k D 1 or k D n 1 there are many possibilities and interesting properties of these graphs. They may have cycles. Furthermore, M is simply connected if and only if the associated graph is a tree. Proposition 7.11 ([13]). Let M1 ; M2 be two C -complete Lorentzian manifolds admitting conformal gradient fields grad f1 ; grad f2 , respectively, each with at least one zero. If F W M1 ! M2 is a conformal diffeomorphism transforming at least one critical point of f1 into a critical point of f2 , then F preserves the trajectories and induces an isomorphism between the two associated graphs including the labeling. On the other hand, a conformal classification in general has to incorporate more than just the combinatorial structure of the associated graph with the labeling .C/ and ./. In addition one needs a time orientation and weights on the edges. The weight is a positive real number (including 1 for free edges) associated with an edge. Somehow the weights correspond to the lengths of the trajectories after a conformal development. Different developments lead to constant ratios of the weights. These constants have to be factorized out. The details can be found in [13]. There remains a discussion of the case of a conformal gradient field which is isotropic on an open set. Here we have the following: Theorem 7.12 (Brinkmann [22], [25]). Assume that .M; g/ is a pseudo-Riemannian manifold of dimension n 3 admitting a non-vanishing and isotropic conformal gradient field, i.e., a non-constant solution f of the equation (27) such that grad f is

282


isotropic on an open subset. Then grad f is in addition parallel, and the metric tensor @ D grad v and can be brought into the form g D dudv C g .u/ where grad f D @u where the .n 2/-dimensional metric g .u/ does not depend on v. Such spaces carrying a parallel isotropic vector field are often called Brinkmann spaces. The transition from a non-isotropic gradient to an isotropic one is further explained in [25]. It corresponds to passing to the limit ˛ ! 0 in the metric g D ˛.u/du2 C dudv C g .u/. We mention here the isotropic case in a generalized Liouville theorem which is the case of a conformal mapping preserving the Ricci tensor. Theorem 7.13 ([71]). Assume that an n-dimensional pseudo-Riemannian manifold .M; g/ admits a conformal mapping F W M ! M such that the conformal factor ' has an isotropic gradient grad ' 6D 0 everywhere. Assume further that F preserves the Ricci tensor and the null-congruence given by the parallel and isotropic vector @v D grad '. Then in certain coordinates u, v, xk .k D 1; : : : ; n 2/ the metric has the form X gij# .u; xk /dx i dx j g D 2dudv C i;j

and, up to an isometry, F has the form

F .u; v; xk / D

1 ; cv C .u; xk /; 1 .u; xk /; : : : ; n2 .u; xk / cu

with a constant c and with a certain function , where for any fixed u, v the transformation .x1 ; : : : ; xn2 / 7! .1 ; : : : ; n2 / is a homothety with respect to the metric g # . The conformal factor of F is the function '.u; v; xk / D u, i.e., F g D u2 g. Conversely, let h be any metric on an .n 2/-dimensional space M admitting a 1-parameter group û of similarities (homothetic transformations) with û h D u2 h. Then on M D RC R M the metric g D 2dudv C û h admits a conformal mapping F such that the conformal factor u has an isotropic gradient grad u D @v . In this case F acts on M by the similarities û .

8 4-dimensional Lorentzian manifolds In General Relativity one considers 4-dimensional spacetimes of 3C1 dimensions which can also be described as 4-dimensional Lorentzian manifoldswith a metric tensor of signature .1; 3/, modelled after the flat Minkowski space R41 with the metric g D dt 2 C dx 2 C dy 2 C dz 2 : For an observer at rest time corresponds to the t -axis and space to the .x; y; z/-part. The local conformal group of this Minkowski space is 15-dimensional. It is generated


283

by four translations, six rotations (that is, the group O.3; 1/), one homothety x 7! cx and four proper conformal mappings. The corresponding conformal vector fields are four infinitesimal translations, six infinitesimal rotations (i.e., the Lie algebra o.3; 1/), the radial vector field V .X / D X and the vector fields V .X / D 2hX; T iX hX; XiT for a fixed vector T . It is well known [96] that the local conformal group of Minkowski space is isomorphic with O.4; 2/=f˙g. It is also well known that any spacetime which is not locally conformally flat has a conformal group which is at most 7-dimensional [50]. The case of dimension 7 is fairly interesting since in this case all Ricci flat metrics (or vacuum spacetimes) can be determined which admit a 7-dimensional conformal group which is not contained in the isometry group, see Theorem 8.3. For further related results cf. [49], [28]. By Theorem 5.5 any 4-dimensional Lorentzian Einstein space (not of constant sectional curvature) is a vacuum spacetime if it admits a non-homothetic conformal vector field V . For vacuum spacetimes in turn we can formulate the following statement: Theorem 8.1. A vacuum spacetime admitting a non-homothetic conformal vector field is locally a pp-wave. Definition 8.2. The class of pp-waves (or plane-fronted waves with parallel rays) in general is given by all Lorentzian metrics on open parts of R4 D f.u; v; x; y/g which are of the form ds 2 D 2H.u; x; y/du2 2dudv C dx 2 C dy 2 with an arbitrary function H , the potential, which does not depend on v. The subclass of plane waves is given by all H of the form H.u; x; y/ D a.u/x 2 C 2b.u/xy C c.u/y 2 ; compare [95]. Isometric, homothetic and conformal vector fields of pp-waves were classified in a kind of a recursive normal form in [84], starting from the possible Killing fields. On the other hand, the possible isometry groups are known from the work of Ehlers–Kundt [32] and Sippel–Goenner [100]. Furthermore it is well known that the isometry group is of codimension at most one in the homothety group, and that in turn the homothety group is of codimension at most one in the conformal group, compare [49]. For a discussion of homothetic transformations with one-dimensional fixed point set see also [2]. At the maximum dimension we have the following result: Theorem 8.3 ([70])). All vacuum spacetimes admitting a 7-dimensional conformal group .together with the vector fields themselves/ can be explicitly determined in terms of elementary functions and a finite number of parameters. Moreover there is one family admitting a non-homothetic conformal vector field.

284


The typical candidate of a hon-homothetic conformal vector field on a pp-wave is the standard special conformal vector field V3 which we already met in Section 3. The flow 1 of V3 is explicitly given by ˆ t .u; v; x; y/ D 12tu u; v.1 2t u/ C t .x 2 C y 2 /; x; y . Any fixed trajectory is a straight line. This vector field is depicted in Figure 1 and Figure 2 below where the .x; y/-plane is reduced to just the .x; 0/-axis2 .

10 8 6 4 2

x 0 _2 _4 _6 _8 10 _10 _ 10

_5

0 0

u

5

10

_10

v

Figure 1. The standard special conformal vector field V3 in the .u; v; x; 0/-slice.

Theorem 8.4 ([70]). Assume that a vacuum pp-wave with metric g D 2H.u; x; y/du2 2dudv C dx 2 C dy 2 admits the standard conformal vector field V3 D u2 @u C 12 .x 2 Cy 2 /@v Cux@x Cuy@y . Assume further that the function H is defined in a neighborhood of x D y D 0 for any fixed u0 6D 0. Then in a neighborhood of u0 H can be written as X H.u; x; y/ D u.nC2/ Pn .x; y/ n0

where Pn denotes a homogeneous polynomial of degree n in the variables x; y which is harmonic, i.e. Pn D 0. Vice versa, any function H of that type admits the standard conformal vector field V3 , compare Section 3. 2

We thank Andreas App for providing these figures in Matlab.


285

10 8 6 4 2

x 0 _2 _4 _6 _8 _10 _10 _ 8

_6

_4

_2

0

2

4

6

8

10

u

Figure 2. The projection of V3 into the .u; 0; x; 0/-plane.

Due to the singularity u D 0 this vector field does not have a zero in the spacetime (except for the flat case). A classification of conformal vector fields with zeros is still not complete. In the case of closed vector fields see Section 7. For non-closed fields there are normal forms under additional assumptions on the Petrov type of the metric, see [49], [101]. Typical results are the following: Proposition 8.5. Let .M; g/ be a spacetime of constant Petrov type D, and let q be a zero of the conformal vector field V . Then the following hold: 1. div V .q/ D 0, 2. after a conformal change of g the vector field becomes isometric, 3. in certain coordinates y0 ; : : : ; y3 we have V D y0 @y0 y1 @y1 ˛y2 @y2 C ˛y3 @y3 where ˛; are real constants. Proposition 8.6. Let .M; g/ be a spacetime of constant Petrov type III, and let q be a zero of the conformal vector field V . Then the following hold: 1. div V .q/ 6D 0, 2. after a conformal change of g the vector field becomes homothetic, 3. in certain coordinates y0 ; : : : ; y3 we have V D div V .q/.3y0 @y0 y1 @y1 C y2 @y2 C y3 @y3 /:

286


Proposition 8.7. Let .M; g/ be a spacetime of Petrov type I or II, and let q be a zero of the conformal vector field V . Then V vanishes identically. It does not seem to be known what happens if the Petrov type degenerates at exactly the zero of the vector field. Possibly there are more cases to be considered. However, examples are still missing. Any conformal vector field V on a vacuum spacetime preserves the Ricci tensor, i.e., LV Ric D 0. This is a trivial case of a so-called Ricci collineation. In more generality, one can consider the case that LV Ric is conformal to the metric. This is a conformal Ricci collineation , as defined in [69]. A conformal Ricci collineation preserves the eigendirections of the Ricci tensor. Further aspects of the geometry of spacetimes can be found in [51] and [29].

9 The transition to the Penrose limit In 1976 R. Penrose introduced in [92] the following construction which associates to any lightlike geodesic on a Lorentzian manifold a plane wave metric. In several recent papers about models for supergravity respectively string theory, in particular regarding the maximally supersymmetric type IIB plane wave background and its relation to AdS5 S 5 the Penrose limit has been discussed intensively, cf. [16], [17] and the survey article [94]. Along a lightlike geodesic W I ! M on a Lorentzian manifold of dimension n which is free of conjugate points it is possible to introduce coordinates .U; V; Y / D .U; V; Y1 ; : : : ; Yn2 / such that the metric nearby the geodesic is of the form n2 n2 X X bi .U; V; Y /d Yi C gij .U; V; Y /d Yi d Yj : g D d V 2d U C a.U; V; Y /d V C 2 iD1

iD1

(34) In these coordinates the lightlike geodesic is of the form .U / D .U; 0; 0/, it is embedded in a congruence of lightlike geodesics U 7! .U; V1 ; Y1 / for any fixed .V1 ; Y1 /. Then we introduce a scaling of coordinates, for any positive let .U; V; Y1 ; : : : ; Yn2 / D .u; 2 v; y1 ; : : : ; yn2 /: In the new coordinates the scaled metric g D 2 g is of the form n2 X 2 2 bi .u; 2 v; y/dyi g D dv 2du C a.u; v; y/dv C 2

C

n2 X

iD1 2

gij .u; v; y/dyi dyj :

iD1

Then Penrose [92] introduced the following construction:

(35)


287

Definition 9.1. With respect to the coordinates .u; v; y1 ; : : : ; yn2 / introduced in Equation (35) the Penrose limit is defined as the metric gN WD lim g D 2dudv C !0

n2 X

gN ij .u/dyi dyj

(36)

i;j D1

with gN ij .u/ D gij .u; 0; 0/. This is a plane wave metric in the so-called Rosen coordinates. Now we investigate whether a conformal vector field .U; V; Y / D A.U; V; Y /@U C B.U; V; Y /@V C

n2 X

Ci .U; V; Y /@Yi

iD1

survives under this limit construction. We express the conformal field 2 in the coordinates .u; v; y/: 2 .u; v; y/ D B.u; 2 v; y/@v C

n2 X

Ci .u; 2 v; y/@yi C 2 A.u; 2 v; y/@u

iD1

and assume that one of the coefficient functions for fixed .u; v; y/: B.u; 2 v; y/, Pn2 iD1 Ci .u; 2 v; y/; 2 A.u; 2 v; y/ has a Taylor expansion around D 0 with a leading term k fk .u; v; y/; fk .u; v; y/ 6D 0 We also assume that k 0 is the minimal N v; y/ WD lim!0 2k .u; v; y/ is a exponent with this property. Then the limit .u; non-trivial vector field. Let L g D g; i.e. D div =n. Since L2k .u;v;y/ 2 g D 2k .u; 2 v; y/ 2 g we obtain in the limit ! 0 that the vector field N is a non-trivial conformal vector field. Therefore the property to be a conformal vector field is a hereditary property, the corresponding argument for Killing fields can be found for example in [16, Chapter 4.3]. We summarize theses considerations in the following Proposition 9.2. For a conformal and analytic vector field D .U; V; Y / of the Lorentzian metric g in coordinates .U; V; Y / adapted to a lightlike geodesic .U / D .U; 0; 0/ by Equation integer k such that the (34) there is a non-negative N v; y/ WD lim!0 2k .u; 2 v; y/ is a conformal vector field on the limit .u; Penrose limit given by Equation (36). The proposition shows the importance of the conformal geometry of plane waves, resp. the description of the conformal vector fields, cf. Section 8. One can transform the Penrose limit in Rosen coordinates given in Equation (36) into Brinkmann coordinates x D .x1 ; x2 ; : : : ; xn /. Then the Penrose limit is of the form n n X X 2 Aij .x1 /xi xj dx1 C dxi2 : (37) gN D 2dx1 dx2 C i;j D3

iD3

288


These is the coordinate form of a plane wave used in the four-dimensional case in Section 8, see for example Definition 8.2. Now the wave profile Aij .x1 / of the plane wave coincides with the only non-vanishing curvature components x1ij1 D R.@ x 1 ; @i ; @j ; @1 /; Aij D R

i; j D 3; 4; : : : n;

of the plane wave metric. The coordinate transformation as well as the curvature computations are explained in detail for example in [44, Chapter VIII.2]. This allows the following invariant interpretation of the Penrose limit: Proposition 9.3 (Blau et al. [17, (2.14)]). Let D .u/ be a lightlike geodesic of a Lorentzian manifold .M; g/. We assume that e1 ; e2 ; e3 ; : : : ; en is a pseudo-orthonormal frame in the tangent space T.0/ M with e1 D 0 .0/, g.e1 ; e2 / D g.ei ; ei / D 1 for all i D 3; 4; : : : ; n and g.ei ; ej / D 0 otherwise and let e1 .u/; e2 .u/; e3 .u/; : : : en .u/ be the parallel transport of .e1 ; : : : ; en / along . Then the Penrose limit of .M; g/ associated to the lightlike geodesic is the plane wave metric n X

2dx1 dx2 C

Aij .x1 /xi xj dx12 C

i;j D3

n X

dxi2

iD3

with the wave profile Aij .x1 / D R.e1 ; ei ; ej ; e1 /: Here R is the curvature tensor of .M; g/. It follows in particular that the Penrose limit of an Einstein manifold is Ricci flat.

10 Conformal vector fields and twistor spinors For a pseudo-Riemannian manifold .M; g/ of signature .k; n k/ with spin structure the tangent bundle acts on the spinor bundle † via the Clifford multiplication X ˝ 2 T M ˝ †M 7! X 2 †M . The spinor bundle carries the spin connection r and a hermitian inner product h ; i satisfying the equations hX ; i D .1/kC1 h; X i; X.h; i/ D hrX ; i C h; rX i: For the details of this construction see [9] and [41] in the Riemannian case and [7] in the pseudo-Riemannian case. A spinor field is called parallel if r D 0, i.e. for all tangent vectors X W rX D 0. The composition of the spin connection and the Clifford product defines the Dirac operator D. If e1 ; : : : ; en is an orthonormal frame with g.ei ; ej / D i ıij ; 1 D D p D 1; pC1 D D n D 1, then D

D

n X iD1

i ei rei :

(38)


289

Definition 10.1. We call a spinor field a twistor spinor if the following twistor equation is satisfied for all tangent vectors X W 1 X D D 0: (39) n Twistor spinors can also be described as the kernel of a differential operator D, called the twistor operator or Penrose operator: It is the composition of the spin connection with a projection onto the kernel of the Clifford product: rX

D

D

n X iD1

C

i ei ˝ rei

C

1 ei D : n

(40)

The Dirac operator D and the twistor operator D are both conformally covariant in the following sense. If gN D exp.4/ g then there is a isometry between the spinor bundles x D DgN 2 †g M ! N 2 †gN M such that for the Dirac operators D D Dg and D x resp. the twistor operators D D Dg and D D DgN the following equations hold, cf. [9, Chapter (1.3), (1.4)], [40]: x N D e .nC1/ D e .n1/ ; D x N D e D e : D Hence the dimension of twistor spinors is a conformal invariant, if is a twistor spinor of the pseudo-Riemannian manifold .M; g/ then e ' N is a twistor spinor of the conformally equivalent metric gN D exp.4/ g. In this sense the twistor equation D D 0 is conformally covariant. Particular twistor spinors are Killing spinors, they satisfy the equation rX D X (41) for a complex number and for all tangent vectors X . Then one can conclude from the relation between the curvature of the spin connection and the Riemannian curvature tensor: Proposition 10.2. Let .M; g/ be a pseudo-Riemannian manifold of signature .k; nk/ with a Killing spinor , i.e. rX D X for some complex number . (a) The scalar curvature S is constant and satisfies S D 4n.n 1/ 2 . (b) The Ricci curvature as a .1; 1/ tensor satisfies ˚

Ric.X / 4 2 .n 1/X D 0: Therefore the complex number is either real or purely imaginary, then we call the Killing spinor either a real Killing spinor or an imaginary Killing spinor. If the manifold is Riemannian then the metric is an Einstein metric. In the pseudo-Riemannian case the traceless Ricci tensor of a manifold carrying a Killing spinor is lightlike. If in addition the Killing spinor is not light-like then the traceless Ricci tensor vanishes, i.e. the manifold is Einstein. On the other hand one can show that a twistor spinor on an Einstein manifold is either parallel or the sum of two Killing spinors:

290


Proposition 10.3. Let .M; g/ be a pseudo-Riemannian Einstein manifold with scalar curvature S carrying a twistor spinor . Then (a) If S 6D 0 then the p twistor spinor D C C is the sum of two Killing with rx ˙ D ˙ 12 S=.n.n 1//. (b) If S D 0 then either or D is a parallel spinor.

˙

We are interested here in the conformal vector field which can be associated to a twistor spinor: Definition 10.4. For a spinor field on a pseudo-Riemannian manifold .M; g/ of index k we call the vector field V defined by g.V ; X / D i kC1 hX ; i for all tangent vectors X the Dirac current. Then we obtain Proposition 10.5. Let .M; g/ be a pseudo-Riemannian manifold of index k with spinor field and Dirac current V . (a) The Dirac current V of a twistor spinor is a conformal vector field. (b) The Dirac current V of a real .resp. imaginary/ Killing spinor is a Killing vector field if k is even .resp. odd/. (c) The Dirac current V of a parallel spinor is a parallel vector field. Proof. We assume that V DV :

is a twistor spinor: We compute the Lie derivative LV g for

LV g.X; Y / D g.rX V; Y / C g.X; rY V / D Xg.V; Y / C Y g.X; V / g.V; ŒX; Y / D X hY ; i Y hX ; i hŒX; Y ; i D hY rX ; i hY ; rX i hX rY ; i hX ; rY i: Then we conclude from the twistor equation (39): ˚

LV g.X; Y / D n hYXD ; i C hY ; XD i C hX YD ; i C hX ; YD i ˚

D n h.X Y C YX /D ; i C .1/kC1 h ; .X Y C YX /D i ˚

D 2n g.X; Y / hD ; i C .1/kC1 h ; D i : Hence V is a conformal vector field with divergence div V D 2b.hD ; i/:

(42)

Here b.a/ for a complex number a denotes the real part of a if k is odd and the imaginary part otherwise. This finishes the proof of part (a). If satisfies Equation (41) then D D n hence by Equation (42) the Dirac current is a Killing vector field if either is real and k is even of is purely imaginary and k is odd. If is parallel then hrY V; Xi D Y h ; X i hV; rY X i D hrY ; X i C h ; XrY i D 0 shows that is parallel.


291

In the Riemannian case it may very well occur that the Dirac current of a twistor spinor vanishes identically. If for example is a parallel spinor on a Riemannian manifold with a Dirac current V which does not vanish identically then V is parallel and the manifold is locally a Riemannian product. Twistor spinors with zeros can be used to present essential conformal vector fields, cf. Section 4. On the doubled spinor bundle E D † ˚ † of a pseudo-Riemannian spin manifold there is a connection r E with the following property: A section . ; '/ of the bundle E is parallel if and only if is a twistor spinor and ' D D . This shows in particular that for a non-trivial twistor spinor with zero p, i.e. .p/ D 0 we have D .p/ 6D 0. Therefore one can show that rV .p/ D 0, hence also div V .p/ D 0. But since a conformal vector field W vanishes identically if for some point p the quantities W .p/ D 0; rW .p/ D 0 and grad div W .p/ D 0 vanish we conclude: Proposition 10.6 ([62]). Let .M; g/ be a pseudo-Riemannian spin manifold with a non-trivial twistor spinor having a zero. If the Dirac current V is non-zero it is an essential conformal vector field. In the Riemannian case we obtain the following consequence: Theorem 10.7 ([62, Theorem A]). If a Riemannian spin manifold .M; g/ carries a twistor spinor with zero and with non-trivial Dirac current V then the manifold is conformally flat. K. Habermann showed in [46] a similar result under an additional curvature and completeness assumption. So the question was whether there are examples of twistor spinors with zeros on Riemannian manifolds which are not conformally flat. The authors show in [64] for dimension n D 4 and for even dimensions n 4 in [67] that there are complete Riemannian spin manifolds carrying twistor spinors with zeros which are not conformally flat. In particular in this case the Dirac current vanishes identically. These examples are conformal compactifications of irreducible and asymptotically locally Euclidean manifolds carrying parallel spinors. This leads to the following Theorem 10.8 ([68, Theorem 1.2]). Let .M; g/ be an n-dimensional Riemannian spin manifold carrying a twistor spinor with non-empty zero set Z WD fp W .p/ D 0g. Then the conformally equivalent Riemannian metric gN D k k4 g on M Z is either flat or locally irreducible and Ricci flat carrying a parallel spinor. In addition corresponding to any zero point the complement M Z of the zero set Z has an end carrying an asymptotically Euclidean coordinate system of order 3. For a construction of compact orbifolds which are not conformally flat and which carry twistor spinors with zero we refer to [14]. In contrast to the Riemannian case in the Lorentzian case the Dirac current is always non-trivial. In addition the twistor spinor as well as the Dirac current can be lightlike. For example the following result in dimension four is well known: Theorem 10.9 (Ehlers–Kundt [32]). A four-dimensional Lorentzian manifold with spin structure carrying a parallel spinor is locally isometric to a pp-metric.

292


In this case the Dirac current is parallel and lightlike. For results on the holonomy of pseudo-Riemannian manifolds with parallel spinors see Baum and Kath [10]. A description of the local geometry of Lorentzian manifolds carrying twistor spinors without zeros up to dimension 7 is given by Baum and Leitner in [11]. These geometries include Brinkmann spaces with special Kähler flag, Fefferman spaces and LorentzianSasaki manifolds. For a survey on these constructions we refer to [8]. The Dirac current V is called non-twisting if the dual one-form ! D V # satisfies d! ^ ! D 0 and twisting if d! ^ ! does not vanish anywhere. Theorem 10.10 (Baum, Leitner [11, Proposition 4.3]). A Lorentzian manifold with spin structure carrying a twistor spinor with lightlike and non-twisting Dirac current is locally conformally equivalent to a Brinkmann space with parallel spinor. The Dirac current also plays an important role in the classification of imaginary Killing spinors on Lorentzian manifolds presented in [80], cf. also [18]. In one of the cases the gradient of the length function h ; i defines a conformal vector field, hence results discussed in Section 7 can be used. In the Riemannian case the same argument was used in [93] to classify manifolds carrying an generalized imaginary Killing spinor satisfying the equation rX D i bX for some real function b and all tangent vectors X. In [79, Theorem 1] it is shown that the zero set of a twistor spinor on a Lorentzian spin manifold consists either of isolated images of lightlike geodesics or of isolated points. In the first case the metric is outside the zero set locally conformally equivalent to a Brinkmann space with parallel spinor. In the second case the metric is outside the zero set locally conformally equivalent to a product metric of the form ds 2 C h, where h is a Riemannian metric carrying a parallel spinor. The second case actually occurs at least in the C 1 -category: Theorem 10.11. (Leitner [81]) There exists a five-dimensional manifold with a C 1 Lorentzian metric carrying a twistor spinor with an isolated zero. The metric is not conformally flat and the Dirac current is causal. It remains open whether such examples exist with a higher order of differentiability. On the other hand there is the following recent result: Theorem 10.12. (Frances [39, Corollary 2]) If an analytic Lorentzian manifold .M; g/ admits a non-zero twistor spinor which has a zero then the manifold is conformally flat. Frances actually proves the following statement about conformal vector fields of Lorentzian manifolds, cf. [39, Theorem 1]: If a smooth Lorentzian manifold of signature .1; n1/, n 3, carries a non-trivial conformal and causal vector field X (i.e. kX k 0) then there is an open and non-empty subset on which the manifold is conformally flat. As a generalization of conformal Killing vector fields one can consider conformal Killing forms, which are studied in detail by Semmelmann [97]. The defining equation is also called conformal Killing–Yano equation. As a particular example for twistor spinors 1 ; 2 on a Riemannian spin manifold the exterior k-form !k .X1 ; : : : ; Xk / D h.X1 ^ ^ Xk /

1;

2i


293

is a conformal Killing form. The twistor equation also allows a supersymmetric interpretation, cf. Alekseevskii et al. [3] and Klinker [59].

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The causal hierarchy of spacetimes Ettore Minguzzi and Miguel Sánchez

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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Elements of causality theory . . . . . . . . . . . 2.1 First definitions and conventions . . . . . . 2.2 Conformal/classical causal structure . . . . 2.3 Causal relations. Local properties . . . . . 2.4 Further properties of causal relations . . . . 2.5 Time-separation and maximizing geodesics 2.6 Lightlike geodesics and conjugate events . .

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3 The causal hierarchy . . . . . . . . . . . . . . . . . . . . . 3.1 Non-totally vicious spacetimes . . . . . . . . . . . . . 3.2 Chronological spacetimes . . . . . . . . . . . . . . . . 3.3 Causal spacetimes . . . . . . . . . . . . . . . . . . . . 3.4 Distinguishing spacetimes . . . . . . . . . . . . . . . 3.5 Continuous causal curves . . . . . . . . . . . . . . . . 3.6 Strongly causal spacetimes . . . . . . . . . . . . . . . 3.7 A break: volume functions, continuous I ˙ , reflectivity 3.8 Stably causal spacetimes . . . . . . . . . . . . . . . . 3.9 Causally continuous spacetimes . . . . . . . . . . . . 3.10 Causally simple spacetimes . . . . . . . . . . . . . . . 3.11 Globally hyperbolic spacetimes . . . . . . . . . . . . .

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1 Introduction Causality is an essential specific tool of Lorentzian geometry, which appears as a fruitful interplay between relativistic motivations and geometric developments. Most of the goals of this theory are comprised in the so-called causal hierarchy of spacetimes: a ladder of spacetimes sharing increasingly better causal properties, each level with some Partially supported by GNFM of INDAM and by MIUR under project PRIN 2005 from Camerino University. Partially supported by MEC-FEDER Grant MTM2007-60731 and J. Andal. Grant P06-FQM-01951.

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specific results. This ladder and its main features were established at the end of the 70s, after the works of Carter, Geroch, Hawking, Kronheimer, Penrose, Sachs, Seifert, Wu and others (essentially, the last introduced level was in [28]) and were collected in the first version of Beem–Ehrlich book (1981) – later re-edited with Easley, [2]. Nevertheless, there are several reasons to write this revision. A first one is that the “folk questions on smoothability” of time functions and Cauchy hypersurfaces, which were left open in that epoch, have been solved only recently [4], [6], [7]. They affect to two levels of the ladder in an essential way – the equivalence between two classical definitions of stable causality and the structure of globally hyperbolic spacetimes. Even more, new results which fit typically on some of the levels, as well as some new viewpoints on the whole ladder, have been developed in the last years. So, we think that the full construction of the ladder from the lowest level to the highest one, may clarify the levels, avoid redundant hypotheses and simplify reasonings. This paper is organized as follows. In Section 2 the typical ingredients of causality are introduced: time-orientation, conformal properties, causal relations, maximizing properties of causal geodesics, etc. Most of this introductory material is well-known and is collected in books such as [2], [27], [39], [40], [56]. Nevertheless, some aspects may be appreciated by specialists, as the introduction of globally hyperbolic neighborhoods (Theorem 2.14), the viewpoint of causal relation I C , J C , E C in M M (Section 2.4), or the conformal properties of lightlike pregeodesics (Theorem 2.36). The conformal invariance of some elements is stressed, even notationally (Remark 2.9). In Section 3 the causal ladder is constructed. The nine levels are developed in subsections, from the lowest (non-totally vicious) to the highest one (globally hyperbolic). Essentially, our aims for each level are: (a) to give natural alternative definitions of the level (see, for example, Definitions 3.11 or 3.59 and further characterizations), with minimum hypotheses (see Definitions 3.63 or 3.70, with Proposition 3.64, Remark 3.72); (b) to check its strictly higher degree of specialization, in a standard way; (c) to explain geometric techniques or specific results of the level (for example, see Theorems 3.3, 3.89, 3.91 or Sections 3.5, 3.7). In particular, we emphasize that only after the solution of the folk questions on smoothability, the classical characterization of causal stability in terms of the existence of a time function can be regarded as truly equivalent to the natural definition (see Theorem 3.56 and its proof). Even more, we detail the consequences of these folk questions for the structure of a globally hyperbolic spacetime (Theorem 3.78). Although the description of the smoothing procedure lies out of the scope of the present review (see [44], in addition to the original articles [4], [6], [7]), the main difficulties are stressed, Remark 3.77. Finally, in Section 4 we explain briefly the recent proposal of isocausality by GarcíaParrado and Senovilla [20]. This yields a partial ordering of spacetimes which was expected to refine the total order provided by the standard hierarchy. Even though, as proven later in [19], this ordering does not refine exactly the standard one, this is an alternative viewpoint, worth to be born in mind. Acknowledgement. The authors would like to thank Professors J. M. M. Senovilla and A. N. Bernal for their careful reading of this article.

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2 Elements of causality theory Basic references for this section are [2], [27], [37], [39], [40], [56], other useful references will be [9], [10], [19], [21], [30], [31], [51]. 2.1 First definitions and conventions Definition 2.1. A Lorentzian manifold is a smooth manifold M of dimension n0 2, endowed with a non-degenerate metric g W M ! T M ˝ T M of signature .; C; : : : ; C/. By smooth M we mean C r0 , r0 2 f3; : : : ; 1g. Except if otherwise explicitly said, the elements in M will be also assumed smooth, i.e., as differentiable as permitted by M (C r0 1 in the case of g, and C r0 3 for curvature tensor R)1 . Manifolds are assumed Hausdorff and paracompact, even though the latter can be deduced from the existence of a non-degenerate metric (recall that the bundle of orthonormal references is always parallelizable; thus, it admits a – positive definite – Riemannian metric, which implies paracompactness [52, Vol. II, Addendum 1], [34]). The following convention includes many of the ones in the bibliography (the main discrepancies come from the causal character of vector 0, which somewhere else is regarded as spacelike [39]), and can be extended for any indefinite scalar product: Definition 2.2. A tangent vector v 2 TM is classified as: • • • • • •

timelike, if g.v; v/ < 0; lightlike, if g.v; v/ D 0 and v ¤ 0; causal, if either timelike or lightlike, i.e., g.v; v/ 0 and v ¤ 0; null, if g.v; v/ D 0; spacelike, if g.v; v/ > 0; nonspacelike, if g.v; v/ 0.

At each tangent space Tp M , gp is a (non-degenerate) scalar product, which admits an orthonormal basis Bp D .e0 ; e1 ; : : : ; en1 /; gp .e ; e / D ı , where ı is Kronecker’s delta and 0 D 1; i D 1 (Greek indexes ; run in 0; 1; : : : ; n 1, while Latin indexes i; j run in 1; : : : ; n 1). Each .Tp M; gp /; p 2 M contains two causal cones. Definition 2.2 is naturally extended to vector fields X 2 X.M / and curves W I ! M (I R interval of extremes,2 1 a < b 1). Nevertheless, when I D Œa; b we mean by timelike, lightlike or causal curve any piecewise smooth curve W I ! M , such that not only the tangent vectors are, respectively, timelike, lightlike or causal, but also the two lateral tangent vectors at each break lie in the same 1 We will not care about problems on differentiability (see the review [50, Section 6.1]). But notice that, essentially, r0 D 2 suffices throughout the paper (the exponential map being only continuous), with the remarkable exception of Section 2.6. Moreover, taking into account that globally hyperbolic neighborhoods make sense for r0 D 1, many elements are extendible to this case, see also [51]. 2 We use as a reflexive relation as in [40], that is, for any set A, A A.

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causal cone. The notion of causal curve will be extended below non-trivially to include less smooth ones, see Definition 3.15. A time-orientation at p is a choice of one of the two causal cones at Tp M , which will be called future cone, in opposition of the non-chosen one or past cone. In a similar way that for usual orientation in manifolds, a smooth choice of time-orientations at each p 2 M (i.e., a choice which coincides at some neighborhood Up with the causal cone selected by a – smooth – causal vector field on Up ) is called a time-orientation. The Lorentzian manifold is called time-orientable when one such time-orientation exists; no more generality is obtained either if smooth choices are weakened in C r ones, r 2 f0; : : : ; r0 1g, or if causal choices are strengthened in timelike ones. As the causal cones are convex, a standard partition of the unity argument yields easily: Proposition 2.3. A Lorentzian manifold is time-orientable if and only if it admits a globally defined timelike vector field X .which can be chosen complete3 /. Recall that this vector field X can be defined to be future-directed at all the points and, then, any causal tangent vector vp 2 Tp M is future directed if and only if g.vp ; Xp / < 0. Easily one has: (a) any Lorentzian manifold admits a time-orientable double covering [40], [39, Lemma 7.17]; and (b) let gR be any Riemannian metric on M and X 2 X.M / non-vanishing, with gR -associated 1-form X [ , then gL D gR

2 X[ ˝ X[ gR .X; X /

is a time-orientable Lorentzian metric. As a consequence, the possible existence of Lorentz metrics can be characterized [39, 5.37], [2, Section 3.1], [40, Section 1]: Theorem 2.4. For any connected smooth manifold, the following properties are equivalent: (1) (2) (3) (4)

M admits a Lorentz metric. M admits a time-orientable Lorentz metric. M admits a non-vanishing vector field X . Either M is non-compact or its Euler characteristic is 0.

Proof. (3) , (4) Well-known result in algebraic topology. (2) , (3) To the right, Proposition 2.3; to the left, comment (b) above. .1/ ) .2/ (The converse is trivial.) The time-orientable double covering .MQ ; g/, Q satisfies (3) and hence (4). So, the latter is satisfied obviously by M . The relevant new ingredient of a spacetime is a time-orientation: 3 X can be chosen complete because, given X and an auxiliary complete Riemannian metric (which exists due to a theorem by Nomizu and Ozeki [38]) it can be replaced by the timelike vector field X=jX jR , which is necessarily complete.


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Definition 2.5. A spacetime .M; g/ is a time-oriented connected Lorentz manifold. The points of M are also called events. Notice that the time-orientation is implicitly assumed in the notation .M; g/ for a spacetime. In principle, M is not assumed to be an orientable manifold. Recall that orientability and time-orientability are logically independent. In fact, one can construct easily time-orientable and non-time-orientable Lorentz metrics on both a Möbius strip (or Klein bottle) and cylinder (or torus) by starting with the metric g on R2 , g.X1 ; X2 / 1;

g.X1 ; X1 / 0 g.X2 ; X2 /;

X1 D cos x @x C senx@y ;

X2 D senx @x C cos x@y ;

by making natural quotients (see Figure 1). Twist and identify (no - nto)

xD0

Twist and identify (no - to) x D 1=2 xD1

x D 3=2

X2 X1

Identify (o - nto) Identify (o - to)

Figure 1. Time-orientability and orientability are logically independent. Here we use the shorthand notation: o D orientable; no D non-orientable; to D time-orientable; nto D non-timeorientable. Figures in which the causal cones are explicitly displayed are standard in causality theory. In this work, if the spacetime is time-oriented the past cones are displayed in black, see Figure 2.

2.2 Conformal/classical causal structure. The following algebraic result (Dajczer et al. criterion) has important consequences for the conformal structure of spacetimes, and has no analog in the positive definite case:

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Proposition 2.6. Let .V; g/ be a real vector space with a non-degenerate indefinite scalar product, and let b be a bilinear symmetric form on V . The following properties are equivalent: 1. b D c g for some c 2 R, 2. b.v; v/ D 0 if g.v; v/ D 0. The proof can be seen also in [2, Lemma 2.1], [56, Appendix D]. Obviously, 1 ) 2, and the converse can be proved in dimension 2 easily; for higher dimensions, the problem is reduced to dimension 2, by grouping suitably the elements of a g-orthonormal basis. By the way, it is also known that any of the following conditions is equivalent to items 1, 2 (this yields bounds on the possible curvatures): (a) there exists a > 0 such that b.v; v/=g.v; v/ a if g.v; v/ ¤ 0, (b) there exists a > 0 such that b.v; v/=g.v; v/ a if g.v; v/ ¤ 0, (c) there exists a > 0 such that jb.v; v/j ajg.v; v/j if g.v; v/ < 0, and (d) there exists a > 0 such that jb.v; v/j ajg.v; v/j if g.v; v/ > 0. In fact, any of these items implies item 2, by using that any lightlike vector can be approximated by both, timelike and spacelike ones. For some algebraic extensions to higher order tensors, see [3]. Two Lorentzian metrics g; g on the same manifold M are called pointwise conformal if g D e 2u g for some function u W M ! R. Proposition 2.6 yields directly: Lemma 2.7. Two Lorentzian metrics g; g on a manifold M of dimension n0 > 2 are pointwise conformal if and only if both have the same lightlike vectors. (The exceptional case n0 D 2 appears because a negative conformal factor keep lightlike vectors unchanged, while exchanges timelike and spacelike vectors.) Two spacetimes on the same manifold M are pointwise conformal if both, their metrics are pointwise conformal and their time-orientations agree at each event. The spacetime .M; g/ is called conformal to the spacetime .M ; g / if there exists a diffeomorphism ˆ W M ! M such that the pull-back spacetime on M obtained inducing the metric and the time-orientation through ˆ is pointwise conformal to .M; g/. Two spacetimes which only differ in the time-orientation are by definition not pointwise conformal and, moreover, they may be also non-conformal (see, for example, Figure 14 at the end). Clearly, the conformal relation is a relation of equivalence in the class of all the spacetimes. The following definition will be revisited in Section 4, in order to discuss what causality means. Definition 2.8. The conformal or classical causal structure of the spacetime .M; g/ is its equivalence class Œ.M; g/ for the conformal relation. Several concepts in Lorentzian geometry do depend on the full metric structure of the spacetime. Examples are the length of a curve, the time-separation between two events (see below), the non-lightlike geodesics or the geodesic completeness of a spacetime. Nevertheless, the conformal structure is particularly rich by itself, and its interplay with the metric becomes specially interesting.


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Remark 2.9. The elements which come only from the conformal structure will be emphasized with the following conventions. For practical purposes, we will work with the relation of equivalence induced by the pointwise conformal relation in the spacetimes on the same M . For the spacetime .M; g/, its pointwise conformal class will be denoted as .M; g/ (g denotes the set of all pointwise conformal metrics to g) where all the spacetimes in the class have the same time-orientation. When we refer to a spacetime as .M; g/, we emphasize that the considered properties hold for any g in the class and, thus, depend only on the conformal structure. The boldface will be extended to classes of equivalence of vectors and curves. So, v denotes the equivalence class of vectors v 0 D ˛v, ˛ > 0 and g.v; w/ is just the sign .1; 0; C1/ of the scalar product g.v; w/. Analogously, if W I ! M is a curve then is the equivalence class of curves coincident with up to a strictly increasing reparametrization. Note that if I is closed (or compact or open) for a representative of , the same holds for any representative W I ! M . Analogously, we say that connects p with q if for a representative, I D Œa; b, .a/ D p and .b/ D q, or write p 2 if p belongs to the image of . If a future-directed causal curve satisfies lim t!b .t / D q (resp. lim t!a .t / D p), where a; b .1 a < b 1/ are the extremes of the interval I , the event q (resp. p) is called the future (resp. past) endpoint of (and the other way round if is past-directed). These concepts are obviously extended to , so one can assume that I is bounded when dealing with the endpoints of . A causal curve without future (resp. past) endpoint is said future (resp. past) inextendible. 2.3 Causal relations. Local properties. Given a spacetime .M; g/ the event p is chronologically (resp. strictly causally; causally; horismotically) related to the event q, denoted p q (resp. p < q; p q; p ! q) if there is a future-directed timelike (resp. causal; causal or constant; causal or constant, but not timelike) curve connecting p with q. If W M , given p; q 2 W , the analogous relations for the spacetime .W; gjW / will be denoted p W q, p <W q, p W q, p !W q. From the viewpoint of set theory, relations ; ; ! are written, regarded as subsets of M M , as: I C D f.p; q/ W p qg;

J C D f.p; q/ W p qg;

E C D f.p; q/ W p ! qg:

Clearly, E ˙ D J ˙ nI ˙ . Note: all the definitions and properties extend naturally to the “minus” sign without further mention; for example, the sets (and binary relations) I , J , and E , are defined changing each .p; q/ above by .q; p/. The chronological future of an event is defined as: I C .p/ D fq 2 M W p qg D 2 .11 .p/ \ I C / D 1 .21 .p/ \ I / where 1 and 2 are the canonical projections to the factors of M M . Analogous expressions hold for the causal future J C .p/ and horismos E C .p/. By using juxtapositions of curves, it is obvious that the relations and are transitive, but ! is not (see Proposition 2.31).

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Every point of a spacetime admits an arbitrarily small (i.e. contained in any given neighborhood) convex neighborhood U , that is, U is a (starshaped) normal neighborhood of any of its points p 2 M . This means that the domain UQ Tp M of the exponential map at p is chosen starshaped and yields a diffeomorphism onto U , expp W UQ ! U . Thus, for any p; q 2 U , there exists a unique geodesic pq W Œ0; 1 ! U which connects p with q. Notice that one also has a diffeomorphism [39, Lemma 5.9] ! 2 TM . between U U and its image on T U , which sends .p; q/ ! expp1 q D pq Such a convex U can be chosen simple, that is, with compact closure Ux included in another open convex neighborhood [40, p. 6]. Given an open subset U by I C .p; U /, J C .p; U /, E C .p; U /, will be denoted the corresponding future elements in U regarded as spacetime. If U is a convex neighborhood the causal relations in U are easily characterized [39, Lemma 14.2]: Proposition 2.10. Let .M; g/ be a spacetime, expp the exponential map at p 2 M , and U a convex neighborhood. Regarding U as a spacetime, given p ¤ q, p; q 2 U : ! D exp1 q is pq 1. q 2 I C .p; U / .resp. q 2 J C .p; U / I q 2 E C .p; U // , p

2. 3. 4. 5.

timelike .resp. causal; lightlike/ and future-pointing. I C .p; U / is open in U .and M /. J C .p; U / is the closure in U of I C .p; U /. Causal relation J C is closed in U U . Any causal curve contained in a compact subset of U has two endpoints.

Notice from the first item (which can be regarded as a consequence of Theorem 2.26 below) that the study of the causal relations in U is reduced to the study of the causal ! character of tangent vectors type pq. Nevertheless, convex neighborhoods depend on the metric structure. The following concept is purely conformal. Definition 2.11. Let U; V be open subsets of a spacetime .M; g/, with V U . V is called causally convex in U if any causal curve contained in U with endpoints in V is entirely contained in V . In particular, when this holds for U D M , V is called causally convex. Remark 2.12. Note that, in this case, if U , V denote, resp., the causal relations in U , V regarded as spacetimes, then the restriction of U to V agrees with V (this property does not characterize causal convexity, as can be checked from U D Ł2 , V D f.t; x/ 2 R2 W jtj; jxj < 1g). There are spacetimes such that the only open subset V ¤ ; of U D M which is causally convex is V D M (see the totally vicious ones in Section 3.1). Nevertheless, at least when U is also convex, the existence of arbitrarily small such V in U , even with further properties, will be shown next. Finally, note that if V is causally convex in U and W is an open set such that V W U , then V is causally convex in W . It is also possible to prove that any point of .M; g/ admits a neighborhood with the best possible causal structure, i.e., which will belong to the top of the ladder, global


307

hyperbolicity (see Section 3.11). Recall first the following result (by g < g 0 we mean that the causal cone of g at each point p is included in the timelike cone of g 0 at p, see Section 3.8). Lemma 2.13. Let .M; g/ be a spacetime. Given p 2 M , and a neighborhood U 3 p, there exists a neighborhood V 3 p, V U and two flat metrics g , g C on V such that g < g < g C . Proof. Take a coordinate neighborhood .Vı ; .t D x 0 ; x 1 ; : : : ; x n1 //, centered at p, P 2 2 .x / < ı , such that the tangent basis Bp D .@0 ; @1 ; : : : ; @n1 / is orthonormal at p according to a representative g of g. Now, recall that the scalar product gp (resp. gpC ) at Tp M , such that Bp D .2@0 ; @1 ; : : : ; @n1 / (resp. BpC D ..1=2/@0 ; @1 ; : : : ; @n1 /) is an orthonormal basis has the cones strictly less (resp. more) open than the cones of gp . By continuity, this property holds in a neighborhood Vı of p, for ı sufficiently small, moreover given U , by taking ı sufficiently small we have Vı UPand so the required P metrics are g D .1=4/dt 2 C i .dx i /2 and g C D 4dt 2 C i .dx i /2 . Let Łn be Lorentz–Minkowski spacetime with natural coordinates .x / D .t D x ; x i /, p ; q 2 Łn , p D .; 0; : : : ; 0/, q D .; 0; : : : ; 0/, > 0. The open neighborhood in Łn , V D I C .p / \ I .q /, satisfies that t 0 is a spacelike Cauchy hypersurface S of V , that is, it is crossed exactly once by any inextendible timelike curve contained in V (see Section 3.11). We will mean by a globally hyperbolic neighborhood of p any coordinate neighborhood .V; x / such that x 0 0 is a Cauchy hypersurface of V . The following result shows that the local structure of a spacetime fulfills all good properties from the viewpoint of causality (see also the study in [29, Section 2]). 0

Theorem 2.14. Let .M; g/ be a spacetime. For any p 2 M and any neighborhood U 3 p there exists an open neighborhood U 0 , p 2 U 0 U , and a sequence of nested globally hyperbolic neighborhoods .Vn ; x /, VnC1 Vn , fpg D \n Vn , all included in U 0 , such that each Vn is causally convex in U 0 . Proof. Consider the metric g C in Lemma 2.13 defined in some neighborhood U 0 U of p. As it is flat, one can find the required sequence of globally hyperbolic neighborhoods .Vn ; x / for g C , Vn U 0 , each one g C -causally convex in U 0 . Nevertheless, any causal curve for g will be timelike for g C and, so, each .Vn ; x / will be both, globally hyperbolic and causally convex for g. Remark 2.15. (1) Of course, in Theorem 2.14 (which is formulated in a conformally invariant way) we can assume U 0 D V1 . Nevertheless, it is clear from the proof that, for any representative g of the conformal class, U 0 can be chosen simple, which leads to the strongest local causal properties. (2) The sequence fVn gn yields a topological basis at p. Thus, an alternative formulation of Theorem 2.14 would ensure the existence of a (simple) U 0 U which admits arbitrarily small globally hyperbolic neighborhoods of p, all of them causally convex at U 0 .

308


(3) It also holds that each obtained neighborhood V D Vn of the sequence satisfies: each q 2 V admits an arbitrarily small neighborhood which is causally convex in V . In fact, this property is one of the alternative definitions of being strongly causal, see Section 3.6. Hence we have also proved that any spacetime .M; g/ is locally strongly causal, as any point p admits an arbitrarily small strongly causal neighborhood V . Also, due to the last observation of Remark 2.12 any open set W V , p 2 W , is a strongly causal neighborhood of p as well. In particular, any spacetime .M; g/ admits arbitrarily small simple strongly causal neighborhoods. 2.4 Further properties of causal relations. None of the properties in Proposition 2.10, but the second one, holds globally. In fact, given a timelike curve connecting the pair .p; q/ there are open neighborhoods U 3 p, V 3 q such that if pQ 2 U , qQ 2 V , then there exists a timelike curve Q connecting pQ and qQ (say, U; V can be chosen as I .p1 / \ Up , I C .q1 / \ Uq , where Up , Uq are convex neighborhoods of p; q which contains p1 , q1 , resp., and these points are chosen such that runs consecutively p, p1 , q1 , q). Summing up, Proposition 2.16. The set I C is open in M M . In what follows we claim that p r and r q (or the other way round) implies p q (see Proposition 2.31 for a more accurate result). In general, J C .p/ INC .p/ but the equality may not hold. Nevertheless, both closures as well as both boundaries (denoted with a dot in what follows) and interiors (denoted Int) coincide. Even more: Proposition 2.17. It holds that JN C D INC , Int J C D I C , and JP C D IPC . Proof. Since I C J C we have INC JN C . Let .p; q/ 2 JN C and let U and V be arbitrarily small neighborhoods of respectively p and q. There are events p 0 2 U , q 0 2 V , such that .p 0 ; q 0 / 2 J C . Take events p 00 2 U \ I .p 0 / and q 00 2 V \ I C .q 0 /. Then p 00 can be connected to q 00 with the composition of a timelike, a causal, and finally a timelike curve, and, as claimed above, it follows p 00 q 00 . Since U and V are arbitrary .p; q/ is an accumulation point for points belonging to I C . We conclude that INC D JN C . Let us show that I C D Int J C from which it follows JP C D IPC . Since I C is open and included in J C , I C Int J C . If .p; q/ 2 Int J C , then chosen normal convex neighborhoods U 3 p, V 3 q, such that U V is included in Int J C , and taken p 0 2 U \ I C .p/, q 0 2 V \ I .q/, then q 0 2 J C .p 0 / and thus q 2 I C .p/, i.e., .p; q/ 2 I C . Definition 2.18. An open subset F (resp. P ) is a future (resp. past) set if I C .F / D F (resp. I .P / D P ). An example of future set is I C .p/ for any p 2 M . We have the following characterization: Proposition 2.19. If F is a future set then Fx D fp W I C .p/ F g, and analogously in the past case.


309

Proof. ( ). If I C .p/ F then p 2 INC .p/ Fx. (). Let p 2 Fx and take any q 2 I C .p/. As I .q/ 3 p is open, I .q/ \ F ¤ ;. Thus, q 2 I C .F / D F , i.e. I C .p/ F . Remark 2.20. Even though the closure of J C in M M induces a binary relation, this is not always transitive. As closedness becomes relevant for different purposes (for example, when one deals with limit of curves) Sorkin and Woolgar [51] defined the Krelation as the smallest one which contains and is: (i) transitive, and (ii) topologically closed. (That is, the corresponding set K C M M which defines the K-relation, is the intersection of all the closed subsets C which contain I C such that .p; q/; .q; r/ 2 C ) .p; r/ 2 C ). Among the applications, some results on positive mass and globally hyperbolic spacetimes with lower order of differentiability (r0 D 1) have been obtained. Notice that, in particular, INC K C , but perhaps there exists .p; q/ 2 K C nJN C . In this case, q … INC .p/ and hence there is a point r 2 I C .q/ not contained in I C .p/. As a consequence .p; q/ 2 K C and r 2 I C .q/ do not imply r 2 I C .p/ (as happens when the causal relation K C is replaced with J C , see Proposition 2.31). In particular, the relation K C does not define a causal space in the sense of Kronheimer and Penrose [30]. Nevertheless, this cannot happen if .M; g/ is causally simple, because then INC D JN C D J C D K C (see Section 3.10); moreover, .I C ; K C / define such a causal space if and only if .M; g/ is causally continuous [15]. Remark 2.21. In general .p; q/ 2 INC does not imply q 2 INC .p/ or p 2 IN .q/ (see Figure 2). For this reason it may be more useful to regard the causal relations as defined in M M , although it is customary to introduce them in M , that is, through I ˙ .p/, J ˙ .p/, E ˙ .p/. q Remove

p

Figure 2. Minkowski spacetime without a spacelike half-line is an example of stably causal noncausally continuous spacetime (see Sections 3.8–3.9). Here .p; q/ 2 INC , but neither q 2 INC .p/, nor p 2 IN .q/.

Recall that we have defined three binary relations ; ; ! (and trivially a fourth one 0 D d.p; q/.

The main Riemannian similarities come from the maximizing properties of causal geodesics, which are consequences of an infinitesimal application of reversed triangle inequality. Concretely, the maximizing properties can be summarized in the following two results (see, for example, [39, Lemma 5.34, 5.9], or around [50, Proposition 2.1]), the first one local (see also Proposition 2.10) and the second global: Theorem 2.26. Let U be a convex neighborhood of .M; g/, and p; q 2 U . Assume there exists a causal curve ˛ W Œ0; b ! U from p to q. Then, the radial segment ! D exp1 .q/ and length pq W Œ0; 1 ! U from p to q .which has initial velocity pq p q ! ! ! jpq D j jg.pq; pq/j /, is causal and, up to reparametrization, maximizes strictly the length among all the causal curves in U which connect p to q. In particular, if pq is lightlike then it is the unique causal curve contained entirely in U which connects p to q.

312


Theorem 2.27. Assume that there exists a causal curve ˛ W Œ0; b ! M which connects p to q, p; q 2 M , with maximum length among all the causal curves which connect p to q in the spacetime .M; g/. Then ˛ is, up to a reparametrization, a causal geodesic without conjugate points .Definition 2.32/ except, at most, the endpoints. That is: (i) the length of a causal geodesic contained in a convex neighborhood is equal to the time-separation (computed in the neighborhood as a spacetime), of its endpoints, and (ii) if a causal curve in the spacetime has a length equal to the timeseparation of its endpoints, then it is, up to a parametrization, a causal geodesic without conjugate points, except at most its endpoints. Recall that if p; q 2 M satisfy p < q and d.p; q/ D 0, then these two properties are conformally invariant. So, Theorem 2.27 implies that any lightlike geodesic (and its first conjugate point) must be conformally invariant, up to a reparametrization. Next, we will see that this can be made much more precise. 2.6 Lightlike geodesics and conjugate events. It is known (see Section 2.3 in [9], p. 367 in this volume) that a curve W I ! M with non-vanishing speed 0 is a pregeodesic (i.e., it can be reparametrized as a geodesic for the Levi-Civita connection r of the spacetime) if and only if it satisfies r 0 0 D f 0

(1)

for some function f W I ! R. Explicitly, the reparametrization is Q .Qs / D .s.Qs // where, for constants sQ0 2 R, s0 2 I , sQ00 ¤ 0, Z s Rt 0 e s0 f .r/dr dt: (2) sQ .s/ D sQ0 C sQ0 s0

If is a lightlike geodesic for g then it satisfies (1) for the Levi-Civita connection r of any conformal metric g D e 2u g, being f D 2 d.uı/ (see [9] or the proof of dt Theorem 2.36 below) and, thus, with the natural choice of sQ00 in (2): Q .Qs / D .s.Qs // lightlike geodesic with Q 0 D e 2u 0 :

(3)

That is, lightlike pregeodesics are (pointwise) conformally invariant and the following definition for the conformal class makes sense. Definition 2.28. Given .M; g/, a lightlike curve is a lightlike geodesic if for a choice of representatives (and hence for any choice), g and , Equation (1) holds. Note that although the concept of lightlike geodesic makes sense given only the conformal structure, the definitions of timelike and spacelike geodesics do not. The fact that two events have a zero time-separation is also conformally invariant and, thus, the following definition makes also sense. Definition 2.29. A lightlike curve connecting two events p and q is maximizing if there is no timelike curve connecting p and q.


313

Recall that this concept is a pure conformal one, but the notion of maximizing for timelike curves depends on the metric. The following result is standard, and relies on the possibility to deform any causal curve which is not a lightlike geodesic without conjugate points in a timelike one (see, for example, [2, Corollary 4.14], [27, Proposition 4.5.10] or [40, Proposition 2.20]). As discussed below Theorem 2.27, all these elements are conformally invariant, and the result is stated consequently. Theorem 2.30. Let .M; g/ be a spacetime. (i) Each two events p; q 2 M connected by a causal curve which is not a maximizing lightlike curve are also connected by a timelike curve. (ii) Any maximizing lightlike curve is a lightlike geodesic of g without conjugate points .i.e., when reparametrized as a lightlike geodesic for any g 2 g, does not have conjugate points/ except at most the endpoints. In fact, the timelike curve in (i) can be chosen arbitrarily close (in the C 0 topology) to . As a straightforward consequence, one has: Proposition 2.31. Two events p; q are horismotically related if and only if they can be joined by a maximizing lightlike geodesic. Thus: (i) If p r and r q then p q .analogously, if p r and r q then p q/. (ii) If r 2 E C .p/ and q 2 E C .r/ then either q 2 E C .p/ or p q. The conformal invariance of conjugate points along lightlike geodesics is not only a consequence of maximizing properties, (which would be applicable only in a restricted way, for example, it would apply only for the first conjugate point) but a deeper one. Next, our aim is to show that the definition of Jacobi field in the lightlike case can be made independent of the metric and indeed depends only on the conformal structure. As a consequence the concept of conjugate point and its multiplicity, depends also only on the conformal structure for lightlike geodesics, while in the timelike case it requires the metric. We begin with the metric-dependent definition of Jacobi field, and show later that it can be made independent of the conformal factor in the lightlike geodesic case. Definition 2.32. Let W I ! M be a geodesic of a spacetime (or any semi-Riemannian manifold), .M; g/. A vector field J on is a Jacobi field if it satisfies the Jacobi equation J 00 C R.J; 0 / 0 D 0 where R is the (Riemann) curvature tensor, R.X; Y / D ŒrX ; rY rŒX;Y . The events p D .sp / and q D .sq /; sp < sq are said to be conjugate (of multiplicity m) if there exist m > 0 independent Jacobi fields such that J.sp / D 0 D J.sq /. As in the (positive-definite) Riemannian case, one has: Lemma 2.33. For any geodesic W I ! M of .M; g/:

314


(i) The variation vector field V of by means of a variation .s; v/ ! v .s/ with geodesic longitudinal curves .at constant v/, is a Jacobi field. (ii) If J is a Jacobi field for then g.J; 0 /.s/ D as C b for suitable constants a and b and all s 2 I . Thus: (a) If J vanishes at the endpoints, then g.J; 0 / D 0. (b) The only Jacobi fields proportional to 0 , J.s/ D f .s/ 0 .s/ satisfy f D cs C d for suitable constants c and d , hence if they vanish at the endpoints they vanish everywhere. (c) If J1 and J2 are two Jacobi fields vanishing at the endpoints and J2 D J1 C f 0 for some function f , then they coincide. As two causal vectors cannot be orthogonal, a straightforward consequence of part (a) is: Proposition 2.34. Let be lightlike and let J be a Jacobi field which vanishes at the endpoints but not everywhere, then J is spacelike and orthogonal to 0 . In particular, no lightlike geodesic W I ! M in a 2-dimensional spacetime admits a pair of conjugate events. Indeed, the last assertion follows because no spacelike vector field J exists which is orthogonal to 0 . In what follows will be always lightlike. We are interested in the case of conjugate points. It is convenient to introduce the space N. 0 / of vector fields over orthogonal to 0 and the quotient space Q of vector fields of N. 0 / defined up to additive terms of type f 0 . If X 2 N. 0 / is a vector field orthogonal to 0 then ŒX 2 Q will denote its equivalence class. Let W N. 0 / ! Q, .X / D ŒX be the natural projection. The covariant derivative, also denoted 0 D r 0 , can be induced on Q by making it to commute with , i.e. ŒX 0 D ŒX 0 . This definition is independent of the representative because: (i) X 0 2 N. 0 /, since g.X 0 ; 0 / D g.X; 0 /0 D 0, (ii) ŒX C f 0 0 D ŒX 0 C .f 0 /0 D ŒX 0 C f 0 0 D ŒX 0 . Even more, the curvature term in the Jacobi equation can be projected to the map R W Q ! Q defined as RŒX D .R.X; 0 / 0 /; (4) which, again, is independent of the chosen representative X . Lemma 2.35. If J 2 N. 0 / is a Jacobi field then ŒJ 2 Q is a Jacobi class, that is, it solves the quotient Jacobi equation ŒJ 00 C RŒJ D 0

(5)

.where the zero must be understood in Q, that is, as the class of any vector field pointwise proportional to 0 /.


315

Conversely, if ŒJN 2 Q is a Jacobi class in the sense of Equation (5) and Jp ; Jq 2 TM are orthogonal to at .sp /, .sq /, sp < sq , with ŒJN p D ŒJp , ŒJN q D ŒJq , then there exist a representative J 2 N. 0 /; ŒJ D ŒJN ; which is a Jacobi field and fulfills the boundary conditions J.sp / D Jp , J.sq / D Jq . In particular, if ŒJN vanishes at the endpoints, then there exists a representative J which vanishes at the endpoints. Proof. The first statement is obvious. For the converse, JQ 00 C R.JQ ; 0 / 0 D h 0 for some suitable function h. Let J be another representative, JQ D J Cf 0 , with f 00 D h. Then J is a Jacobi field, and given the initial conditions, fp D f .sp / and fq D f .sq /, function Z sq Z s 0 Z s Z s0 s sp 00 0 00 fq fp hds ds C fp C hds ds 0 ; f .s/ D sq sp sp sp sp sp solves the problem. These lemmas imply that in order to establish whether two events p and q are conjugate along a lightlike geodesic (and its multiplicity, i.e., the dimension of the space of Jacobi fields vanishing at the endpoints) it is easier to look for Jacobi fields vanishing at the endpoints in the quotient space Q, as the reduced Equation (5) collects the relevant information. Theorem 2.36. The quotient Jacobi equation (5) is invariant under conformal transformations, that is: if g D e 2u g, the curve is a lightlike g-geodesic, Q is its parametrization as a g -geodesic .given by (3)/, J 2 N. 0 /, and JQ 2 N.Q 0 / is the corresponding reparametrization of J on Q , then ŒJ satisfies Equation (5) on .taking RŒJ from (4)/ if and only if ŒJQ satisfies Equation (5) on Q .where R ŒJQ is defined Q D .R .XQ ; Q 0 /Q 0 /, and R denotes the curvature tensor of g /. as R ŒX Thus, the concept of conjugate events p and q along a lightlike geodesic , and its multiplicity, is well defined for the conformal structure .M; g/. Proof. We will put ŒX Q D ŒrQ0 X , X 0 D r 0 X and use index notation as in [56, Appendix D], a; b; c; d D 0; : : : ; n1, (see [9] for more intrinsic related computations). It is proved in that reference: c ra X c D ra X c C Cab Xb; c c where Cab D 2ı.a @b/ u gab g cd @d u, which implies that if X 2 N. 0 /.D N.Q 0 / up to reparametrizations), c ŒXQ D e 2u ŒX 0 C Cab X b . 0 /a D e 2u ŒX 0 C u0 X D e 2u .ŒX 0 C u0 ŒX /;

and in particular

ŒX Q Q D e 4u .ŒX 00 C u00 ŒX .u0 /2 ŒX /:

(6)

We use the transformation of the Riemann tensor under conformal transformations (see, for example, [56]) d d .R /dcab D Rdcab 2ıŒa rb @c u C 2g de gcŒa rb @e u 2.@Œa u/ıb @c u d ef C 2.@Œa u/gbc g df @f u C 2gcŒa ıb g .@e u/.@f u/:

316


Using a0 J a D a0 0a D 0, .R /dcab . Q/c J a . Q/b d d D e 4u fRdcab 2ıŒa rb @c u C 2g de gcŒa rb @e u 2.@Œa u/ıb @c u d ef C 2.@Œa u/gbc g df @f u C 2gcŒa ıb g .@e u/.@f u/g. 0 /c J a . 0 /b

D e 4u fRdcab ıad rb @c u C .@b u/ıad @c ug. 0 /c J a . 0 /b C f . 0 /d ; for a suitable function f . This equation reads (up to reparametrizations) R ŒJ D e 4u .RŒJ u00 ŒJ C .u0 /2 ŒJ /; which together with Equation (6) for X D J , gives the thesis.

3 The causal hierarchy As explained in the Introduction, the aim of this section is to construct the causal ladder, a hierarchy of spacetimes according to strictly increasing requirements on its conformal structure. Essentially, some alternative characterizations of each level will be studied, as well as some of its main properties, checking also that each level is strictly more restrictive than the previous one. At the top of this ladder globally hyperbolic spacetimes appear. Even though somewhat restrictive, this last hypothesis is, in some senses, as natural as completeness for Riemannian manifolds. Even more, according to the Strong Cosmic Censorship Hypothesis, the natural (generic) models for physically meaningful spacetimes are globally hyperbolic ones. So, these spacetimes are the main target of causality theory, and it is important to know exactly the generality and role of their hypotheses. Most of the levels are related to the non-existence of travels to the past either for observers travelling through timelike curves (“grandfather’s paradox”), or for light beams, or for certain related curves. It is convenient to distinguish between the following notions, especially in the case of causal geodesics: Definition 3.1. Let W Œa; b ! M be a piecewise-smooth curve with non-vanishing velocity at any point: (a) is a loop (at p) if .a/ D .b/ D p; (b) is closed if it is smooth and 0 .a/ D 0 .b/ (following our convention in Remark 2.9 for vectors). (c) is periodic if it is closed with 0 .a/ D 0 .b/ Recall that if is a lightlike geodesic, the properties of being closed or periodic are conformal invariant; moreover, such a closed can be extended to a complete geodesic if and only if it is periodic (see in these proceedings [9]). For non-lightlike geodesics, the notions of closed and periodic become equivalent.


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Globally hyperbolic + Causally simple + Causally continuous + Stably causal + Strongly causal + Distinguishing + Causal + Chronological + Non-totally vicious Figure 4. The causal ladder.

3.1 Non-totally vicious spacetimes. Recall that if p p then there exist a timelike loop at p and, giving more and more rounds to it, one finds d.p; p/ D 1. Even more, if this property holds for all p 2 M , then I C .p/; I .p/ are both, open and closed. So, one can check easily the following alternative definitions. Definition 3.2. A spacetime .M; g/ is called totally vicious if it satisfies one of the following equivalent properties: (i) d.p; q/ D 1 for all p; q 2 M . (ii) I C .p/ D I .p/ D M for all p 2 M . (iii) Chronological relation is reflexive: p p for all p 2 M . Accordingly, a spacetime is non-totally vicious if p 6 p for some p 2 M . Of course, it is easy to construct non-totally vicious spacetimes. Nevertheless, totally vicious ones are interesting at least from the geometric viewpoint, and sometimes even in physical relativistic examples (Gödel spacetime is the most classical example). Let us consider an example. A spacetime .M; g/ is called stationary if it admits a timelike Killing vector field K; classical Schwarzschild, Reissner Nordström or Kerr spacetimes (outside the event horizons) are examples of stationary spacetimes. This definition depends on the metric g, but the fact that K is conformal Killing depends only on the conformal class g. Moreover, if K is timelike and conformal Killing then it selects a unique representative g K of g such that g K .K; K/ 1 (and then K will

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be Killing for g K , as the conformal factor through the integral lines of K must be equal to 1; see also, for example, [43, Lemma 2.1]). Theorem 3.3 ([47]). Any compact spacetime .M; g/ which admits a timelike conformal Killing vector field K is totally vicious. Sketch of proof. In order to prove that each p 2 M is crossed by a timelike loop, it is enough to prove that there exists a timelike vector field X with periodic integral curves. Recall that K is Killing not only for the selected metric g K D g=g.K; K/ in the conformal class g, but also for the associated Riemannian metric gR : gR .u; v/ D g K .u; v/ C 2g K .u; K/g K .v; K/ for all u; v 2 Tp M , p 2 M . Now, let G be the subgroup generated by K of the isometry group Iso.M; gR /. Then x satisfies: its closure G x is compact, because so is Iso.M; gR / (recall that gR is Riemannian and M is • G compact). x is abelian, because so is G. • G x is a k-torus, k 1, and there exists a sequence of subgroups • As a consequence, G Sm diffeomorphic to S 1 which converges to G. Finally, notice that the corresponding infinitesimal generator Km of Sm (which is Killing for gR ) have periodic integral curves and, for big m, are timelike for g. Thus, one can choose X D Km for large m. Remark 3.4. The result is sharp: if K is allowed to be lightlike in some points there are counterexamples, see Figure 5. 3.2 Chronological spacetimes Definition 3.5. A spacetime .M; g/ is called chronological if it satisfies one of the following equivalent properties: (i) No timelike loop exists. (ii) Chronological relation is irreflexive, i.e., p q ) p ¤ q. (iii) d.p; p/ < 1 (and then equal to 0) for all p 2 M . A chronological spacetime is clearly non-totally vicious (see Definition 3.2 (iii)) but the converse does not hold, as Figure 5 shows. Notice that this example is compact and, in fact, as a general fact: Theorem 3.6. No compact spacetime .M; g/ is chronological. ˚ Proof.˚ Recall the open covering of M : I C .p/ ; p 2 M . Take a finite subrecovassume that, ering I C .p1 / ; I C .p2 / ; : : :; I C .pm / and, without loss of generality, if i ¤ j then pi … I C pj (otherwise, I C .pi / I C pj , and I C .pi / can be removed). Then, p1 2 I C .p1 /, as required.

319


t D 1=2 I C .p/

Identify p

t D 1=2 x D 1=3

x D 2=3

xD1

Identify

Figure 5. Non-totally vicious and non-chronological torus with a Killing vector field K D @ t . The vector field K is timelike everywhere except when x D 1=3, 2=3, where it is lightlike.

3.3 Causal spacetimes Definition 3.7. A spacetime .M; g/ is called causal if it satisfies one of the following equivalent properties: (i) No causal loop exists. (ii) Strict causal relation is irreflexive, i.e., p < q ) p ¤ q. The following possibility is depicted in Figure 6. C

C nfpg

p

Identify

Non-causal spacetime

Causal but non-distinguishing spacetime

Figure 6. Chronological non-causal cylinder, and causal but non-distinguishing spacetime obtained by removing fpg. If one also removed the vertical half line below p, a causal pastdistinguishing but non-future distinguishing spacetime would be obtained.

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Theorem 3.8. A chronological but non-causal spacetime .M; g/ admits a closed lightlike geodesic. Proof. Take a causal loop at some p 2 M . If were not a lightlike geodesic loop then p p (Theorem 2.30), in contradiction with chronology condition. And if were not closed, run it twice to obtain the same contradiction. Now, recall relation !./ in Definition 2.22. Theorem 3.9. In any causal spacetime .M; g/: x !./ y , x ! y. Proof. .(/. If x ! y, x ¤ y, then x and y are connected by a (non-necessarily unique) maximizing lightlike geodesic contained in J C .x/ \ J .y/. Taken x < p < q < y, the points p and q must lie on a unique maximizing lightlike geodesic , which will also cross x and y (otherwise, there would be a broken causal curve joining x with y, and hence y 2 I C .x/). Thus, J C .p/ \ J .q/ is nothing but the image of a portion of , which can be either homeomorphic to a segment joining p to q, or to a circumference (the latter excluded by the causality of .M; g/). .)/. If x !./ y, x ¤ y, there are .pn ; qn /, x < pn < qn < y, pn ! x, qn ! y, such that J C .pn / \ J .qn / is linearly ordered; in particular, x < y. But clearly x 6 y because, otherwise, as I C is open, qn 2 I C .pn / for large n. That is, the open set I C .pn /\I .qn / would be non empty, which clearly makes J C .pn /\J .qn / non-isomorphic to Œ0; 1. 3.4 Distinguishing spacetimes. The set of parts of M , i.e., the set of all the subsets of M , will be denoted P .M /. Here it is regarded as a point set, but it will be topologized later (see Proposition 3.38). The equivalence between some alternative definitions of distinguishing is somewhat subtler than in previous cases [30], [50]. So, we need the following previous result, which is proved below. Lemma 3.10. The following properties are equivalent for .M; g/: (i) I C .p/ D I C .q/ .resp. I .p/ D I .q// ) p D q. (ii) The set-valued function I C .resp. I / W M ! P .M /, p ! I C .p/ .resp. p ! I .p//, is one to one. (iii) Given any p 2 M and any neighborhood U 3 p there exists a neighborhood V U , p 2 V , which distinguishes p in U to the future .resp. past/ i.e. such that any future-directed .resp. past-directed/ causal curve W I D Œa; b ! M starting at p meets V at a connected subset of I .or, equivalently, if p D .a/ and .b/ 2 V then is entirely contained in V /. (iv) Given any p 2 M and any neighborhood U 3 p there exists a neighborhood V U , p 2 V , such that J C .p; V / D J C .p/ \ V .resp. J .p; V / D J .p/ \ V /. Definition 3.11. A spacetime .M; g/ is called future .resp. past/ distinguishing if it satisfies one of the equivalent properties in Lemma 3.10. A spacetime is distinguishing if it is both, future and past distinguishing.


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Proof of Lemma 3.10 for the future case. (i) , (ii) and (iii) ) (iv): Trivial. No (i) ) no (iii). Let p ¤ q but I C .p/ D I C .q/, take U 3 p such that q 62 Ux and any V 3 p, V U . Then, choose p 0 2 V; p V p 0 and any q 0 62 U; q 0 ¤ q; on a future-directed timelike curve 1 which joins q with p 0 . The required is obtained by joining p; q 0 with a future-directed timelike curve 0 , and then q 0 and p 0 through 1 . No (iii) ) no (i). Let U 3 p be a neighborhood where (iii) does not hold, that is, every V U intersects a suitable (V -dependent) future inextendible causal curve starting at p in a disconnected set of its domain I . Take the sequence fVn gn of nested globally hyperbolic neighborhoods in Theorem 2.14. They will be causally convex in some U 0 U and we can assume U D U 0 (if (iii) does not hold for the pair .p; U / then it does not hold for the pair .p; U 0 /, U 0 U ), being U also with closure contained in a simple neighborhood W . For each Vn , the causal curve n which escapes Vn and then returns Vn also escapes U (because of causal convexity) and then returns to some point in the boundary qn 2 UP which is the last one outside U , and to another point pn 2 Vn . As W was simple, fqn g ! q 2 UP , up to a subsequence. Even more, q 2 J .p; W /, because qn 2 J .pn ; W /, .qn ; pn / ! .q; p/, and J is closed on any convex neighborhood (see Proposition 2.10). Thus, I C .p/ I C .q/. Moreover, let q 0 2 I C .q/ then, for large n, .p /qn q 0 , that is q 0 2 I C .p/, I C .q/ I C .p/. No (iii) ) No (iv). Follow the reasoning in the last implication, with the same assumptions on U , and assuming that such a V as in (iv) exists. Notice that connecting the obtained q 2 J .p; W / (satisfying I C .p/ D I C .q/) with p by means of the unique geodesic in W , one point qV 2 .V \ /nfpg will also satisfy I C .p/ D I C .qV /. But, as U is convex, J C .p; U / ¤ J C .qV ; U / (use Proposition 2.10) and, even more4 , this holds arbitrarily close to qV . Concretely, .J C .p/ \ V D/ J C .p; V / 6 I C .qV ; V / . I C .qV / \ V J C .p/ \ V /, a contradiction. Remark 3.12. (1) One can give easily another two alternative characterizations of being distinguishing, say (iii0 ), (iv0 ), just by replacing causal curves and futures in (iii), (iv) by timelike curves and chronological futures. (2) Notice that, Lemma 3.10 also allows to define in a natural way what means to be distinguishing at p. In this case, for any neighborhood U of p, a neighborhood V which distinguishes p in U satisfies (iii) (and, thus, (iv)) for future and past causal curves. Notice also that, given U , one can find another neighborhood U 0 and a sequence of nested neighborhoods Vn U 0 such that \n Vn D fpg and each Vn is causally convex in U 0 (see also Theorem 2.14). (3) Recall that if V future-distinguishes p in U then it also future-distinguishes in U any other point q on a future causal curve starting at p contained in V . (4) Obviously, any past or future distinguishing spacetime is causal (if p; q lie on the same closed causal curve then I ˙ .p/ D I ˙ .q/), but the converse does not hold (Figure 6). A remarkable property of distinguishing spacetimes (complementary to Proposition 3.19 below) is the following [33]. These statements can be strengthened, as any convex subset U is, in fact, causally simple and, thus, any open neighborhood of U is not only distinguishing, but stably (and strongly) causal. 4

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Proposition 3.13. Let .M1 ; g 1 /, .M2 ; g 2 / be two spacetimes, .M1 ; g 1 / distinguishing, and f W M1 ! M2 a diffeomorphism which preserves , that is, such that: p q , f .p/ f .q/. Then .M2 ; g 2 / is distinguishing and g 1 D f g 2 . Proof. Let us show that .M2 ; g 2 / is distinguishing. First note that since f is bijective it preserves also 0: t .qj / D m.I .qj // m.I .p// C m.U / D t .p/ C : Thus, the previous two lemmas yields directly: Proposition 3.41. The following properties are equivalent for a spacetime: (i) The set valued map I .resp. I C / is .outer/ continuous. (ii) The volume function t .resp. t C / is continuous. 3.7.3 Reflectivity. Continuity of I ˙ (and, thus, t ˙ ) can be also characterized in terms of reflectivity. Lemma 3.42. Given any pair of events .p; q/ 2 M M the following logical statements are equivalent: (i) I C .p/ I C .q/ ) I .p/ I .q/ .resp. I .p/ I .q/ ) I C .p/ I C .q//, (ii) q 2 INC .p/ ) p 2 IN .q/ .resp. q 2 IN .p/ ) p 2 INC .q//, (iii) q 2 IPC .p/ ) p 2 IP .q/ .resp. q 2 IP .p/ ) p 2 IPC .q//. 6

One can check this for any admissible measure (and it is obvious for any regular measure, as the explicitly constructed m), see [44, Lemma 3.7] for details.

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Proof. (Equivalence in the past case). (i) , (ii). Trivial from the equivalences: (a) I C .q/ I C .p/ , q 2 INC .p/, and (b) I .p/ I .q/ , p 2 IN .q/. (ii) , (iii). To the right, recall: q 2 IPC .p/ ) q 2 INC .p/ but .p; q/ … I C ) p 2 N I .q/ but .p; q/ … I C ) p 2 IP .q/. For the converse: q 2 INC .p/ ) q 2 IPC .p/ or .p; q/ 2 I C ) p 2 IP .q/ or .p; q/ 2 I C ) p 2 IN .q/. Definition 3.43. A spacetime .M; g/ is past .resp. future/ reflecting at q 2 M if any of the corresponding equivalent items (i), (ii), (iii) in Lemma 3.42 holds for the pair .p; q/ for every p 2 M . A spacetime is past .resp. future/ reflecting if it is so at any q 2 M , and reflecting if it is both, future and past reflecting. Remark 3.44. Notice that if the items of Lemma 3.42 are required for .p; q/, for every q 2 M , a different property, say .past/ pseudo-reflectivity at p, would be obtained. Even though pseudo-reflectivity and reflectivity would be equivalent as spacetime properties .i.e. with no reference to a single point/, they are different as properties for a single event (as can be checked in Figure 8), the former not to be considered in what follows. Another characterization of reflectivity is the following. Proposition 3.45. A spacetime .M; g/ is past reflecting at q .resp. future reflecting at p/ if and only if .p 0 ; q/ 2 INC ) p 0 2 IN .q/ .resp. .p; q 0 / 2 INC ) q 0 2 INC .p//: An analogous result holds with IN replaced with IP. Proof. (Past case). Assume the spacetime is past reflecting at q and let .p; q/ 2 INC , then there are sequences pn ! p, qn ! q, qn 2 I C .pn /. Take any s 2 I .p/, so that p 2 I C .s/ and for large n, qn 2 I C .s/ which implies q 2 INC .s/. By using past reflectivity at q, s 2 IN .q/ and taking the limit s ! p, p 2 IN .q/. Conversely, assume that .p 0 ; q/ 2 INC ) p 0 2 IN .q/ and consider any p such that q 2 INC .p/. Then, .p; q/ 2 INC which implies p 2 IN .q/, that is, the spacetime is past reflecting at q. Lemma 3.46. The following properties are equivalent: (i) I .resp. I C / is outer continuous at p, (ii) the spacetime is past .resp. future/ reflecting at p. Proof. (i) ) (ii). Let I be outer continuous at q, and assume there is a p such that q 2 INC .p/ but p … IN .q/. By outer continuity there is a neighborhood V 3 q such that for every q 0 2 V , p … IN .q 0 /, but since q 2 INC .p/ there is q 0 2 V such that .p; q 0 / 2 I C , a contradiction. (ii) ) (i). Let the spacetime be past reflecting at p and assume by contradiction that I is not outer continuous. Then there is a compact K, K \ IN .p/ D ;, and a sequence pn ! p such that K \ IN .pn / ¤ ;. Taken rn 2 K \ IN .pn /, up to a subsequence rn ! r 2 K, and for any s 2 I .r/ we have for large n, pn 2 I C .s/, which implies p 2 INC .s/. By using reflexivity at p, s 2 IN .p/, and making s ! r, r 2 IN .p/, a contradiction because r 2 K.


333

The set R of points that do not comply these conditions has been studied in detail. The set R is a suitable union of null geodesics without past or future endpoint [14, Proposition 1.7]. Moreover, no point of R is isolated [55], and optimal bounds for its dimension are known [28], [12]. From Lemma 3.46, obviously: Proposition 3.47. The following properties are equivalent for .M; g/: (i) The set valued map I .resp. I C / is .outer/ continuous. (ii) The spacetime is past .resp. future/ reflecting. 3.8 Stably causal spacetimes. Volume and time functions are essential in this and following levels. We start discussing their relations with previous ones. 3.8.1 Time-type functions and characterization of some levels Definition 3.48. Let .M; g/ be a spacetime. A (non-necessarily continuous) function t W M ! R is: • A generalized time function if t is strictly increasing on any future-directed causal curve . • A time function if t is a continuous generalized time function. • A temporal function if t is a smooth function with past-directed timelike gradient rt. Notice that a temporal function is always a time function (d.t ı.s/=ds/ D g.P .s/; rt / > 0), but even a smooth time function may be non-temporal. From Remark 3.36, volume functions are not far from being generalized time ones. In fact, the next two theorems characterize this property. Theorem 3.49. A spacetime .M; g/ is chronological if and only if t .resp. if and only if t C / is strictly increasing on any future-directed timelike curve. Proof. ((). Obvious. ()). If p q but t .p/ D t .q/, necessarily almost all the points in the open subset I C .p/ \ I .q/ lie in I .p/. Thus, any point r in I C .p/ \ I .q/ \ I .p/ satisfies p r p. Remark 3.50. Notice that, as t is also constant on any causal loop, causal spacetimes cannot be characterized in this way. Figure 6 gives an example of causal nondistinguishing spacetime for which t is constant along a causal curve (the central almost closed circle). Theorem 3.51. A spacetime .M; g/ is past .resp. future/ distinguishing if and only if t .resp. t C / is a generalized time function.

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Proof. ()). To prove that t is strictly increasing on any future-directed causal curve, assume that p < q, p ¤ q, but t .p/ D t .q/. Then, almost all the points of I .q/ are included in I .p/. Choose a sequence fqn gn I .p/ \ I .q/ converging to q. Recall that, necessarily then I .qn / I .p/ for all n, and I .q/ D [n I .qn /. But this implies I .q/ I .p/ and, as the reversed inclusion is obvious, the spacetime is non-past distinguishing. ((). If I .p/ D I .q/ with p ¤ q, choose a sequence fpn g I .p/ which converges to p, and a sequence of timelike curves n from q to pn . By construction, the limit curve of the sequence starting at q is a (non-constant) causal curve and I .p/ I ..t // I .q/ for all t. Thus, the equalities in the inclusions hold, and t is constant on . 3.8.2 Stability of causality and chronology. Stable causality is related with the simple intuitive ideas that the spacetime must remain causal after opening slightly its lightcones, or equivalently, under small (C 0 fine) perturbations of the metric. Surprisingly, this is equivalent to the existence of time and temporal functions. More precisely, let Lor.M / be the set of all the Lorentzian metrics on M (which will be assumed time-orientable in what follows, without loss of generality). A partial (strict) ordering < is defined in Lor.M /: g < g 0 if and only if all the causal vectors for g are timelike for g 0 . Notice that this ordering is naturally induced in the set Con.M / of all the classes of pointwise conformal metrics on M . Even more, it induces naturally a topology in Con.M /, the interval topology, which admits as a subbasis the subsets type U g1 ;g2 D fg W g1 < g < g2 g where g1 ; g2 2 Con.M /, g1 < g2 . Remarkably, the interval topology coincides with the topology induced in Con.M / from the C 0 fine topology on Lor.M /. Roughly, the C 0 topology on Lor.M / can be described by fixing a locally finite covering of M by open subsets of coordinate charts with closures also included in the chart. Now, for any positive continuous function ı W M ! R and g 2Lor.M / one defines Uı .g/ Lor.M / as the set containing metrics gQ such that, in the fixed coordinates at each p, jgij .p/ gij .p/j < ı.p/ (in order to define the C r topology on Lor.M /, this inequality is also required for the partial derivatives of gij up to order r). A basis for the C 0 -fine topology is defined as the set of all such Uı .g/ constructed for any ı and g (see [44], [2], [41] for more detailed descriptions of this topology). Then, the (quotient) C 0 -topology in Con.M / is defined as the finer one such that the natural projection Lor.M / !Con.M /; g ! g is continuous. A way to define directly the C 0 -topology on Con.M / which shows the relation with the interval one is as follows [2], [32]. Fix an auxiliary Riemannian metric gR , and, for each g 2 Con.M /, define the gR -unit lightcone at p 2 M as Cp.R/ D fv 2 Tp M W g.v; v/ D 0; gR .v; v/ D 1g:


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Now, if j jR is the natural gR -norm, one define naturally the distance of any vector w 2 Tp M to Cp.R/ as usual: dR .w; Cp.R/ / D Minfjw vjR W v 2 Cp.R/ g: Given a second gQ 2Con.M / with associated gR -unit lightcone CQp.R/ , the maximum and minimum distances between the lightcones are, respectively: .R/ Q .R/ Q M jg gj R .p/ D MaxfdR .v; Cp / W v 2 Cp g; .R/ Q .R/ Q m jg gj R .p/ D MinfdR .w; Cp / W w 2 Cp g:

Notice that

Q m 0 < jg gj R

,

either g < gQ or gQ < g.

Now, for any positive continuous function ı W M ! R, let U ı .g/ D fgQ 2 Con.M / W 0 Q M jg gj R < ıg. The sets Uı .g/ yields a basis for the C topology. Definition 3.52. A spacetime .M; g/ is stably causal if it satisfies, equivalently: (i) There exists gQ 2Con.M / such that g < gQ and gQ is causal. (ii) There exists a neighborhood U of g in the quotient C 0 topology such that all the metrics in U are causal. Remark 3.53. (1) The equivalence of both definitions is clear because, if gQ is causal, then so are all the spacetimes with smaller lightcones, and these spacetimes constitute a C 0 neighbourhood. (2) A property of a metric g is called C r stable (r D 0; 1; : : : ; 1) if it holds for a C r neighborhood of g. As the C r topologies for r > 0 are finer than the C 0 one, stable causality means that the metric of the spacetime is not only causal, but also that this property is stable in all the C r topologies. Proposition 3.54. .C 0 / stable chronology and stable causality are equivalent properties for any spacetime .M; g/. Proof. Obviously, the latter implies the former. Let us show than non-stably causal implies non-stably chronological. Indeed, if the spacetime is non-stably causal, any g 1 > g admits a closed causal curve 1 . But since this is also true for any g 2 such that g < g 2 < g 1 , then the corresponding 2 is a closed timelike curve with respect to g 1 . Thus, any g 1 > g admits a closed timelike curve. A nice property of bidimensional spacetimes is the following. Theorem 3.55. Any simply connected 2-dimensional spacetime .M; g/ is stably causal. Proof. As M has 0 Euler characteristic (Theorem 2.4), necessarily M must be homeomorphic to R2 . Obviously, it is enough to prove that any spacetime constructed on R2 is causal. Otherwise, by closing if necessary the lightlike cones in a tubular neighborhood of , we can assume that there exists a lightlike closed curve , which (regarded

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as Jordan’s curve) bounds a domain D. Thus, taking any timelike vector field X , we P have g.X; 0 / never vanishes, i.e., X must point out either outwards or inwards D. Thus, a standard topological argument says that X must vanish on some point of D, a contradiction. 3.8.3 Time and temporal functions. The following characterization of stable causality in terms of time-type functions (see Definition 3.48) becomes specially useful. Nevertheless, it has been proved with rigor only recently [6], [44]. Theorem 3.56. For a spacetime .M; g/ the following properties are equivalent: (i) To be stably causal. (ii) To admit a time function t . (iii) To admit a temporal function T . Proof. (Sketch with comments; see [44, Section 4] for detailed proofs and discussions.) (iii) ) (i) As causality is conformally invariant, choose the representative g of g with g.rT ; rT / D 1. Now, the metric can be written as g D d T 2 C h; where h is the restriction of g to the bundle orthogonal to rT (up to natural identifications). Then, consider the one parameter family of metrics g D d T 2 C h;

> 0:

Clearly, T is still a temporal function for each g . Thus, g is always causal, and g D g1 < g2 , as required. (i) ) (ii) (Hawking [26], see also [27, Proposition 6.4.9] or [44, Theorem 4.13]). The fundamental idea is that, even though the past volume function t may be noncontinuous (it is only a generalized time-function), an “average” of such functions for a 1-parameter family of metrics g will work if g satisfies: (i) g0 D g, (ii) g is causal, for all 2 Œ0; 2, and (iii) < 0 ) g < g0 . Concretely, one checks that the following function is a time function: Z 1 t .p/ D t .p/d ; 0

t

where is the past volume function for, say, g D g C . =2/.gQ g/, 2 Œ0; 2 (gQ is chosen causal with g < g). Q (ii) ) (iii) This has been one of the “folk questions” on smoothability of the theory of causality until its recent solution [6]. It becomes crucial because, otherwise, the implication (ii)) (i) was also open. We refer to the detailed exposition in [44, Section 4.6] (see also the comments on smoothability for globally hyperbolic spacetimes below, especially Remark 3.77). Proposition 3.57. Stable causality implies strong causality.


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Proof. Let t be a time function and let us see that condition (ii) in Lemma 3.21 holds at any p 2 M . Let U 3 p a neighborhood and assume, without loss of generality, that U is simple, its closure is included in another simple neighborhood UQ , and t .p/ D 0. For any q 2 U put qC D Minft .r/ W r 2 J C .q; UQ / \ UP g; q D Minft .r/ W r 2 J .q; UQ / \ UP g; the variation of q˙ with q is continuous because UQ is convex. As UP is compact, q < t .q/ < qC , in particular, p ; pC > 0. Thus, for a small neighborhood S . From the compactness of S U , one has q ; qC > 0 for all q 2 W W 3 p; W ˙ S g > 0. The required neighborhood is V D S , necessarily W WD Minfq W q 2 W W 1 W \ t .W =2; W =2/. In fact, if a future-directed timelike curve starts at some q 2 V and leaves U at some point qU , then t .qU / W ; thus, cannot return to V . Remark 3.58. (1) Stable causality implies strong causality but the converse does not hold (see Figure 9). (2) Between strong and stable causality, an infinite set of levels can be defined by using Carter’s “virtuosity” [10].

Remove

Remove

Identify

Remove

Figure 9. An example of strongly causal non-stably causal spacetime. By opening slightly the causal cones there appear closed causal curves.

3.9 Causally continuous spacetimes. Taking into account the characterizations of the continuity of I ˙ (Proposition 3.41, 3.47) as well as the behavior of t ˙ in distinguishing spacetimes (Theorem 3.51), the following definitions of causal continuity (which can be also combined with the characterizations of reflectivity, Lemma 3.42, Proposition 3.45) hold. Definition 3.59. A spacetime .M; g/ is causally continuous if (equivalently, and for any admissible measure): (i) Maps I ˙ W M ! P are: (a) one to one, and (b) continuous (i.e., .M; g/ is reflecting, Lemma 3.46). (ii) .M; g/ is: (a) distinguishing, and (b) with continuous volume functions t ˙ .

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(iii) The volume functions t ˙ are time functions Remark 3.60. Trivially, totally vicious spacetimes have continuous I ˙ . Even more, they are also continuous in the causal non-distinguishing spacetime of Figure 6 (notice that the removed point in the circle does not affect to function t ˙ ). Thus, the injectivity of these maps (i.e., the hypotheses “distinguishing”) is truly necessary for this level of the ladder. Recall that a causally continuous spacetime not only admits a time function, but also the past and future volume functions are time functions. In particular: Proposition 3.61. Any causally continuous spacetime is stably causal. Remark 3.62. (1) The converse does not hold, as the example in Figure 8 shows. (2) Until stable causality, all the levels in the hierarchy of causality, except nontotally vicious, were inherited by open subsets7 . This is not the case neither for causal continuity (as the counterexample in figures 8 shows, being obtained from an open subset of R2 ) nor for the remaining levels of the ladder. 3.10 Causally simple spacetimes. There are different characterizations of causal simplicity (Proposition 3.68), we will start by the simplest one. Definition 3.63. A spacetime .M; g/ is causally simple if it is: (a) causal, and (b) J C .p/; J .p/ are closed for every p 2 M . Typically, the condition of being distinguishing is imposed directly in the definition of causal simplicity instead of causality, but the former can be deduced from the latter [8, Section 2], as proven next. Nevertheless, “causality” cannot be weakened in “chronology”, see Remark 3.72(1). Proposition 3.64. Conditions (a) and (b) in Definition 3.63 imply that the spacetime is distinguishing. Proof. Otherwise, if p ¤ q and, say I C .p/ D I C .q/, any sequence fqn g ! q, with q qn shows q 2 INC .q/ D INC .p/ D JN C .p/ D J C .p/. Thus, p < q and, analogously, q < p, i.e., the spacetime is not causal. Condition (b) has also the following consequence. Proposition 3.65. If a spacetime satisfies that J C .p/ .resp. J .p// is closed for every p, then I .resp. I C / is outer continuous. Thus, condition (b) in Definition 3.63 implies the reflectivity of .M; g/. 7

A counterexample for total-viciousness can be obtained from Figure 5, taking the open region determined by 1=3 < x < 2=3.


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Proof. Recall first the equivalence between outer continuity and reflectivity (Proposition 3.47), and let us prove the characterization of Lemma 3.42 (ii) (for the future case). As now IN˙ .p/ D J ˙ .p/, we have: q 2 INC .p/ ) p 2 J .q/ D IN .p/. Remark 3.66. (1) By Propositions 3.64 and 3.65, causally simple implies causally continuous, but the converse does not hold. A counterexample can be obtained just by removing a point to Ł2 . On the other hand, a spacetime may have closed J .p/ for every p but non-closed J C .q/ for some q (Figure 8). (2) Even though these spacetimes are almost at the top of the causal hierarchy, a metric in the pointwise conformal class of a causally simple spacetime may have a time-separation d with undesirable properties (see Figure 10). For example: (a) For some p; q, perhaps d.p; q/ D 1. (b) Even if 0 < d.p; q/ < 1, perhaps no causal geodesic connects p and q. (c) d may be discontinuous. This will be remedied in the last step of the hierarchy. t

Remove

q

Remove

˛ > =4 x s p r Figure 10. An example of causally simple non-globally hyperbolic spacetime, with a general metric conformal to the usual one, g D 2 .t; x/.dt 2 C dx 2 /, p D .0; 1/, q D .0; 1/, > 0. If D 1 then d.p; q/ D 2, but no geodesic connects them (Remark 3.66, case (2) (b)), while if 2 D 1=.t 2 C x 2 /, then d.p; q/ D C1 (case (2) (a)). If 2 D 1=.x C 1/2 then d is discontinuous (case (2) (c)) as d.p; q/ < C1, but d.p; q 0 / D C1 for q 0 q (because the connecting causal curves can approach a finite segment on the left-hand side border). The causal diamond J C .r/ \ J .s/ is not compact and there are inextendible causal curves which, being “created by the naked singularity”, pass through s.

Property (b) of Definition 3.63 can be also characterized in different ways. Lemma 3.67. Let J C .p/ and J .p/ be closed for every p 2 M , then: (1) J C .K/ and J .K/ are closed for every compact K M . (2) J C .and hence J /, regarded as a subset of M M , is closed.

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Proof. (1) Otherwise if, say, q 2 JN C .K/nJ C .K/ there exist sequences qn ! q, pn qn , pn 2 K, where, up to a subsequence, pn ! p 2 K. Thus, .p; q/ 2 INC and, by using Proposition 3.45 (recall Proposition 3.65), q 2 INC .p/ D J C .p/ J.K/. (2) Obviously, J C INC and, for the converse, use again .p; q/ 2 INC ) q 2 C IN .p/ D J C .p/. Thus, on the basis of these results, we have the following characterization. Proposition 3.68. A spacetime .M; g/ is causally simple if it is causal and satisfies one of the following equivalent properties: (i) J C .p/ and J .p/ are closed for every p 2 M . (ii) J C .K/ and J .K/ are closed for every compact set K. (iii) J C is a closed subset of M M . Finally, notice that causal relations can be obtained now “starting at chronology” (Definition 2.22) Theorem 3.69. In a causally simple spacetime8 , x ./ y , x y. Proof. To the left, it is trivial in any spacetime. So, let x ./ y. Since I C .y/ I C .x/, y 2 INC .x/ D JN C .x/ D J C .x/, where J C is the usual causal relation. 3.11 Globally hyperbolic spacetimes. There are at least four ways to consider global hyperbolicity: (1) by strengthening the notion of causal simplicity, (2) by using Cauchy hypersurfaces, (3) by splitting orthogonally the spacetime, and (4) by using the space of causal curves connecting each two points. We will regard (1) as the basic definition and will study subsequently the other approaches, as well as some natural results under them. 3.11.1 Strengthening causal simplicity Definition 3.70. A spacetime .M; g/ is globally hyperbolic if: (a) it is causal, and (b) the intersections J C .p/ \ J .q/ are compact for all p; q 2 M . Following [8, Section 3], the next result yields directly that a globally hyperbolic spacetime (according to our Definition 3.70) is causally simple. Proposition 3.71. Condition (b) implies both, J C .p/ and J .p/ are closed for all p 2 M. Proof. Assume that J C .p/ is not closed and choose r 2 JN C .p/nJ C .p/ and q 2 I C .r/. Take a sequence frn g ! r with rn 2 I C .p/ for all n (Proposition 2.17), and notice that rn q up to a finite number of n (Proposition 2.16). Thus, frn gn J C .p/ \ J .q/, but converges to a point out of this compact subset, a contradiction. 8

Notice that, for this result, one can define x ./ y , either I C .y/ I C .x/ or I .x/ I .y/.


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Remark 3.72. (1) As stressed in [8], the full consistency of the causal ladder yields that any globally hyperbolic spacetime is not only causally simple but also strongly causal. This last hypothesis is usually imposed in the definition of global hyperbolicity, instead of causality, but becomes somewhat redundant. Notice that causality does not follow from property (b) and cannot be weakened. Indeed, there are chronological non-causal spacetimes which satisfy it (see Figure 6). (2) The open subset M D f.t; x/ 2 Ł2 W 0 < xg shows that a causally simple spacetime may be non-globally hyperbolic. Notice that the two conditions in Definition 3.70 are natural from the physical (even philosophical) viewpoint: 1. Causality avoids paradoxes derived from trips to the past (grandfather’s paradox). For example, one cannot “send a laser beam which describes a causal loop in the spacetime and kills him/herself”. 2. The compactness of the diamonds J C .p/ \ J .q/ can be interpreted as “there are no losses of information/energy in the spacetime”. In fact, otherwise one can find a sequence frn gn J C .p/ \ J .q/ with no converging subsequence. Taking a sequence of causal curves f n gn , each one joining p, rn , q, the limit curve p starting at p cannot reach q.This can be interpreted as something which is suddenly lost or created in the boundary of the spacetime (see Figure 10). That is, a singularity (this sudden loss/creation) is visible from q – there are “naked singularities”. 3.11.2 Cauchy hypersurfaces and Geroch’s theorem. Recall that a subset A M is called achronal (resp. acausal) if it is not crossed twice by any timelike (resp. causal) curve. The following notions are useful in relation to Cauchy hypersurfaces. Definition 3.73. Let A be an achronal subset of a spacetime .M; g/. • The domain of dependence of A is defined as D.A/ D D C .A/ [ D .A/, where D C .A/ (resp. D .A/) is defined as the set of points p 2 M such that every past (resp. future) inextendible causal curve through p intersects A. • The Cauchy horizon of A is defined as H.A/ D H C .A/ [ H .A/, where x C .A/nI .D C .A// D fp 2 D x C .A/ W I C .p/ does not meet H C .A/ D D C D .A/g, and H .A/ is defined dually. One can check that, if A is a closed subset, then DP C .A/ D A [ H C .A/. Recall that D.A/ can be interpreted as the part of the spacetime predictable from A. A Cauchy hypersurface is defined as an achronal subset from where the full spacetime is predictable: Definition 3.74. A Cauchy hypersurface of a spacetime .M; g/ is, alternatively: (i) A subset S M which is intersected exactly once by any inextendible timelike curve.

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(ii) An achronal subset S , with D.S/ D M . (iii) An achronal subset S , with H.S/ D ;. Some properties of any such Cauchy hypersurface S are the following: 1. Necessarily, S is a closed subset and an embedded topological hypersurface. 2. The spacetime M is the disjoint union M D I .S / [ S [ I C .S /. 3. Any inextendible causal curve crosses S and, if S is spacelike (at least C 1 ) then crosses S exactly once (in general S may be non-acausal because may intersect S in a segment, i.e., in the image of an interval Œc; d ; c < d ). 4. If K is compact then J ˙ .K/ \ S is compact. In what follows, a function t W M ! R (in particular, a time or temporal one, according to Definition 3.48) will be called Cauchy if its levels Sc D t 1 .c/ are Cauchy hypersurfaces; without loss of generality, we can assume that Cauchy functions are onto. Notice that the levels of a Cauchy time function are necessarily acausal Cauchy hypersurfaces. The characterization of global hyperbolicity in terms of Cauchy hypersurfaces comes from the following celebrated Geroch’s theorem [22]. Theorem 3.75. .M; g) is globally hyperbolic if and only if it admits a Cauchy hypersurface S. Even more, in this case: (i) the spacetime admits a Cauchy time function; (ii) all Cauchy hypersurfaces are homeomorphic to S , and M is homeomorphic to R S . The implication to the left is a (non-trivial) standard computation written in many references (for example, [39], [56]). For the implication to the right and the last assertion, recall first the following result: Lemma 3.76. In a globally hyperbolic spacetime, the continuous function t .p/ m.I .p// t .p/ D log C D log t .p/ m.I C .p//

(8)

satisfies: lim t ..s// D 1;

s!a

lim t ..s// D 1

s!b

(9)

for any inextendible future-directed causal curve W .a; b/ ! M . Proof. It is sufficient to check: lim t ..s// D 0;

s!a

lim t C ..s// D 0:

s!b

Reasoning for the former, it is enough to show that, fixed any compact subset K, then K \ I ..s0 // D ; for some s0 2 .a; b/ (and, thus, for any s < s0 ), see [44] for details. Choose any point on the curve, q D .c/ for some c 2 .a; b/, and assume by contradiction the existence of a sequence pj D .sj /; sj ! a; sj 2 .a; c/, with an associate sequence rj 2 K \ I .pj /. Up to a subsequence, frj g ! r, and choosing


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p r, one has p pj < q, and j.a;c lies in the compact subset J C .p/ \ J .q/. That is, is totally imprisoned to the past, in contradiction with strong causality, (see Proposition 3.29). Proof of Theorem 3.75 .necessity only/. As t in Lemma 3.76 is a time function, each level Sc is an acausal hypersurface. In order to check that any inextendible timelike curve crosses Sc (thus proving (i)), recall that can be reparametrized on all R with t , and (9) will also hold under any increasing continuous reparametrization of . Thus, assuming that this reparametrization has been carried out, .c/ 2 Sc . For assertion (ii), it is enough to choose a complete timelike vector field X , (Proposition 2.3) and project the full spacetime onto S by using its flow. 3.11.3 The folk questions on smoothability and the global orthogonal splitting. The statements of the results in Geroch’s theorem and its proof, suggest obvious problems on the smoothability of S and t. In fact, these questions were regarded as “folk problems” because, on one hand, some proofs were announced and rapidly cited (see [5, Section 2] for a brief account) and, on the other, smoothability results yield useful simplifications and applications commonly employed. Nevertheless, they have remained fully open until very recently. Remark 3.77. The solution to the problems on smoothability in [4], [6], [7] involves technical procedures very different to the expected approaches in previous attempts. These approaches can be summarized as: (a) To smooth the Cauchy hypersurface S or the (Cauchy) time function t by using covolution [49]. The difficulty comes from the fact that, even when S; t are smooth, the tangent to S or the gradient of t may be degenerate, that is, close hypersurfaces or functions to S; t may be non-Cauchy or non-time functions. Therefore, S; t must be smoothed by taking into account that a C 1 approximation may be insufficient. (b) To choose an admissible measure m such that the volume functions t C , t are directly not only continuous but also smooth [13]. Nevertheless, notice that those stably causal spacetimes which are not causally continuous, cannot admit continuous t C , t , but they do admit temporal time functions (Theorem 3.56). As a summary on these questions, assume that .M; g/ be globally hyperbolic. 1. Must a .smooth/ spacelike Cauchy hypersurface S exist? This is the simplest smoothability question, posed explicitly by Sachs and Wu in their review [42, p. 1155]. One difficulty of this problem (which makes useless naive approaches based on covolution) is the following. Even if a Cauchy hypersurface S is smooth at some point p, the tangent space Tp S may be degenerate; so, the smoothing procedure of S must “push” Tp S in the right spacelike direction. The existence of one such S implies that the spacetime is not only homeomorphic but also diffeomorphic to RS. Physical applications appear because spacelike Cauchy hypersurfaces are essential for almost any global problem in General Relativity (initial value problem for Einstein equation, singularity theorems, mass...), see [46]. For example, from the foundational viewpoint, they are necessary for the well-posedness

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of the initial value problem, as there is no a general reasonable way to pose well these conditions if the Cauchy hypersurface is not spacelike (or, at least, smooth). This smoothability problem was solved in [4]. The idea starts recalling the following result, interesting in its own right (see also [18]): Let S be a Cauchy hypersurface. If a closed subset N M is a embedded spacelike .at least C 1 / hypersurface which lies either in I C .S / or in I .S / then it is achronal. If N lies between two disjoint Cauchy hypersurfaces S1 ; S2 .N I C .S1 / \ I .S2 // then it is a Cauchy hypersurface .see Figure 11/. Thus, as Geroch’s theorem ensures the existence of such S1 ; S2 , the crux is to find a smooth function t with a regular value c such that Sc D t 1 .c/ lies between S1 and S2 , and rt is timelike on Sc . D.N / L2

N S

S2 t

N S1

(A)

(B)

Figure 11. (A). The embedded spacelike hypersurface N is achronal. because it lies in I C .S /. But it is not (extendible to) a spacelike Cauchy hypersurface. (B). Now, as N M lies between two disjoint Cauchy hypersurfaces S1 ; S2 , it would be a Cauchy hypersurface if it were closed.

2. Must a Cauchy temporal function T exist? This question is relevant not only as a natural extension of Geroch’s, but in much more depth, because in the affirmative case the smooth splitting R S of the spacetime can be strengthened in such a way that the metric has no cross terms between R and S (see (10) below for the explicit expression). This splitting is useful from practical purposes and also to introduce different techniques (Morse theory [54], variational methods [35], quantization...) Notice that the constructive proof of Geroch’s Cauchy time function may yield a non-smooth one (Figure 12). The freedom to choose an admissible measure m may suggest that, perhaps, a wise choice of m will yield directly a smooth Geroch’s function. Nevertheless, the related problem of smoothability in stably causal spacetimes (Theorem 3.56) suggest that this cannot be the right approach (in this case, even t ˙ may be non-continuous). The problem was solved affirmatively in [6] by different means,

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based on the construction of “time step functions”. We also refer to [44] for a sketch of these ideas. v

S

J C .p/ p p

1

1

u

Figure 12. M Ł2 (coord. u; v). M D f.u; v/ 2 Ł2 W juj; jvj < 2gnf.u; v/ 2 Ł2 W u; v 1g; p D .0; 1/; p D .0; 1 /. Diagonal S is a Cauchy hypersurface. For the natural g-measure, t C .p / D 2 C t C .p/ when > 0, and t C is not smooth.

3. If a spacelike Cauchy hypersurface S is prescribed, does a Cauchy temporal function T exist such that one of its levels is S? This question has natural implications in classical General Relativity (even though was proposed explicitly by Bär, Ginoux and Pfäffle in the framework of quantization). For example, for the initial value problem, one poses initial data on a prescribed hypersurface which will be, a posteriori, a Cauchy hypersurface S of the solution spacetime. Now, in order to solve Einstein equation, one may assume that the spacetime will admit an orthogonal splitting as (10) below, with S one of the slices and being ˇ, and the evolved metric gT , the unknowns. This problem was solved affirmatively in [7]. Notice that even a non-smooth Cauchy (resp. acausal Cauchy) hypersurface S can be regarded as a level of a time (resp. Cauchy time) function t as follows. I C .S / and I .S /, regarded as spacetimes, are globally hyperbolic and, thus, we can take Cauchy temporal functions TS ˙ on I ˙ .S /. Now, the required function is:

exp.T

t .p/ D

S C .p//

0 exp.TS .p//

for all p 2 I C .S /, for all p 2 S, for all p 2 I .S /.

The function t is also smooth (and a Cauchy temporal function) everywhere except at most in S D t 1 .0/. Nevertheless (replacing, if necessary, t by a function obtained technically by modifying t around S), one can assume that t is smooth even if S is not. Nevertheless, in this case the gradient of t on S will be 0 and, thus, t will not be a true Cauchy temporal function. Now, the crux is to show that, if S is spacelike, then it is

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possible to modify t in a neighborhood of S, making its gradient everywhere timelike, and maintaining its other properties. 4. Under which circumstances a spacelike submanifold A .with boundary/ can be extended to a spacelike .or, at least, smooth/ Cauchy hypersurface? As the previous question, this one is solved in [7] and has a natural classical meaning (but it was posed by Brunetti and Ruzzi motivated by quantization). Notice that an obvious requirement for p A is achronality; moreover, compactness becomes also natural (the hyperbola t D x 2 C 1 would yield a counterexample, see Figure 11). Even more: any compact achronal K M , can be extended to a Cauchy hypersurface. In fact, M 0 D M n.I C .K/ [ I .K// would be a (possibly non-connected) globally hyperbolic spacetime and, then, would admit a Cauchy hypersurface S 0 ; the required Cauchy hypersurface of M would be SK D S 0 [ K. Nevertheless, the corresponding Cauchy hypersurface SA for the (smooth, compact, achronal) submanifold A, may be non-smooth and even non-smoothable, see Figure 13. But it is possible to prove that if A is not only achronal but also acausal, then SA can be modified in a neighborhood of AP to make it not only smooth but also spacelike.

q

A

p

Figure 13. The canonical Lorentzian cylinder (R S 1 , g D dt 2 C d 2 ) with the spacelike hypersurface A D f. =2; / W 2 Œ0; 4=3g. The spacelike achronal (but non-acausal) hypersurface A can not be extended to a smooth Cauchy hypersurface, although by adding the null geodesic segment between p and q one obtains a continuous Cauchy hypersurface SA .

As a summary of all these problems, it is possible to prove: Theorem 3.78. A spacetime .M; g/ is globally hyperbolic if and only if it admits a .smooth/ spacelike Cauchy hypersurface S. In this case it admits a Cauchy temporal function T and, thus, it is isometric to the smooth product manifold

R S; h ; i D ˇ d T 2 C gT

(10)


347

where ˇ W R S ! R is a positive smooth function, T W R S ! R the natural projection, each level at constant T , ST , is a spacelike Cauchy hypersurface, and gT is a Riemannian metric on each ST , which varies smoothly with T . Even more, if S a prescribed .topological/ Cauchy hypersurface then there exists a smooth Cauchy function W M ! R such that S is one of its levels .S D S0 /. If, additionally: • S is also acausal then function becomes a smooth Cauchy time function. • If S is spacelike .and thus smooth and acausal/, then can be modified to obtain a Cauchy temporal function T W M ! R such that S D T 1 .0/. Finally, if A M is a compact achronal subset then it can be extended to a Cauchy hypersurface. If, additionally, A is acausal and a smooth spacelike submanifold with boundary, then it can be extended to a spacelike Cauchy hypersurface S A. 3.11.4 The space of causal curves. The first definition of global hyperbolicity was given by Leray [31], and involves the compactness of the space of causal curves which connects any two points. More precisely, consider two events p; q of the spacetime .M; g/, and let C.p; q/ be the set of all the continuous curves which are future-directed and causal (according to Definition 3.15) and connect p with q, under the convention in Remark 2.9, i.e., two such curves are regarded as equal if they differ in a strictly monotonic reparametrization. For simplicity, .M; g/ will be assumed to be causal, and we will consider the C 0 topology9 on C.p; q/, that is, a basis of open neighborhood of 2 C.p; q/ is constructed by taking all the curves in C.p; q/ contained in an open neighborhood U of the image of . Theorem 3.79. A spacetime .M; g/ is globally hyperbolic if and only if: (i) it is causal, and (ii) C.p; q/ is compact for all p; q 2 M . Proof. .(/ Let frn gn be a sequence in J C .p/ \ J .q/ and n be a causal curve from p to q trough rn for each n. Up to a subsequence f n gn converges to a curve 2 C.p; q/. So, chosen any neighborhood U M of with compact closure Ux , all n .3 rn / lie in U for large n and, up to a subsequence, frn g ! r 2 Ux . But necessarily r 2 . J C .p/ \ J .q//, as required. .)/ See for example [27, p. 208–209]. Remark 3.80. In fact, hypothesis (i) is somewhat redundant, because it is possible to define a natural topology on C.p; q/ even if the spacetime is not causal. But in this case, if there were a closed causal curve , parametrizing by giving more and more rounds, a sequence of (non-equivalent) causal curves would be obtained, and the compactness assumption of C.p; q/ would be violated for this natural topology. 9 This sense of C 0 topology agrees with the C 0 -limit of curves, described in Definition 3.31. Even though this notion of limit had specially good properties for strongly causal spacetimes, we will not need a priori this hypothesis but only causality (recall also that the two extremes of the curves are fixed). Nevertheless, a posteriori, we will work with globally hyperbolic spacetimes, where strong causality holds.

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With this notion of global hyperbolicity at hand, it is not difficult to prove the main properties of the time-separation d of a globally hyperbolic spacetime. Recall that d is not conformally invariant, but the properties below will be so. Lemma 3.81. Let .M; g/ be globally hyperbolic and p < q. Consider sequences: fpk g ! p;

fqk g ! q;

pk qk

Then, for any sequence k of causal curves, each one from pk to qk , there exists a limit in the C 0 topology which joins p to q. Proof. Choose p1 p; and q q1 and, for large n, construct a sequence of causal curves fn gn starting at p1 , going to pn , running qn and arriving at q1 . Then, use the compactness of C.p1 ; q1 /. Remark 3.82. From the properties in Section 3.6.3, is also a limit curve of the sequence, and L. / limm L.k /. Theorem 3.83. In any globally hyperbolic spacetime .M; g/: (1) d is finite-valued. (2) (Avez–Seifert [1], [48]) Each two causally related points can be joined by a causal geodesic which maximizes time-separation. (3) d is continuous. Proof. (1) Cover J C .p/ \ J .q/ with a finite number m of convex neighbourhoods Uj such that each causal curve which leaves Uj satisfies: (i) it never returns to Uj , (ii) its length is 1. Then d.p; q/ m. (2) Take a sequence of causal curves k with lengths converging to d.p; q/ and use Lemma 3.81 (this also yields an alternative proof of (1)). (3) Otherwise (taking into account that d is always lower semi-continuous) there are sequences fpk g ! p; fqk g ! q; pk qk with d.pk ; qk / d.p; q/ C 2ı for some ı > 0. Choose causal curves k from pk to qk satisfying L.k / d.pk ; qk / ı: Then the limit yields the contradiction: L. / lim supL.k / d.p; q/ C ı > d.p; q/: Remark 3.84. (1) The finiteness of d holds for all the time-separations of metrics in g. In fact, the following characterization is classical: A strongly causal spacetime .M; g/ is globally hyperbolic if and only if the timeseparation d of any metric g conformal to g is finite.


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To check it, notice that when .M; g/ is not globally hyperbolic, there is a sequence fk gk C.p; q/ which has a limit curve starting at p with no final endpoint. The conformal factor must be taken diverging fast along a neighborhood of (see [2, Theorem 4.30] for details). (2) The existence of connecting causal geodesics in Avez–Seifert result can be made more precise: there exists a d -maximizing geodesic in each causal homotopy class and, if p q, there is also a maximizing timelike geodesic in all the timelike homotopy classes included in each causal homotopy class, see the detailed study in [36, Section 2]. 3.11.5 An application to closed geodesics and static spacetimes. Next, we will see some simple applications of the properties of globally hyperbolic spacetimes for the geodesics of some spacetimes. We refer to [45] for more results and extended proofs, especially regarding static spacetimes. Proposition 3.85. If the universal covering .MQ ; g/ Q of a totally vicious spacetime .M; g/ is globally hyperbolic, then .M; g/ is geodesically connected through timelike geodesics .i.e., each p; q 2 M can be connected through a timelike geodesic/. Proof. By lifting to MQ any timelike curve which connects p; q, one obtains two chronologically related points p; Q qQ 2 MQ . So, they are connectable by means of a (maximizing) timelike geodesic Q , which projects in the required one. Now, recall that a static spacetime is a stationary one such that the orthogonal distribution to its timelike Killing vector field K is integrable. Locally, any static spacetime looks like a standard static spacetime, i.e., the product R S endowed with the warped metric g D ˇdt 2 C gS , where gS is a Riemannian metric on S and ˇ is a function which depends only on S . If K is complete, any simply connected static spacetime is standard static, in particular: Q of a compact static spacetime is standard Lemma 3.86. The universal covering .MQ ; g/ static. These spacetimes have a good causal behaviour: Proposition 3.87. Any standard static spacetime .M; g/ is causally continuous, and the following properties are equivalent: (i) .M; g/ is globally hyperbolic. (ii) The conformal metric gS D gS =ˇ is complete. (iii) Each slice t Dconstant is a Cauchy hypersurface. In particular, the universal covering of a compact static spacetime is globally hyperbolic. Proof. For the first assertion, it is enough to prove past (and analogously future) reflectivity I C .q/ I C .p/ ) I .p/ I .q/. Put p D .tp ; xp /; q D .tq ; xq /. Assuming the first inclusion, it is enough to prove p D .tp ; xp / 2 I .q/, for all

350


> 0. As q WD .tq C ; xq / 2 I C .p/, there exists a future-directed timelike curve .s/ D .s; x.s//; s 2 Œtp ; tq C joining p and q . Then, the future-directed timelike curve .s/ D .s ; x.s// connects p and q, as required. The equivalences (i)–(iii) follows from standard computations valid for warped product spacetimes [2, Theorems 3.67, 3.69]. In particular, a standard static spacetime will be globally hyperbolic if gS is complete and ˇ is bounded (or at most quadratic). These conditions hold in the universal covering of a compact static spacetime, proving the last sentence. Thus, Proposition 3.85, 3.87, and Theorem 3.3 yields [47]: Theorem 3.88. Any compact static spacetime is geodesically connected through timelike geodesics. For closed geodesics, let us start with the following well-known result by Tipler [53] (in Beem’s formulation [2]), later extended by Galloway [17]. Theorem 3.89. Any compact spacetime .M; g/, regularly covered by a spacetime .MQ ; g/ Q which admits a compact Cauchy hypersurface S , contains a periodic timelike geodesic. Proof. Take a timelike loop in M and a lift Q W Œ0; 1 ! MQ . Let W MQ ! MQ be a deck transformation which maps Q .0/ in Q .1/. The function f W S ! R p ! d.p; .p// admits a maximum p0 (necessarily, f .p0 / > 0). The maximizing timelike geodesic from p0 to .p0 / projects not only onto a geodesic loop, but also to a closed one (otherwise, a closed curve with bigger length could be obtained by means of a small deformation). Remark 3.90. The compactness of S cannot be removed (Guediri’s counterexample, see [24] and references therein). Nevertheless, it can be replaced by the existence of a class of conjugacy C of the fundamental group which contains a timelike curve and satisfies one of the following two conditions (see [47]): (a) C is finite. (b) The deck transformations satisfy a technical property of compatibility with an orthogonal globally hyperbolic splitting (roughly, .t; x/ D .t C T ; S .x// for some T 2 R and some automorphism S of S ), which is always satisfied in the case of compact static spacetimes. Thus, this possibility (b) yields [47]: Theorem 3.91. Any compact static spacetime admits a closed timelike geodesic.


351

4 The “isocausal” ladder 4.1 Overview. Up to now, the causal structure of a spacetime is related to two notions: (a) its conformal structure, and (b) its position in the causal hierarchy. Nevertheless, in order to understand “when two spacetimes share the same causal structure” one can argue that the first one is too restrictive, and the latter too weak. For example: (a) most modifications of a Lorentzian metric around a point (say, any non-conformally flat perturbation of Minkowski spacetime in a small neighbourhood) imply a different conformal structure; but, one may have a very similar structure of future and past sets for all points, and (b) all globally hyperbolic spacetimes belong to the same level of the hierarchy, but clearly the causality of, say, Lorentz–Minkowski and Kruskal spacetimes behave in a very different way. It is not easy to find an intermediate notion, because “same causal structure” suggests “same causal relations ; 0 .sab / for TM ˚ T M . By refining the cover fUa g if necessary, we can assume it to be convex. The closed 3-form H is locally exact, that is, HjUa D dB .a/ and we define ˇ .ab/ D .B .a/ B .b/ /jUa \Ub 2 2 .Ua \ Ub /: By design, the 2-forms ˇ .ab/ are closed. Trivialising TM over Ua or Ub , we can think of these 2-forms as maps Ua \ Ub ! ƒ2 Rn . We indicate the trivialisation we use by the subscript a or b, that is for p 2 Ua \ Ub , ˇ .ab/ .p/ D Œa; p; ˇa.ab/ .p/ D Œb; p; ˇb.ab/ .p/; so that ˇa.ab/ D sab ˇb.ab/ . This gives rise to the twisted transition functions .ab/

ab D Sab ı e 2ˇb

.ab/

D e 2ˇb

ı Sab :

Metric bundles of split signature and type II supergravity

481

Indeed, whenever Ua \ Ub \ Uc 6D 0, we have .ab/

ab ı bc D Sab ı e 2ˇb D Sab ı e

.bc/

ı e 2ˇb

ı Sbc

.a/ .b/ .b/ .c/ 2.Bb Bb CBb Bb /

D Sab ı Sbc ı e

.a/ .c/ 2.Bc Bc /

ı Sbc .ac/

D Sac ı e 2ˇc

1 D ca :

We define the generalised tangent bundle by a E D E .H / D Ua .Rn ˚ Rn /= ab : Up to isomorphism, this bundle only depends on the closed 3-form H . Indeed, assume 0 we are given a different convex cover fUa0 g together with locally defined 2-forms B .a/ 2 0 2 .Ua0 / such that HjUa0 D dB .a/ , resulting in a new family of transition functions .a/ 0 D Sab ı exp.2ˇb0 /. Now on the intersection Va D Ua \ Ua0 we have d.BjV ab a 0

.a/ / D 0, hence BjV a

0

.a/ .a/ BjV BjV D dG .a/ a a

for G .a/ 2 1 .Va /:

One readily verifies the family Ga D exp.dGa.a/ / to define a gauge transformation, i.e. 0 ab D Ga1 ı ab ı Gb on Vab 6D ;: 0 In particular, the bundles defined by the families ab and ab respectively are isomorphic, where the isomorphism is provided by

Œa; p; vsab 7! Œa; p; Ga .v/s0 : ab

Since the transition functions ab take values in SO.n; n/C , the invariant orientation and inner product . ; / on Rn ˚ Rn make sense globally and turn E into an oriented pseudo-Riemannian vector bundle. Again, we can consider reductions inside this SO.n; n/C -structure, for instance to SO.n; 0/ SO.0; n/. Definition 4.1. A generalised Riemannian metric for the generalised tangent bundle E .H / is a reduction from its structure group SO.n; n/C to SO.n; 0/ SO.0; n/. Since the group SO.n; 0/ SO.0; n/ preserves a decomposition of Rn ˚ Rn D V ˚V into a positive and negative definite subspace V C and V , we can equivalently define a generalised Riemannian metric by the choice of a maximally positive definite subbundle V C E .H /. This bundle provides a splitting of the exact sequence C

i

! TM ! 0: 0 ! T M ! E

(17)

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F. Witt

Here, i W T M ! E is the canonical inclusion: Any differential form transforms and is acted on trivially by exp.ˇ .ab/ /, therefore defining a section of E . under sab Identifying T M with its image i.T M / in E , T M \ V C D f0g, for T M is isotropic. Hence the projection W E ! TM restricted to V C is injective, so that a generalised Riemannian structure defines a splitting of the exact sequence (17). We obtain a lift from TM to V C which in accordance with the notation used in the previous section we denote by X C , for X a vector field X 2 .TM /. Locally, X corresponds to smooth maps Xa W Ua ! Rn such that Xa D sab Xb , while for XaC W Ua ! Rn ˚ Rn , the relation XaC D ab XbC holds. As before, V C is obtained as the graph of a linear isomorphism PaC W Rn ! Rn , so XaC D Xa ˚ PaC Xa : From the transformation rule on fXaC g, we deduce PbC sba : ˇa.ab/ D PaC sab

(18)

The symmetric part ga D .PaC C .PaC /> /=2 is therefore positive definite, and since ˇ .ab/ is skew-symmetric, the symmetrisation of the right hand side vanishes. Hence 1 C PbC sba C .PaC /> sab .PbC /> sba / D ga sab ga sab D 0; .P sab 2 a so that the collection ga W Ua ! ˇ2 Rn of positive definite symmetric 2-tensors patches together to a globally defined metric. Conversely, a Riemannian metric g induces a generalised Riemannian structure on E .H /: The maps Pa D Ba.a/ C ga induce local lifts of TM to E which give rise to a global splitting of (17). Proposition 4.2. A generalised Riemannian structure is characterised by the datum .g; H /, where g is a Riemannian metric and H a closed 3-form. Remark 4.3. (i) From (18) we also conclude that the skew-symmetric part of Pa is .a/ C C Ba.a/ . Hence we have local isomorphisms VjU D e 2B DjU , where TM ˚ T M D a a C D ˚ D . In this way, we can think of the local model of a generalised Riemannian metric as a B-field transformed Riemannian metric. (ii) Of course, the negative definite subbundle V also defines a splitting of (17). The lift of a vector field X is then induced by Xa D Xa ˚ Pa Xa with Pa D ga C Ba.a/ . For generalised SU.m/-, G2 - or Spin.7/-structures, we need to speak about E spinor fields. This shall occupy us next. 4.2 Spinors. As discussed in Section 2.2, we first need to exhibit a spin structure, i.e. a Spin.n; n/C -valued family of functions Q ab satisfying (4) and covering ab , that is 0 ı Q ab D ab . In the situation present, we can make a canonic choice: Exponentiating ˇ .ab/ to Spin.n; n/C and considering GL.n/C as its subgroup, we define .ab/

Q ab D Szab • e ˇb

.ab/

D e ˇa

• Szab :


483

The even and odd spinor bundles associated with E D E .H / are S .E /˙ D

a

Ua S˙ = Q ab :

a

An E -spinor field is thus represented by a collection of smooth maps a W Ua ! S˙ with a D Q ab • b . Since as a vector space, S˙ D ƒ Rn , it is tempting to think of chiral E -spinors as even or odd differential forms. As we have already p remarked (cf. (13)), sab 2 GL.n/ acts on b via det sab • sab

b , where sab denotes the induced action of GL.n/ on forms, so that the family f a g does not transform as a differential form. To remedy this, we pick a nowhere vanishing n-vector field , that is, a collection of smooth maps a W Ua ! ƒn Rn , a D 2 a 0 , where the coefficient of a in C 1 .Ua / is assumed to be strictly positive. The notation 2 a is introduced to ease notation in the subsequent computations. Since is globally defined, 2 the coefficients under 2 a D det sab b . We then define an isomorphism transform ev;odd L W S .E/˙ ! .M / by .a/

La W . a W Ua ! S˙ / 7! .e Ba ^ a a W Ua ! ƒev;odd Rn /: We need to show that this transforms correctly under the action of the transition functions sab on ƒev;odd T M , using the fact that a D Q ab • b . Indeed, we have on Ua \Ub 6D ; p .b/ .b/ sab .e Bb ^ b b / D a det sab e Ba ^ sab

b p .a/ .ab/ D a e Ba ^ e ˇa ^ det sab sab

b .a/

D a e Ba ^ Q ab • b .a/

D a e Ba ^ a ; ı Lb . In the same vein, the Spin.n; n/-invariant form or equivalently, La ı Q ab D sab h ; i induces as above a globally defined inner product on .S / by

1

h ; i D ŒL . / ^ L . /n ; where is a nowhere vanishing n-vector field. The fact that the generalised tangent bundle is obtained out of twisting the transition functions of TM ˚ T M with closed B-fields bears an important consequence. Given the choice of with local coefficients 2 a , we define a parity reversing map d W S .E/˙ ! S .E/ ;

.d /a D 1 a da .a a /;

(19)

where da is the usual differential applied to forms Ua ! ƒ Rn , i.e. .d˛/a D da ˛a .

484

F. Witt

This definition gives indeed rise to an E -spinor field, for p .ab/ ˇa ^ det sab sab db .b b / Q ab • .d /b D 1 b e .ab/ ˇa D 1 ^ sab .b b / a da e p .ab/ ˇa D 1 ^ det sab sab

b / a da .a e D 1 a da .a a / D .d /a :

The operator d squares to zero and therefore induces a differential complex on S .E/˙ . As a corollary of the next proposition, we deduce that this complex actually computes the so-called twisted cohomology, where one replaces the usual differential d of the de Rham cohomology by the twisted differential dH D d C H ^. Proposition 4.4. Let 2 S .E/ . Then L .d / D dH L . /: Proof. This follows from a straightforward local computation: L .d /a D La .d /a .a/

D e Ba ^ a .d /a .a/

D e Ba ^ da .a a / .a/

D dHa .e Ba ^ a a / D dHa La . a /: In presence of a generalised Riemannian metric, we can make a canonic choice for , namely we pick the dual g of the Riemannian volume form volg . In this case, we write Lg D L and dg D d . Moreover, we obtain again an operator Gz D volV • for which we find as above: Proposition 4.5. The action of Gz D volV • on S ˙ .E / is given by ( n even: ?g L. / L.Gz • / D ; n odd: ?g L. /

b b e

where g is the Riemannian metric associated with V C . If the underlying manifold is spinnable, we can again identify bispinors with E spinors via the map Œ ; L1 Œ; G ev;odd W n .TM / ˝ n .TM / ! ev;odd .M / ˝ C ! S .E /˙ ˝ C :


485

A vector field X 2 .TM / acts on TM -spinor fields via the inclusion TM ,! Cliff.TM; g/ and Clifford multiplication. On the other hand, we can lift X to sections X ˙ of V ˙ which act on E -spinor fields via the inclusion V ˙ ,! Cliff.E / and Clifford multiplication. As in the previous section, we find that these actions are compatible in the following sense. Proposition 4.6. We have ŒX ‰L ˝ ‰R G D .1/n.n1/=2 X C • Œ‰L ˝ ‰R G ;

E

Œ‰L ˝ Y ‰R G D Y • Œ‰L ˝ ‰R G : In particular, we get as a Corollary 4.7. Let ‰L;R 2 n . (i) If n D 2m and ‰L;R are chiral, then for ‰R 2 ˙ Gz • Œ‰L ˝ ‰R G D ˙.1/m.m1/=2 i m Œ‰L ˝ ‰R G : (ii) If n D 2m C 1, then

E

Gz • Œ‰L ˝ ‰R G D .1/m.m1/=2 i mC1 Œ‰L ˝ ‰R G : Now it is clear how we can describe reductions to SU.m/ SU.m/, G2 G2 or Spin.7/ Spin.7/. Example. We consider a generalised SU.3/-structure. In dimension 6, a reduction from Spin.6/ to SU.3/ is induced by a unit spinor field, and as we saw in Section 2.2, there is no obstruction against existence. A reduction from the generalised Riemannian metric structure given by .g; H / to SU.3/ SU.3/ can be therefore characterised in terms of two unit spinors .‰L ; ‰R /, giving rise to the E -spinor fields

0 D ŒA.‰L / ˝ ‰R G ;

1 D Œ‰L ˝ ‰R G :

The corresponding differential forms are just L. 0 / D e i! ;

L. 1 / D

(cf. Section 3.2).

5 The field equations The second step of the G-structure ansatz consists in recovering the type II field equations (11)and (12) by an integrability condition on the algebraic objects defining the G-structure. Here, we will deal with the case F D 0. For a treatment with non-trivial R-R-fields in the case of SU.3/- and G2 -structures, see [23]. Throughout this section, r denotes the Levi-Civita connection. By a generalised G-structure, we shall mean a generalised structure characterised by a collection of decomposable bispinors such as generalised SU.m/-, G2 - or Spin.7/-structures.

486

F. Witt

5.1 Integrable generalised G -structures Definition 5.1. Let f 0 ; : : : ; l g be a collection of E -spinors defining a generalised G-structure. Then this structure is called integrable if d i D 0;

d Gz • i D 0;

i D 0; : : : ; l

(20)

for some 2 .ƒn TM / Example. Consider a Riemannian manifold .M 2m ; g/ whose holonomy is contained in SU.m/. As a consequence, M carries an SU.m/-structure with d! D 0 and d D 0. Then the corresponding straight structure characterised by 0 and 1 is also integrable, for d 0;1 D 0 if and only if d ŒA.‰/ ˝ ‰ D .1/m.mC1/=2 de i! D 0;

d Œ‰ ˝ ‰ D d D 0

by Proposition 4.4. From this point of view, the integrability condition (20) generalises the holonomy condition of classical SU.m/-structures. If the generalised G-structure induces a Riemannian metric, then we can define the dilaton field 2 C 1 .M / via D e 2 g : We can then write (20) as a form equation dH e L. i / D 0;

1

˙dH e ? L. i / D 0;

i D 0; : : : ; l:

Remark 5.2. The appearance of the dilaton field may seem artificial. However, there are two reasons to it: Firstly, we will prove a no-go theorem in the next section which asserts that a constant dilaton field implies H D 0 if the structure is integrable. In conjunction with the theorem we are going to prove in a moment, this means that the only integrable generalised structures which occur in that case are straight structures. Secondly, we can derive the integrability condition (20) on generalised SU.3/- and G2 structures from Hitchin’s variational principle which requires an a priori identification of E -spinors with forms and thus the choice of some dilaton field [23], [29]. The following theorem links integrability of SU.m/-structures into the supersymmetry equations (1) and (2). For generalised G2 - and Spin.7/-structures, see [29]. Theorem 5.3. Let . 0 ; 1 / D .ŒA.‰L /˝‰R G ; Œ‰L ˝‰R G / be a generalised SU.m/structure and 2 C 1 .M /. Then dH e ŒA.‰L / ˝ ‰R D 0;

dH e Œ‰L ˝ ‰R D 0

holds, i.e. the generalised SU.m/-structure is integrable, if and only if the equations

1 1 d H ‰L D 0; rX ‰L .X xH / ‰L D 0; 4 2 1 1 rX ‰R C .X xH / ‰R D 0; d C H ‰R D 0; 4 2 hold.

(21)


487

Proof. For computational purposes, it will be convenient to consider the equation d Œ‰L ˝‰R D .˛CH /^Œ‰L ˝‰R ; d ŒA.‰L /˝‰R D .˛CH /^ŒA.‰L /˝‰R for a 1-form ˛ instead of the dilaton. The key for solving this set of equations is the decomposability of the spinor: it makes the spinor and its associated differential form “self-dual” in the sense of Corollary 4.7. From (16) ?Œ‰L ˝ ‰R D .1/m.m1/=2 i m Œ‰L ˝ ‰R , so that

4

4

d ? Œ‰L ˝ ‰R D d .1/m.m1/=2 i m Œ‰L ˝ ‰R D .1/m.m1/=2 i m .˛ C H / ^ Œ‰L ˝ ‰R

(22)

and similarly for ŒA.‰L / ˝ ‰R . We recall that

G

ŒA.‰L / ˝ ‰R D ŒA.‰L / ˝ ‰R ;

D

Œ‰L ˝ ‰R D .1/m Œ‰L ˝ ‰R ;

as well as the general rules for forms R 2 .M 2m /, namely c f y D .1/m ?R; ?R

z ?.˛ ^ R/ D ˛x?R;

y D d R; z dR

From these we deduce

d R D ? d ? R:

D

d Œ‰L ˝ ‰R D .˛ C H /xŒ‰L ˝ ‰R ; d ŒA.‰L / ˝ ‰R D .˛ C H /xŒA.‰L / ˝ ‰R : From Proposition 3.4 follows immediately a technical lemma we need next. Lemma 5.4. Let ˛ be a 1-form. Its metric dual will be also denoted by ˛. Then

F F

1 .1/m Œ˛ ‰1 ˝ ‰2 Œ‰1 ˝ ˛ ‰2 2 1 ˛xŒ‰1 ˝ ‰2 D .1/m Œ˛ ‰1 ˝ ‰2 C Œ‰1 ˝ ˛ ‰2 2

˛ ^ Œ‰1 ˝ ‰2 D

In particular, if ek defines a local orthonormal basis, then for a 3-form H we find H ^ Œ‰1 ˝ ‰2 D

X .1/m

ek ‰1 ˝ .ek xH / ‰2 H ‰1 ˝ ‰2 8 k X 1

C ‰1 ˝ H ‰2 .ek xH / ‰1 ˝ ek ‰2 8 k

X .1/m

H ‰ 1 ˝ ‰2 C H xŒ‰1 ˝ ‰2 D ek ‰1 ˝ .ek xH / ‰2 8 k X 1

C ‰1 ˝ H ‰2 .ek xH / ‰1 ˝ ek ‰2 : 8 k

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F. Witt

Moreover, since d D

P

ek ^ rek and d D

P

ek xrek , we get

F F F F

1 z 1 ˝ ‰2 / ; .1/m ŒD.‰1 ˝ ‰2 / ŒD.‰ 2 1 z 1 ˝ ‰2 / ; d Œ‰1 ˝ ‰2 D .1/m ŒD.‰1 ˝ ‰2 / C ŒD.‰ 2 z on .T / ˝ .T / , given locally by with the twisted Dirac operators D and D X ek rek ‰1 ˝ ‰2 C ek ‰1 ˝ rek ‰2 D.‰1 ˝ ‰2 / D X D D‰1 ˝ ‰2 C ek ‰1 ˝ rek ‰2 ; X z 1 ˝ ‰2 / D D.‰ rek ‰1 ˝ ek ‰2 C ‰1 ˝ ek rek ‰2 X D rek ‰1 ˝ ek ‰2 C ‰1 ˝ D‰2 : d Œ‰1 ˝ ‰2 D

Note that r and Œ ; commute, since r is metric. As a result,

D.‰L ˝ ‰R / D .˛ C H / ^ Œ‰L ˝ ‰R C .1/m .˛ C H /xŒ‰L ˝ ‰R

D A.‰L / ˝ ‰R D .1/m .˛ C H / ^ .˛ C H /x ŒA.‰L / ˝ ‰R

(23) z L ˝ ‰R / D .1/m .˛ C H / ^ Œ‰L ˝ ‰R C .˛ C H /xŒ‰L ˝ ‰R D.‰

z D.A.‰ L / ˝ ‰R / D .˛ C H / ^ C.˛ C H /x ŒA.‰L / ˝ ‰R : Using the previous lemma to compute the action of ˛ C H on Œ‰L ˝ ‰R the two first equations of (23) become X ek ‰L ˝ rek ‰R D‰L ˝ ‰R C (24) 1X 1 D ˛ ‰L ˝ ‰R C ek ‰L ˝ .ek xH / ‰R H ‰L ˝ ‰R 4 4 X DA.‰L / ˝ ‰R C ek A.‰L / ˝ rek ‰R (25) X 1 1 D ˛ A.‰L / ˝ ‰R C ek A.‰L / ˝ .ek xH / ‰R H A.‰L / ˝ ‰R : 4 4 Contracting (24) from the left hand side with q.ej ‰L ; / gives 1 0 D q ej ‰L ; D‰L C ˛ ‰L C H ‰L ‰R 4 X C q.ej ‰L ; ek ‰L /.rek ek xH / ‰R : We apply the conjugate linear operator A which commutes with the Levi-Civita connection r since it is Spin.n/-invariant, and get 1 0 D q ej ‰L ; D‰L C ˛ ‰L C H ‰L A.‰R / 4 (26) X 1 C q.ej ‰L ; ek ‰L / rek ek xH A.‰R / 4 k

489


Moreover, applying A ˝ A to (25) yields D‰L ˝ A.‰R / C

X

ek ‰L ˝ rek ‰R D

1X ek ‰L ˝ .ek xH / A.‰R / 4 1 H ‰l ˝ A.‰R / 4

as ( X m even: ŒA.‰/ ˝ A.ˆ/ D m O q ‰; eK A.ˆ/ eK D m O Œˆ ˝ ‰ D m odd: K

3

Œ‰ ˝ ˆ;

C

Œ‰ ˝ ˆ:

Contracting again with q.ej ‰L ; / on the left gives 1 0 D q ej ‰L ; D‰L C ˛ ‰L C H ‰L A.‰R / 4 X 1 C q.ej ‰L ; ek ‰L / rek ek xH A.‰R /: 4

(27)

k

Adding (26) and (27) yields 1 0 D Re q ej ‰L ; D‰L ˛ ‰L H ‰L A.‰R / 4 X 1 C Re q.ej ‰L ; ek ‰L / rek ek xH A.‰R /: 4

(28)

Now the real part of q.ej ‰L ; ek ‰L / vanishes unless j D k when it equals 1. This implies rej A.‰R / D

1 .ej xH / A.‰R / 4 1 Re q ej ‰L ; D‰L C ˛ ‰L C H ‰L A.‰R /; 4

so that rX ‰R D

1 1 .X xH / ‰R Re q X ‰L ; D‰L C ˛ ‰L C H ‰L ‰R : 4 4

We contract with q.A.‰R /; / to see that Re q.ej ‰L ; : : : / D 0 as the remaining terms are purely imaginary, hence (upon applying A) 1 rX ‰R C .X xH / ‰R D 0: 4 Using the second set of equations in (23) gives 1 rX ‰L .X xH / ‰L D 0: 4

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F. Witt

As a result, we find D‰R;L D 3H=4 ‰R;L . On the other hand, (24) now reads 1 D‰L ˛ ‰L H ‰L ˝ ‰R D 0; 4 so by contracting with q.‰R ; / from the right hand side we obtain 1 D‰L ˛ ‰L H ‰L D 0; 4 which by the previous yields

1 ˛ H ‰L D 0: 2

Similarly, we obtain .˛ C 12 H / ‰R D 0. Remark 5.5. If H D 0, then const, and we get two parallel spinors for the Levi-Civita connection, leaving us with two possibilities: Either the spinors coincide at one and thus at any point, or the two spinors are linearly independent everywhere, in which case we may assume that they are orthogonal. In either scenario, the holonomy reduces to the intersection of the stabilisers of ‰L and ‰R inside Spin.n/, giving rise to a well-defined classical G-structure. We therefore refer to these solutions as classical. 5.2 Geometric properties. We now study some geometric properties of integrable G-structures by using the formulation given in (21). In fact, most statements are valid for geometries defined by one parallel spinor, i.e., we suppose to be given a solution ‰ to 1 1 rXH ‰ D rX ‰ C .X xH / ‰ D 0; d C H ‰ D 0: (29) 4 2 The key assumption here is that H is closed, as we will presently see. To start with, we first compute the Ricci tensor. By results of [10], the Ricci endomorphism RicH of r H with H closed is given by RicH .X / ‰ D .rXH H / ‰; and relates to the metric Ricci tensor through 1 1 Ric.X; Y / D RicH .X; Y / C d H.X; Y / C g.X xH; Y xH /: 2 4

(30)

Consequently, the scalar curvature S of the Levi-Civita connection is S D SH C

3 kH k2 ; 2

where S H is the scalar curvature associated with r H . Since ‰ is parallel with respect to r H , the dilatino equation implies RicH .X / ‰ D rXH .H ‰/ D 2.rXH d/ ‰;


491

hence RicH .X/ D 2rXH d. Now pick a frame that satisfies rei ej D 0 at a fixed point, or equivalently, reHj ek D ek xej xH=2. As the connection r H is metric, we obtain RicH .ej ; ek / D 2g.reHj d; ek / D 2ej :g.d; ek / C g.d; reHj ek / D 2ej :ek : C ek xej xH:=2: The first summand is minus twice H , the Hessian of evaluated in the basis fek g. Consequently, RicH .X; Y / D 2H .X; Y / X xY xH=2, hence S H D 2, where . / D Trg H . / is the Riemannian Laplacian. In the situation where we have two spinors ‰L;R parallel with respect to the connections r ˙H , we obtain from (30) Ric.X; Y / D

1 1 RicH .X; Y / C RicH .X; Y / C g.X xH; Y xH /; 2 4

and thus the Theorem 5.6. The Ricci tensor Ric and the scalar curvature S of a metric of an integrable generalised G-structure are given by 1 Ric.X; Y / D 2H .X; Y / C g.X xH; Y xH /; 2

S D 2 C

3 kH k2 : 2

Remark 5.7. Note that Theorem 5.6 corrects an error in [29], where in Theorem 4.9 and Proposition 5.7 the scalar curvature was stated to be S D 2 C 3=4 kH k2 . Closeness of the torsion implies two striking no-go theorems. The first one is this: if the dilaton is constant, then we get a classical solution, i.e. H D 0. Theorem 5.8. If there exists a spinor field ‰ satisfying (29), then 4 kdk2 C2 kH k2 D 0: Consequently, S D 5 C 6 k k2 , and const implies H D 0. In particular, there are no non-classical homogeneous solutions. Proof. The equations (29) imply D‰ D 3H=4 ‰ D 3d=2 ‰. Taking the Dirac operator of the dilatino equation, we obtain (cf. for instance (1.27) in [2]) D.d ‰/ D d D‰ C d d' ‰ 2

D

X

d.ek / rek ‰

3 1 kdk2 C ‰ C dxH ‰; 2 2

(31)

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and with H D

P

.ek xH / ^ .ek xH /=2,

X 1 1 1 ek xH rek ‰ D. H ‰/ D H D‰ C d H ‰ 2 2 2 X 3 1 1 D H H ‰ C d H ‰ C .ek xH / .ek xH / ‰ 8 2 4 (32) 1 1 H 3 3 kH k2 ‰ D .kH k2 2 H / ‰ C d H ‰ C 8 2 2 2 1 1 3 D kH k2 ‰ C d H ‰ H ‰: 8 2 4 Adding (31) and (32) yields 1 3 1 H 3 2 2 Re q ‰; D.d C H / ‰ D q ‰; ‰ D 0: kdk C kH k 2 2 8 4 Note that q.˛ p ‰; ‰/ is purely imaginary for p 2.4/ and real for p 0.4/. On the other hand, q.‰L ; H ‰L / D S H q.‰; ‰/=2 by Corollary 3.2 in [10]. But this equals by the computation above, from which the assertion follows. The second no-go theorem states that any integrable G-structure over a compact manifold has vanishing torsion and is therefore classical. To see this, we recall the relation X DH ‰ C H D‰ D d H ‰ 2 H ‰ 2 ek xH reHk ‰ taken from Theorem 3.3 in [10]. Now assuming a solution to (29), 3 H D‰ D H H ‰ D 4

3 3 kH k2 C H ‰; 4 2

and

3 1 D.H ‰/ D kH k2 Cd H H ‰ 4 2 by the proof of the previous theorem. Hence

DH ‰ C H D‰ D

3 kH k2 Cd H C H ‰ D .d H 2 H / ‰; 2

and consequently H ‰ D 12 kH k2 ‰. On the other hand, q.‰; H ‰/ D S H D 2. If M is compact, integration of 2q. H ‰; ‰/ gives Z Z 2 kH k D 4 D 0; M

M

whence H D 0. Theorem 5.9. On compact manifolds, any integrable generalised G-structure is classical. Remark 5.10. There are compact examples of generalised G2 - and Spin.7/-structures satisfying (21) for non-closed H [29]. For a compact example of a generalised G2 structure satisfying an inhomogeneous version of (20), see [9].


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References [1] Baum, H., Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten. Teubner-Texte Math. 41, B. G. Teubner Verlagsgesellschaft, Leipzig 1981 463 [2] Baum, H., Friedrich, T., Grunewald, R., and Kath, I., Twistors and Killing Spinors on Riemannian manifolds. Teubner Texte Math. 124, Teubner Verlag, Stuttgart, Leipzig 1991. 491 [3] Berger, M., Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83 (1955), 279–330. 463 [4] Bergshoeff, E., Kallosh, R., Ortin, T., Roest, D., and Van Proeyen, A., New formulations of D = 10 supersymmetry and D8 - O8 domain walls. Classical Quantum Gravity 18 (2001), 3359–3382. 472 [5] Bernard, D., Sur la géométrie différentielle des G-structures. Ann. Inst. Fourier 10 (1960), 151–270. 470 [6] Cardoso, G., Curio, G., Dall’Agata, G., Lüst, D., Manousselis, P., and Zoupanos, G., NonKähler string backgrounds and their five torsion classes. Nucl. Phys. B 652 (2003), 5–34. 456, 470, 471 [7] Chiossi, S., and Salamon, S., The intrinsic torsion of S U.3/ and G2 structures. In Differential geometry, Valencia 2001, World Scientific Publishing, River Edge, NJ, 115–133. 470 [8] Courant, T., Dirac manifolds. Trans. Amer. Math. Soc. 319 (1990), 631–661. [9] Fino, A., and Tomassini, A., Generalized G2 -manifolds and SU(3)-structures. Internat. J. Math., to appear; preprint 2007, arXiv:math/0609820v2. 458, 492 [10] Friedrich, T., and Ivanov, S., Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6 (2002), 303–335. 490, 492 [11] Fulton, W., and Harris, J., Representation Theory. Grad. Texts in Math. 129, Springer-Verlag, New York 1991. 460 [12] Gauntlett, J., Martelli, D., Pakis, S., and Waldram, D., G-structures and wrapped NS5branes. Comm. Math. Phys. 247 (2004), 421–445. 458 [13] Gauntlett, J., Martelli, D., and Waldram, D., Superstrings with intrinsic torsion. Phys. Rev. D 69 (2004), 086002. 456, 470 [14] Gmeiner, F., and Witt, F., Calibrated cycles and T-duality. Comm. Math. Phys. , to appear; preprint, 2007, arXiv:math. dg/0605710. 477, 478 [15] Harvey, F., Spinors and Calibrations. Perspect. Math. 9, Academic Press, Boston, MA, 1990. 463 [16] Hitchin, N., The moduli space of special Lagrangian submanifolds. Annali Scuola Sup. Norm. Pisa Sci. Fis. Mat. 25 (1997), 503–515. 479, 480 [17] Hitchin, N., Lectures on special Lagrangian submanifolds. In Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc., Providence, RI, 2001, 151–182. 480 [18] Hitchin, N., Generalized Calabi-Yau manifolds. Quart. J. Math. Oxford Ser. 54 (2003), 281–308. 457, 458, 473, 480

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[19] Hitchin, N., Brackets, forms and invariant functionals. Asian J. Math. 10 (3) (2006), 541–560. 458 [20] Huybrechts, D., Generalized Calabi-Yau structures, K3 surfaces, and B-fields. Internat. J. Math. 16 (1) (2005), 13–36. 457 [21] Ivanov, P., and Ivanov, S., SU(3)-instantons and G(2), Spin(7)-heterotic string solitons. Comm. Math. Phys. 259 (2005), 79–102. 470 [22] Jeschek, C., and Witt, F., Generalised G2 -structures and type IIB superstrings. J. High Energy Phys. 0503 (2005), 053. 457, 478 [23] Jeschek, C., and Witt, F., Generalised geometries, constrained critical points and RamondRamond fields. Preprint, 2005; arXiv:math.DG/0510131. 458, 471, 485, 486 [24] Kapustin, A., Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1 (2004), 49–81. 457 [25] Kobayashi, S., and Nomizu, K., Foundations of differential geometry. Vol. I, Wiley Classics Lib., John Wiley & Sons, New York 1996. 459 [26] Lawson, H., and Michelsohn, M.-L., Spin geometry. Princeton Math. Ser. 38, Princeton University Press, Princeton, NJ, 1989. 463 [27] Moore, J., Lectures on Seiberg–Witten invariants. Lecture Notes in Math. 1629, SpringerVerlag, Berlin 2001. 459 [28] Wang, M., Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7 (1) (1989), 59–68. 463 [29] Witt, F., Generalised G2 -manifolds. Comm. Math. Phys. 265 (2006), 275–303. 458, 459, 473, 476, 477, 479, 486, 491, 492 [30] Witt, F., Special metric structures and closed forms. DPhil thesis, University of Oxford, 2005; arXiv:math.DG/0502443. 457

Einstein metrics with 2-dimensional Killing leaves and their physical interpretation Gaetano Vilasi

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

2

Metrics of .G2 ; 2/-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 2.1 Geometric aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

3

Global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 3.1 -complex structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 3.2 Global properties of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

4

Examples . . . . . . . . . . . . . . . 4.1 Algebraic solutions . . . . . . . 4.2 Info-holes . . . . . . . . . . . . 4.3 A star “outside” the universe . . 4.4 Kruskal–Szekeres type solutions

5

Metrics of .G2 ; 1/-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

6

Metrics of .G2 ; 0/-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

7

Physical properties of .G2 ; 2/-type metrics . . . . . . . . . . . . . . . . . . . . . . . 510 7.1 The standard linearized theory . . . . . . . . . . . . . . . . . . . . . . . . . . 510 7.2 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

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Appendix: The Petrov classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

1 Introduction The aim of this paper is to illustrate some interesting and, in a sense, surprising physical properties of special solutions of Einstein field equations belonging to the larger class of Einstein metrics invariant for a non-Abelian Lie algebra of Killing vector fields generating a 2-dimensional distribution. Some decades ago, by using a suitable generalization of the Inverse Scattering Transform, Belinsky and Sakharov [8] were able to determine 4-dimensional Ricci-flat Lorentzian metrics invariant for an Abelian 2-dimensional Lie algebra of Killing vector fields such that the distribution D ? orthogonal to the one, say D, generated by the Killing fields is transversal to D and Frobenius-integrable. This success opened immediately several interesting questions:

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• Integrable systems, beyond a Lax pair, generally exhibit a recursion operator which is responsible for the construction of a sequence of conserved functionals and which can be naturally interpreted as a mixed tensor field on the carrier phase space. Such a tensor field has a vanishing Nijenhuis torsion and special spectral properties which allow to generalize [17], [18], [19], [27], to infinite dimensional manifolds, the classical Liouville theorem on complete integrability. How many conserved functionals do exist in this integrable model and what is their geometric significance? • From a physical point of view, 4-dimensional Lorentzian Ricci-flat metrics represent gravitational fields and deserve special attention when they have a wave-like character. Indeed, presently there are, worldwide, many efforts to detect gravitational radiation, not only because a direct confirmation of their existence is interesting per se but also because new insights on the nature of gravity and of the Universe itself could be gained. Gravitational waves, that is a propagating warpage of space time generated from compact concentrations of energy, like neutron stars and black holes, have not yet been detected directly, although their indirect influence has been seen and measured with great accuracy. Thus, as a first step, it has been natural to consider [40] the problem of characterizing all gravitational fields g admitting a Lie algebra G of Killing fields such that: I. the distribution D, generated by vector fields of G , is 2-dimensional; II. the distribution D ? , orthogonal to D is integrable and transversal to D. As we will see in Section 5 and 6, the condition of transversality can be relaxed. This case, when the metric g restricted to any integral (2-dimensional) submanifold (Killing leaf) of the distribution D is degenerate, splits naturally into two sub-cases according to whether the rank of g restricted to Killing leaves is 1 or 0. Sometimes, in order to distinguish various cases occurring in the sequel, the notation .G ; r/ will be used: metrics satisfying the conditions I and II will be called of .G ; 2/-type; metrics satisfying conditions I and II, except the transversality condition, will be called of .G ; 0/-type or of .G ; 1/-type according to the rank of their restriction to Killing leaves. According to whether the dimension of G is 3 or 2, two qualitatively different cases can occur. Both of them, however, have in common the important feature that all manifolds satisfying the assumptions I and II are in a sense fibered over -complex curves[42]. When dim G D 3, assumption II follows from I and the local structure of this class of Einstein metrics can be explicitly described. Some well-known exact solutions [37], [44], e.g. Schwarzschild, belong to this class. A 2-dimensional G is either Abelian (A2 ) or non-Abelian (G2 ) and a metric g satisfying I and II, with G D A2 or G2 , will be called G -integrable. The study of A2 -integrable Einstein metrics goes back to Einstein and Rosen [21], Kompaneyets [25], Geroch [22], Belinsky, Khalatnikov, Zakharov [7], [8], Verdaguer [45]. Recent results can be found in [14]. The greater rigidity of G2 -integrable metrics, for which some partial results can be

Einstein metrics with 2-dimensional Killing leaves

497

found in [2], [16], [23], allows an exhaustive analysis. It will be shown that the ones of .G ; 2/-type are parameterized by solutions of a linear second order differential equation on the plane which, in its turn, depends linearly on the choice of a -harmonic function (see later). Thus, this class of solutions has a bilinear structure and, as such, admits two superposition laws. All the possible situations, corresponding to a 2-dimensional Lie algebra of isometries, are described by the following table where a non-integrable 2-dimensional disD ?, r D 0

D ?, r D 1

D ?, r D 2

G2

integrable

integrable

integrable

G2

semi-integrable

semi-integrable

semi-integrable

G2

non-integrable

non-integrable

non-integrable

A2

integrable

integrable

integrable

A2

semi-integrable

semi-integrable

semi-integrable

A2

non-integrable

non-integrable

non-integrable

tribution which is part of a 3-dimensional integrable distribution has been called semiintegrable and in which the cases indicated with bold letters have been essentially solved [40], [41], [42], [13], [14], [4]. In Section 2, 4-dimensional metrics of .G2 ; 2/-type invariant for a non-Abelian 2-dimensional Lie algebra are characterized from a geometric point of view. The solutions of corresponding Einstein field equations are explicitly written. The construction of global solutions is described in Section 3 and some examples are given in Section 4. Section 5 and 6 are devoted to metrics of .G ; 1/-type and of .G ; 0/-type respectively. In Section 7 the case in which the commutator of generators of the Lie algebra is of light-type is analyzed from a physical point of view. Harmonic coordinates are also introduced. Moreover, the wave-like character of the solutions is checked through the Zel’manov and the Pirani criterion. The canonical, the Landau–Lifshitz and the Bel energy-momentum pseudo-tensors are introduced and a comparison with the linearised theory is performed. Realistic sources for such gravitational waves are also described. Eventually, the analysis of the polarization leads to the conclusion that these fields are spin-1 gravitational waves.

2 Metrics of .G2 ; 2/-type In the following, we will consider 4-dimensional manifolds and Greek letters take values from 1 to 4; the first Latin letters take values from 3 to 4, while i , j from 1 to 2. Moreover, Kil .g/ will denote the Lie algebra of all Killing fields of a metric g while Killing algebra will denote a sub-algebra of Kil .g/. Moreover, an integral (2-dimensional) submanifold of D will be called a Killing leaf , and an integral (2-dimensional) submanifold of D ? orthogonal leaf.

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2.1 Geometric aspects • Semiadapted coordinates. Let g be a metric on the space-time M (a connected smooth manifold) and G2 one of its Killing algebras whose generators X , Y satisfy ŒX; Y D sY , s D 0; 1. The Frobenius distribution D generated by G2 is 2-dimensional and in the neighborhood of a non-singular point a chart .x 1 ; x 2 ; x 3 ; x 4 / exists such that XD

@ ; @x 3

Y D exp.sx 3 /

@ : @x 4

From now on such a chart will be called semiadapted (to the Killing fields). • Invariant metrics. It can be easily verified [40], [41] that in a semiadapted chart g has the form g D gij dx i dx j C 2.li C smi x 4 / dx i dx 3 2mi dx i dx 4 C .s 2 .x 4 /2 2sx 4 C / dx 3 dx 3 C 2. sx 4 / dx 3 dx 4 C dx 4 dx 4 ;

i D 1; 2; j D 1; 2

with gij , mi , li , , , arbitrary functions of .x 1 ; x 2 /. • Killing leaves. Condition II allows to construct semi-adapted charts, with new coordinates .x; y; x 3 ; x 4 /, such that the fields e1 D @=@x, e2 D @=@y, belong to D ? . In such a chart, called from now on adapted, the components li ’s and mi ’s vanish. As it has already been said, we will call Killing leaf an integral (2-dimensional) submanifold of D and orthogonal leaf an integral (2-dimensional) submanifold of D ? . Since D ? is transversal to D, the restriction of g to any Killing leaf, S , is non-degenerate. Thus, .S; gjS / is a homogeneous 2-dimensional Riemannian manifold. Then, the Gauss curvature K.S / of the Killing leaves is constant (depending on the leave). In the chart (p D x 3 jS , q D x 4 jS ) one has 2 Q Q 2; Q 2 2s q gjS D .s 2 q Q C /dp Q C 2.Q s q/dpdq C dq

Q ; where ; Q , Q being the restrictions to S of ; ; , are constants, and Q 2 .Q 2 Q / K.S/ D s Q 1 : 2.1.1 Einstein metrics when g.Y; Y / ¤ 0. In the considered class of metrics, vacuum Einstein equations, R D 0, can be completely solved [40]. If the Killing field Y is not of light type, i.e. g.Y; Y / ¤ 0, then in the adapted coordinates .x; y; p; q/ the general solution is g D f .dx 2 ˙ dy 2 / C ˇ 2 .s 2 k 2 q 2 2slq C m/dp 2 C 2.l skq/dpdq C kdq 2 (1) where f D 4˙ ˇ 2 =2s 2 k, and ˇ.x; y/ is a solution of the tortoise equation ˇ C A ln jˇ Aj D u .x; y/ ;


499

where A is a constant and the function u is a solution either of the Laplace or the 2 d’Alembert equation, 4˙ u D 0, 4˙ D @2xx ˙ @yy , such that .@x u/2 ˙ .@y u/2 ¤ 0. 2 The constants k, l, m are constrained by km l D 1, k ¤ 0 for Lorentzian metrics or by km l 2 D ˙1, k ¤ 0 for Kleinian metrics. Ricci-flat manifolds of Kleinian signature possess a number of interesting geometrical properties and undoubtedly deserve attention in their own right. Some topological aspects of these manifolds were studied for the first time in [29], [30] and then in [28]. In recent years the geometry of these manifolds has seen a revival of interest. In part, this is due to the emergence of some new applications in physics. Canonical form of metrics when g.Y; Y / ¤ 0. The gauge freedom of the above solution, allowed by the function u, can be locally eliminated by introducing the coordinates .u; v; p; q/, the function v.x; y/ being conjugate to u.x; y/, i.e. 4˙ v D 0 and ux D vy ; uy D vx . In these coordinates the metric g takes the form gD

exp uˇ A .du2 ˙ dv 2 / C ˇ 2 .s 2 k 2 q 2 2slq C m/dp 2 C 2.l skq/dpdq C kdq 2 2 2s kˇ

with ˇ .u/ a solution of ˇ C A ln jˇ Aj D u. Normal form of metrics when g.Y; Y / ¤ 0. In geographic coordinates .#; '/ along Killing leaves one has gjS D ˇ 2 Œd # 2 C F .#/d' 2 ; where F .#/ is equal either to sinh2 # or cosh2 #, depending on the signature of the metric. Thus, in the normal coordinates .r D 2s 2 kˇ, D v; #; '/, the metric takes the form (local “Birkhoff’s theorem”)

A A 1 2 d2 ˙ 1 dr C "2 r 2 Œd # 2 C F .#/d' 2 (2) r r where "1 D ˙1, "2 D ˙1. The geometric reason for this form is that, when g.Y; Y / ¤ 0, a third Killing field exists which together with X and Y constitute a basis of so.2; 1/. The larger symmetry implies that the geodesic equations describe a non-commutatively integrable system [39], and the corresponding geodesic flow projects on the geodesic flow of the metric restricted to the Killing leaves. The above local form does not allow, however, to treat properly the singularities appearing inevitably in global solutions. The metrics (1), although they all are locally diffeomorphic to (2), play a relevant role in the construction of new global solutions as described in [41], [42]. g D "1

1

2.1.2 Einstein metrics when g.Y; Y / D 0. If the Killing field Y is of light type, then the general solution of vacuum Einstein equations, in the adapted coordinates .x; y; p; q/, is given by g D 2f .dx 2 ˙ dy 2 / C Œ.w .x; y/ 2sq/dp 2 C 2dpdq;

(3)

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where D Aˆ p C B with A; B 2 R, ˆ is a non-constant harmonic function of x and y, f D .rˆ/2 jj=, and w.x; y/ is solution of the -deformed Laplace equation: ˙ w C .@x ln jj/@x w ˙ .@y ln jj/@y w D 0; where C ( ) is the Laplace (d’Alembert) operator in the .x; y/-plane. Metrics (3) are Lorentzian if the orthogonal leaves are conformally Euclidean, i.e. the positive sign is chosen, and Kleinian if not. Only the Lorentzian case will be analyzed and these metrics will be called of .G2 ; 2/-isotropic type. In the particular case s D 1, f D 1=2 and D 1, the above (Lorentzian) metrics are locally diffeomorphic to a subclass of the vacuum Peres solutions [35], that for later purpose we rewrite in the form g D dx 2 ˙ dy 2 C 2dudv C 2.';x dx C ';y dy/du:

(4)

The correspondence between (3) and (4) depends on the special choice of the function '.x; y; u/ (which, in general, is harmonic in x and y arbitrarily dependent on u); in our case x ! x; y ! y; u ! u; v ! v C ' .x; y; u/ with h D ';u . In the case D const, the -deformed Laplace equation reduces to the Laplace equation; for D 1, in the harmonic coordinates system .x; y; z; t/ defined [11], for jz tj ¤ 0, by

„ x D x;

y D y; z D 12 Œ.2q w .x; y// exp .p/ C exp .p/ ; t D 12 Œ.2q w .x; y// exp .p/ exp .p/ ; the Einstein metrics (3) take the particularly simple form g D 2f .dx 2 ˙ dy 2 / C dz 2 dt 2 C d .w/ d.ln jz t j/:

(5)

This shows that, when w is constant, the Einsteinpmetrics given by equation (5) are static and, under the further assumption ˆ D x 2, they reduce to the Minkowski one. Moreover, when w is not constant, gravitational fields (5) look like a disturbance propagating at light velocity along the z direction on the Killing leaves (integral twodimensional submanifolds of D).

3 Global solutions Here we will give a coordinate-free description of previous local Ricci-flat metrics, so that it becomes clear what variety of different geometries, in fact, is obtained. We will see that with any of the obtained solutions there is associated a pair .W ; u/, consisting of


501

a -complex curve W and a -harmonic function u on it. If two solutions are equivalent, then the corresponding pairs, say .W ; u/ and .W 0 ; u0 /, are related by an invertible holomorphic map ˆ W .W ; u/ ! .W 0 ; u0 / such that ˆ .u0 / D u. Roughly speaking, the moduli space of the obtained geometries is surjectively mapped on the moduli space of the pairs .W ; u/. Further parameters, distinguishing the metrics we are analyzing, are given below; before that, however, it is worth to underline the following common peculiarities of these metrics: • They have, in the adapted coordinates, a block diagonal form whose upper block does not depend on the last two coordinates so that orthogonal leaves are totally geodesic. • They possess a non-trivial Killing field. Geodesic flows, corresponding to metrics admitting 3-dimensional Killing algebras, are non-commutatively integrable. The existence of a non-trivial Killing field is obvious from the description of model solutions given in the next section. For what concerns geodesic flows, they are integrated explicitly for model solutions in the next section, and the general result follows from the fact that any solution is a pullback of a model solution. Solutions of the Einstein equations previously described manifest an interesting common feature. Namely, each of them is determined completely by a choice of 1) a solution of the wave, or the Laplace equation, and either by 20 / a choice of the constant A and one of the branches, for ˇ as function of u, of the tortoise equation ˇ C A ln jˇ Aj D u; (6) if g .Y; Y / ¤ 0, or by 2 / a choice of a solution of one of the two equations 2 @y @2x C y @y x @x w D 0; 2 @y C @2x C y @y C x @x w D 0; 00

D 0;

(7)

4 D 0;

(8)

in the case g .Y; Y / D 0. They have a natural fibered structure with the Killing leaves as fibers. The wave and Laplace equations mentioned above in 1/, are in fact defined on the 2-dimensional manifold W which parameterizes the Killing leaves. These leaves themselves are 2dimensional Riemannian manifolds and, as such, are geodesically complete. For this reason the problem of the extension of described local solutions is reduced to that of the extension of the base manifold W . Such an extension should carry a geometrical structure that gives an intrinsic sense to the notion of the wave or the Laplace equation and to equations (7) and (8) on it. A brief description of how this can be done is the following.

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3.1 -complex structures. It is known there exist three different isomorphism classes of 2-dimensional commutative unitary algebras. They are C D RŒx=.x 2 C 1/;

R.2/ D RŒx=.x 2 /;

R ˚ R D RŒx=.x 2 1/:

Elements of this algebra can be represented in the form a C b, a; b 2 R, with 2 D 1; 0, or 1, respectively. For a terminological convenience we will call them -complex numbers. Of course, -complex numbers for 2 D 1 are just ordinary complex numbers. Furthermore, we will use the unifying notation R2 for the algebra of -complex numbers. For instance C D R2 for 2 D 1. In full parallel with ordinary complex numbers, it is possible to develop a -complex analysis by defining -holomorphic functions as R2 -valued differentiable functions of the variable z D x C y. Just as in the case of ordinary complex numbers, the function f .z/ D u.x; y/ C v.x; y/ is -holomorphic iff the -Cauchy–Riemann conditions hold: ux D vy ; uy D 2 vx : (9) The compatibility conditions of the above system requires that both u and v satisfy the -Laplace equation , that is 2 uxx C uyy D 0;

2 vxx C vyy D 0:

Of course, the -Laplace equation reduces for 2 D 1 to the ordinary Laplace equation, while for 2 D 1 to the wave equation. The operator 2 @2x C @y2 will be called the -Laplace operator. In the following a -complex structure on W will denote an endomorphism J W D .W / ! D .W / of the C 1 .W / module D .W / of all vector fields on W , with J 2 D 2 I , J ¤ 0; I , and vanishing Nijenhuis torsion, i.e., ŒJ; J F N D 0, where Œ ; F N denotes for the Frölicher–Nijenhuis bracket. A 2-dimensional manifold W supplied with a -complex structure is called a -complex curve. Obviously, for 2 D 1 a -complex curve is just an ordinary 1-dimensional complex manifold (curve). By using the endomorphism J the -Laplace equation can be written intrinsically as d.J du/ D 0; where J W ƒ1 .W / ! ƒ1 .W / is the adjoint to J endomorphism of the C 1 .W / module of 1-forms on W . Given a 2-dimensional smooth manifold W , an atlas f.Ui ; î /g on W is called -complex iff i) î W Ui ! W , Ui is open in R2 , ii) the transition functions ˆj1 ı î are -holomorphic. Two -complex atlases on W are said to be equivalent if their union is again a -complex atlas.


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A class of -complex atlases on W supplies, obviously, W with a -complex structure. Conversely, given a -complex structure on W there exists a -complex atlas on W inducing this structure. Charts of such an atlas will be called -complex coordinates on the corresponding -complex curve. In -complex coordinates the endomorphism J and its adjoint J are described by the relations J.@y / D 2 @x ;

J.@x / D @y ; J .dx/ D 2 dy;

J .dy/ D dx:

If 2 ¤ 0, the functions u and v in the equation (9) are said to be conjugate. Alternatively, a -complex curve can be regarded as a 2-dimensional smooth manifold supplied with a specific atlas whose transition functions .x; y/ 7! . .x; y/; .x; y// satisfy the -Cauchy–Riemann relations (9). As it is easy to see, the -Cauchy–Riemann relations (9) imply that @2 2 @2 D

x2

1 .@2 2 @2x /; 2 y2 y

x2

1 .@y2 2 @2x / C y @y 2 x @x : 2 2 y

and also .@2 2 @2 / C @ 2 @ D

This shows that equation (7) (respectively, (8)) is well defined on a -complex curve with 2 D 1 (respectively, 2 D 1). The manifestly intrinsic expression for these equations is d.J dw/ D 0: We will refer to it as the -deformed -Laplace equation. A solution of the -Laplace equation on W will be called -harmonic. We can see that in the case 2 ¤ 0 the notion of conjugate -harmonic function is well defined on a -complex curve. In addition, notice that the metric field d 2 2 d 2 , being -conjugate with , is canonically associated with a -harmonic function on W . A map ˆ W W1 ! W2 connecting two -complex curves will be called -holomorphic if ' ı ˆ is locally -holomorphic for any local -holomorphic function ' on W2 . Obviously, if ˆ is -holomorphic and u is a -harmonic function on W2 , then ˆ .u/ is -harmonic on W1 . It is worth noting that the standard -complex curve is R2 D f.x C y/g, and the standard -harmonic function on it is given by x, whose conjugated is y. The pair .R2 ; x/ is universal in the sense that for a given -harmonic function u on a -complex curve W there exists a -holomorphic map ˆ W W ! R2 defined uniquely by the relations ˆ .x/ D u and ˆ .y/ D v, v being conjugated with u.

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3.2 Global properties of solutions. The above discussion shows that any global solution, that can be obtained by matching together local solutions described in Section 2, is a solution whose base manifold is a -complex curve W and which corresponds to a -harmonic function u on W . A solution of Einstein equations corresponding to W R2 , u x will be called a model. Notice that there exist various model solutions due to various options in the choice of parameters appearing in 20 / and 200 / at the beginning of this section. An important role played by the model solutions is revealed by the property [42] that Any solution of the Einstein equation which can be constructed by matching together local solutions described in Section 2 is the pullback of a model solution via a -holomorphic map from a -complex curve to R 2 . We distinguish between the two following qualitatively different cases: I. metrics admitting a normal 3-dimensional Killing algebra with 2-dimensional leaves; II. metrics admitting a normal 2-dimensional Killing algebra that does not extend to a larger algebra having the same leaves and whose distribution orthogonal to the leaves is integrable. It is worth mentioning that the distribution orthogonal to the Killing leaves is automatically integrable in Case I [41]. In Case II the 2-dimensionality of the Killing leaves is guaranteed by proposition 2 of [41]. Any Ricci-flat manifold .M; g/ we are analyzing is fibered over a -complex curve W W M ! W ; whose fibers are the Killing leaves and as such are 2-dimensional Riemann manifolds of constant Gauss curvature. Below, we shall call the Killing fibering and assume that its fibers are connected and geodesically complete. Therefore, maximal (i.e., non-extendible) Ricci-flat manifolds, of the class we are analyzing in the paper, are those corresponding to maximal (i.e., non-extendible) pairs .W ; u/, where W is a -complex curve and u is -harmonic function on W . 3.2.1 Case I. Here the Killing algebra G is isomorphic to one of the following: so.3/, so.2; 1/, Kil.dx 2 ˙ dy 2 /, A3 (see Section 7 in [41]), and the Killing fibering splits in a canonical way into the Cartesian product. This product structure can be interpreted as a flat connection in , determined uniquely by the requirements that its parallel sections are orthogonal to the Killing leaves and the parallel transports of fibers are their conform equivalences (with respect to the induced metrics). In that sense one can say that the Killing fibering is supplied canonically with a conformally flat connection. This is a geometrically intrinsic way to describe the Killing fibering. A discrete isometry group acting freely on a fiber of the Killing fibering can be extended fiber-wise to the whole of M due to the canonical product structure mentioned


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above. Conversely, a locally isometric covering W Sz ! S of a fiber S allows to z ! M which copies fiber by fiber. So any construct a locally isometric covering M homogeneous 2-dimensional Riemannian manifold can be realized as typical fiber of a Killing fibering. Denote by .†; g† / a homogeneous 2-dimensional Riemannian manifold, whose Gauss curvature K .g† /, if different from zero, is normalized to ˙1. Denote by .W ; u/ the pair constituted by a -complex curve W and a -harmonic function u on W . Denote also by 1 (respectively, 2 ) the natural projection of M D W † on W (respectively, on †). Then, the above data determine the Ricci-flat manifold .M; g/ with M D W †;

g D 1 gfug C 1 .ˇ 2 /2 .g† /

(10)

where ˇ D ˇ.u/ is implicitly determined by u via the equation ˇ C A ln jˇ Aj D u and gfug D

ˇA .du2 2 dv 2 /; ˇ

(11)

(12)

A being an arbitrary constant and D ˙1 . Only in the case A D 0 the equation (11) determines the function ˇ .u/ uniquely: ˇ u and g is flat. Thus, from equation (11) one can see that for A ¤ 0 there are up to three possibilities for ˇ D ˇ.u/ that correspond to the intervals of monotonicity of u.ˇ/. For instance, for A > 0 these are 1; 0Œ, 0; AŒ, and A; 1Œ. In these regions the metric (12) is regular and has some singularities along the curves ˇ D 0 and ˇ A D 0. Thus, Any Ricci-flat 4-metric admitting a normal Killing algebra isomorphic to so .3/ or so .2; 1/ with 2-dimensional leaves is of the form (10) [42]. In the case of normal Killing algebras isomorphic to Kil.dx 2 ˙ dy 2 / consider Ricci-flat manifolds M of the form M D W †;

g D 1 .gŒu / C 1 .u/2 .g† /;

(13)

where .†; g† / is a flat 2-dimensional manifold and 1 gŒu D p .du2 2 dv 2 / u with D ˙1. Thus, Any Ricci-flat 4-metric, admitting a normal Killing algebra which is isomorphic to Kil.dx 2 C dy 2 / and with 2-dimensional Killing leaves, is either of the form (13) or flat [42].

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3.2.2 Case II. In this case, a coordinate-free description of global solutions, obtained in a local form in [41], is as follows. Let .W ; u/ be as before, and w be a solution of the equation d.u J .dw// D 0. Consider the flat indefinite Euclidean plane .R2 ; d 2 d 2 / introduced at the end of Section 3 of [41]. Then the direct product M D W R2 can be supplied with the following Ricci-flat metric gD

1

gŒu C

1 .u/2 .d 2

2

d / C

1 .uw/2

d d

2

;

(14)

where 1 W M D W R2 ! W and 2 W M D W R2 ! R2 are natural projections and 1 gŒu D p .du2 dv 2 / u with D ˙1. In the above construction one can substitute the quotient R2 =T for R2 , where T denotes the discrete group acting on R2 generated by a transformation of the form . ; / ! . C a; C a/, a 2 R. Let now .W ; u/ be as before but w is a -harmonic function on W . Then M D W R2 carries the Ricci-flat metric d d 2 2 2 2 2 2 g D 1 1 .du dv / C 2 2 .d d / C 1 .w/2 (15)

with i D ˙1. We have [42]: Any Ricci-flat 4-metric, admitting a non-extendible 2- dimensional non-commutative Killing algebra, is either of the form (14) or (15) with Killing leaves of one of two types or R2 or R2 =T .

4 Examples In this section, we illustrate the previous general results with a few examples using the fact that any solution can be constructed as the pullback of a model solution via a -holomorphic map ˆ of a -complex curve W to R2 . Recall that in the pair .W ; u/, describing the so obtained solution, u D Re ˆ. 4.1 Algebraic solutions. Let W be an algebraic curve over C, understood as a complex curve with 2 D 1. With a given meromorphic function ˆ on W a pair .Wˆ ; u/ is associated, where Wˆ is W deprived of the poles of ˆ and u the real part of ˆ. A solution (metric) constructed over such a pair will be called algebraic. Algebraic metrics are generally singular. For instance, such a metric is degenerate along the fiber 11 .a/ (see Section 3) if a 2 W is such that da u D 0.


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4.2 Info-holes. Space-times corresponding to algebraic metrics and, generally, to metrics with signature equal to 2 constructed over complex curves ( 2 D 1), exhibit the following interesting property: for a given observer there exists another observer which can be never contacted. By defining an info-hole (information-hole) of a given point a to be the set of points of the space-time whose future does not intersect the future of a, the above property can be paraphrased by saying that the info-hole of a given point of such a space-time is not empty. In fact, consider a metric of the form (10) constructed over a complex curve whose standard fiber † is a 2-dimensional manifold supplied with an indefinite metric of constant Gauss curvature equal to 1. For our purpose, it is convenient to take for † the hyperboloid x12 C x22 x32 D 1, supplied with the induced metric gj† D dx12 C dx22 dx32 j† . The light-cone of gj† at a given point b is formed by the pair of rectilinear generators of † passing through b. Since the geodesics of g project via 2 into geodesics of gj† , it is sufficient to prove the existence of info-holes for the 2-dimensional space-time .†; gj† /. To this purpose, consider the standard projection of R3 D f.x1 ; x2 ; x3 /g onto R2 D f.x1 ; x2 /gThen, j† projects † onto the region x12 C x22 1 in R2 and the rectilinear generators of † are projected onto tangents to the circle x12 C x22 D 1. Suppose that the time arrow on † is oriented according to increasing value of x3 . Then the future region F .b/ † of the point b 2 †, b.1; 1; ˇ/, ˇ > 0, projects onto the domain defined by D D fx 2 R2 W x_1>1, x2 > 1g, and the future region of any point b 0 2 †, such that .b 0 / 2 D 0 and x3 .b 0 / > 0, does not intersect F .b/. By obvious symmetry arguments the result is valid for any point b 2 †. 4.3 A star “outside” the universe. The Schwarzschild solution shows a “star” generating a space “around” itself. It is an so.3/-invariant solution of the vacuum Einstein equations. On the contrary, its so.2; 1/-analogue shows a “star” generating the space only on “one side of itself”. More precisely, the fact that the space in the Schwarzschild universe is formed by a 1-parametric family of “concentric” spheres allows one to give a sense to the adverb “around”. In the so .2; 1/-case the space is formed by a 1-parameter family of “concentric” hyperboloids. The adjective “concentric” means that the curves orthogonal to hyperboloids are geodesics and metrically converge to a singular point. This explains in what sense this singular point generates the space only on “one side of itself”. 4.4 Kruskal–Szekeres type solutions. We describe now a family of solutions which are of the Kruskal–Szekeres type [46], namely, that are characterized as being maximal extensions of the local solutions determined by an affine parametrization of null geodesics, and also by the use of more than one interval of monotonicity of u .ˇ/. Consider the -complex curve ˚

W D .z D x C y/ 2 R2 W y 2 x 2 < 1 ; 2 D 1;

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and the -holomorphic function ˆ W W ! R2 :

ˇ

ˇ

ˇx C y ˇ ˇ : ˆ.z/ D A ln jAjz 2 D A ln jA.x 2 y 2 /j C ln ˇˇ xyˇ

Thus, in the pair .W ; u/ the -harmonic function u is given by u D A ln jA.x 2 y 2 /j: Let us decompose W in the following way: W D U1 [ U2 where ˚

U1 D .z D x C y/ 2 R2 W 0 y 2 x 2 < 1 ; ˚

U2 D .z D x C y/ 2 R2 W y 2 x 2 0 : Consider now the solution defined as the pull back with respect to ˆjU1 and ˆjU2 of the model solutions determined by the following data: in the case of ˆjU1 , G D so.3/ or G D so.2; 1/, characterized by ± .#/ D sin2 # or ± .#/ D sin h2 # respectively, 1 D 2 D 1, A > 0, and for ˇ.u/ the interval 0; A; in the case of ˆjU2 the same data except for ˇ.u/ which belongs to the interval ŒA; 1Œ. The case ± .#/ D sin2 #, corresponding to so.3/, will give the Kruskal–Szekeres solution. The case ± .#/ D sinh2 #, corresponding to so.2; 1/, will differ from the previous one in the geometry of the Killing leaves, which will now have a negative constant Gaussian curvature. The metric g has the following local form g D 4A3

ˇ exp A .dy 2 dx 2 / C ˇ 2 d # 2 C ± .#/d' 2 ˇ

the singularity ˇ D 0 occurring at y 2 x 2 D 1.

5 Metrics of .G2 ; 1/-type In this case there exists a 1-dimensional distribution, tangent to Killing leaves S , that associates to a point p 2 M the kernel Cp of the tensor gjSp on Sp at p. This distribution, called characteristic, will be denoted by C D fCp g and we shall refer to Cp as the characteristic direction at p [13], [14]. A vector field C belonging to the characteristic distribution will be also called characteristic. In such an instance, we write C 2 C and similarly for other distributions we shall deal with. The 3-dimensional distribution C? , orthogonal to C, obviously contains the 2dimensional distribution D ? , orthogonal to D. In its turn D ? contains the characteristic distribution C. Observe that the intersection of distributions D and D ? is exactly the characteristic distribution (i.e., Dp \ Dp? D Cp for all p 2 M).


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Thus, a vector field U 2 D ? which is transversal to D generates, together with a characteristic field C 2 C, the distribution D ? . ? ? While C is the intersection of distributions D and D , the distribution C is?the ? ? span of them in the sense that Cp D span Dp ; Dp for all p 2 M. Therefore, C is generated by X; Y and U . Since the distribution C? is completely integrable [13], by the Frobenius theorem there exists a unique 3-dimensional integral submanifold of the distribution C? passing through a point a 2 M . Denote it by Pa . Then, an adapted local chart in a neighborhood of a point a 2 M , can be constructed as follows. First, choose a curve .v/, v belonging to an interval I R through a and transversal to all submanifolds P.v/ ’s. Next consider the local flow f'u g generated by a nowhere vanishing field U 2 D ? defined in a neighborhood of a and transversal to Killing leaves. Then .u; v/ ! 'u . .v// with .u; v/ ranging in a “small” domain U R2 is a parametric surface † in M. By construction, † is transversal to Killing leaves. ˚

The local chart .x; y; u; v/ D .Ax ı By ı 'u /. .v// with fAx g and By being local flows generated by X and Y , respectively, and .x; y; u; v/ ranging over a suitable “small” domain in R4 is called almost adapted. For any almost adapted chart .x; y; u; v/ it holds: • X D @x ; Y D exp .sx/ @y ; ˚

• vector fields @u and @v are invariant with respect to flows fAx g and By ; • @u 2 D ? . In this case it is always possible to normalize the vector field @u . An almost adapted chart .x; y; u; v/ will be called adapted if r D g.@u ; @u / D ˙1. In an adapted chart .x; y; u; v/ the most general metric of .G2 ; 1/-type has the form g D "0 ..syaCb/dxady/2 C2.sycCl/dxdvC2cdydvC"1 du2 C2mdudvCndv 2 where "0 ; "1 D ˙1 and a, b, c, l, m are functions in .u; v/. The distribution D ? is integrable if and only if au D f a; bu D f b: Explicit Ricci-flat metrics are given in [13], [14].

6 Metrics of .G2 ; 0/-type First, note that this case is characterized by the fact that D ? D D. Thus, the condition II of the introduction is satisfied automatically and any 1-dimensional distribution C tangent to Killing leaves may be “declared” as a characteristic one. If, moreover, C is preserved by the algebra G , i.e. ŒX; Z; ŒY; Z 2 C for all Z 2 C , then one obtains a geometrically privileged chart just by replacing C for C in the construction of the previous section. The only difference coming out in the considered context is that

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the component r D g.@u ; @u / is not necessarily nowhere vanishing and, so, can not be normalized to ˙1 in such a case. Thus we call adapted a local chart which is constructed according to the scheme of the previous section in which C is replaced by C . We shall keep the notation .x; y; u; v/ for adapted coordinates in the considered context as well; this time @u 2 C Then in an adapted chart .x; y; u; v/ the most general metric has the form [13], [14], g D 2dxdu C 2.syb C a/dxdv C 2bdydv C cdu2 C 2ldudv C mdv 2 ; with a, b, c, l, m arbitrary functions of .u; v/. Being the .1; 1/-Ricci component not vanishing, R11 D 1=2s 2 , the above metrics are not Ricci-flat.

7 Physical properties of .G2 ; 2/-type metrics From a physical point of view, only Lorentzian metrics will be analyzed in the following, even if Ricci-flat manifolds of Kleinian signature appear in the ‘no boundary’ proposal of Hartle and Hawking [24] in which the idea is suggested that the signature of the space-time metric may have changed in the early universe. So, assuming that particles are free to move between Lorentzian and Kleinian regions some surprising physical phenomena, like time travelling, would be observable (see [1] and [38]). Some other examples of Kleinian geometry in physics occur in the theory of heterotic N D 2 string (see [33] and [5]) for which the target space is four dimensional. The analysis will be devoted to metrics of .G2 ; 2/-type, when the vector field Y , i.e. the commutator ŒX; Y , is of light-type: g .Y; Y / D 0. The wave character of gravitational fields (3) has been checked by using covariant criteria. In the following we will shortly review the most important properties of these waves which will turn out to have spin-1. In the first part of the section the standard theory of linearized gravitational waves will be shortly described. In the second part, the theoretical reality of spin-1 gravitational waves will be discussed. 7.1 The standard linearized theory. The standard analysis of linearized theory and the issue of the polarization will be analyzed. In particular, the usual transversetraceless gauge in the linearized vacuum Einstein equations and the (usually implicit) assumptions needed to reduce to this gauge play an important role: the generality of the usual claim “the graviton has spin 2” (that, of course, is strictly related to the possibility of achieving this special gauge in any “reasonable” physical situation) is strictly related to these assumptions. Thus, it is quite useful to discuss the physical and mathematical hypothesis leading to this result to check if there are physical interesting situations in which they are not fulfilled. Here, only the case of gravitational waves propagating on flat space-time will be considered since the principal ingredients needed in the following are already present in this case. Let us consider a generic metric g D C h where is the Minkowski metric and h can be thought as a perturbation. We formally write the inverse metric as


511

P g D C k with k D n2N k.n/ and k.n/ is of order hn and k.1/ D h . Then we have for the Christoffel symbols D and

1 .h ; Ch ; h ; / 2

D

X

.n/

;

n2N .n/

where D k .n1/ and k .0/ D .1 Thus the Ricci tensor may be written as X .n/ R D R ; n

with

.n/

.n/

.n/ D ; ;

C R

X

mCm0 Dn

.m/ .m0 /ˇ .m/ .m0 /ˇ ˇ ˇ :

The harmonicity condition reads 0 D D g g D

X

.n/ :

(16)

n

Up to now, we made no hypothesis on h . The gravitational field is said to be weak (in M 0 ) if there exists a (harmonic) coordinate system and a region M 0 M of space-time in which the following conditions hold: jh j 1;

jh;˛ j 1:

(17)

In the space-time regions where the linearized theory can be applied, one can take into account only terms which are of first order in h. In particular only the term .1/ D .

h /; C .

h /;

. h /; (18) contributes to the sum (16). In this approximation the components of Ricci tensor read .1/ R

1

h ; .h; C h ; / 2

and, because of the harmonicity condition .1/ D 0, the Einstein equations reduce to the well-known wave-equations h D 0: (19) Thus, we see that the harmonicity condition has a key role in deriving equation (19). Up to now, apart from the harmonicity condition, no special assumptions either on the form or on the analytic properties of the perturbation h has been done. Then, a natural 1

In fact, is intrinsically of the first order in h.

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question arises: is the original gauge freedom of the theory completely fixed or is there a residual gauge that can simplify the form of h ? The answer is, of course, well known: the residual gauge transformations are nothing but the so called gauge transformations of the linearized theory, that is, the coordinates transformations with the following two properties: (i) they preserve the harmonicity of the coordinates system and, then, the form of equations (19); (ii) they preserve the “weak character” of the gravitational field, namely, conditions (17). It is worth to stress here that it makes sense to talk about “spin” or “polarization” of a gravitational waves only in a harmonic coordinates system: the point is that the concepts of “spin” or “polarization” to be well defined need a genuine Lorentz-invariant equation of motion and the linearized Einstein equations reduce to equation (19) only if the harmonic gauge is imposed. It is commonly believed that, with a suitable gauge transformation with the above properties (1) and (2), one can always kill the “spin-1” components of the gravitational waves. However, even if not explicitly declared, the standard textbook analysis of the polarization is performed for square integrable solutions of the wave-equation (19) but, as we will see in the following, some very interesting solutions do not belong to this class. To make this point clear, now we will briefly describe the standard analysis to kill the spin-1 components stressing the role played by the square integrability assumption. Indeed, square integrable solutions of equations (19) can be always Fourier expanded in terms of plane-wave functions with a (real) light-like vector wave k . The i standard plane-wave solutions of equations (19) are h D e e i C e e with D k x , k being the propagation direction vector fulfilling k k D 0, the harmonicity condition reduces to 1 k e D k e ; 2

(20)

while the gauge transformations of the linearized theory in this case are 0 D e C k l C k l ; e ! e

k l D 0;

(21)

l being a real vector too. It is easy to see that the symmetry group of this equation, which encodes the harmonic nature of the coordinate system, reduces to linear transformations and more precisely to Poincaré transformations [47]. This characteristic is, of course, essential for a meaningful definition of polarization of gravitational waves. In particular, it allows to show the spin contents of a gravitational perturbation. Namely, a real propagation direction vector k can always be chosen in the following form k D .1; 0; 0; 1/, so that the gravitational perturbation is propagating along the z-axis. Then it is trivial to show (see, for example, [47]) that the spin-1 components of e are eit (where i; j D x; y) while the spin-2 are eij (that is, the spin-2 components are the ones with


513

both index in the plane orthogonal to the propagation direction). Moreover, it can be easily shown that with a suitable transformation (21) the eit components can be killed. From the formal point of view this means that the system of equations for l , eit C ki l t C k t li D 0 with k D .1; 0; 0; 1/, has always a solution [20]. A different but equivalent procedure makes use of the so-called energy-momentum pseudo-tensors. 7.1.1 The energy-momentum pseudo-tensors. The definition of momentum and energy associated with a gravitational field is an intrinsically controversial problem because these quantities are connected to the space-time translation invariance, whereas the group of invariance of general relativity is much bigger. With this cautionary remark in mind, various definitions are available which apply to different physical situations (for a recent discussion see [3]). The canonical energy-momentum pseudo-tensor. A commonly accepted definition is based on the canonical energy-momentum pseudo-tensor [20]: @L L; @ g˛ˇ g @.@ g˛ˇ /

(22)

L D g

(23)

D where

is the Ricci scalar deprived of terms containing the second derivative of the metric. Owing to not being a real tensor, we do not get, in general, a clear result independent of the coordinate system. But there is one special case in which we do get a clear result; namely when the waves are all moving in the same direction. If the h are assumed to be square integrable we may consider the general case in which they are all functions of the single variable r D k x , the k ’s being real constants satisfying k k k k D 0. We then have h; D u k ; where u D @r h . The harmonic condition .1/ D 0 gives

u k D

1 uk ; 2

(24)

with u D u D u . It is easily seen that the action density L, defined by expression (23), vanishes on account of harmonic coordinates (24) and condition and k k D 0. There is a

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corresponding result for the electromagnetic field, for which the action density also vanishes in the case of waves moving only in one direction. It is easy to show that

1 1 D u˛ˇ u˛ˇ u2 k k : 2 2

We have the result that transforms like a tensor under those transformations that preserve the character of the field consisting only of waves moving in the direction k so that h remain functions of the single variable r D k x . To understand the physical significance of the above expression, let us go back to the case of waves moving in the direction x 3 , so that k0 D 1, k1 D 0, k2 D 0, k3 D 1; by using the harmonic conditions we get 00 D

1 .u11 u22 /2 C .u12 / 2; 4

03 D 00 :

We see that the energy density is positive definite and the energy flows in the direction x 3 with the velocity of light. To discuss the polarization of waves, we introduce the infinitesimal rotation operator R in the .x 1 ; x 2 )-plane. Applied to u˛ˇ , it has the effect Ru11 D 2u12 ; Ru12 D u22 u11 ; Ru22 D 2u12 : Thus, R .u11 C u22 / D 0

and

R .u11 u22 / D 4u12 :

Since u11 Cu22 is invariant, while i R has eigenvalues ˙2 when applied to u11 u22 or u12 , the components of u˛ˇ that contribute to the energy correspond to spin-2. Then, for the wave solutions of the linearised Einstein equations the energy density 00 is expressed [20] as the sum of squares of derivatives of some metric components which do represent the physical degrees of freedom of the metric. Under a transformation preserving the propagation direction and the harmonic character of the coordinates system, in particular a rotation in the .x 1 ; x 2 /-plane, the physical components of the metric transform like a spin-2 field. It is well known that in general in equation (22) is not a tensor field but it does transform as a tensor field under those transformations which preserve the character of the field of consisting only of waves moving in the z direction, so that the g remain functions of the single variable x 3 x 4 . Thus, within the linearised theory, the canonical energy-momentum pseudo-tensor is a good tool to study the physical properties of gravitational waves. Thus, it has been shown that, for square integrable perturbations, one can always choose the transverse-traceless gauge. In other words, it is always possible to eliminate,


515

with a suitable gauge transformation, the components of the perturbations with one index in the propagation direction [47], i.e. square integrable perturbations of spin-1 do not exist. However, for complex k such that k k D 0 (for example .1; a; ˙i a; 1/ with a 2 R), this is not true anymore, that is, the above system of equations has in general no solution as can be easily checked. Thus, in this case, the spin-1 components are not pure gauge. A complex k could look quite strange; however this simply corresponds to split the four-dimensional d’Alembert operator .4/ in a two-dimensional d’Alembert operator in the .x 3 ; x 4 /-plane plus a Laplace operator .2/ in the .x 1 ; x 2 /-plane: .4/

D .2/ C .2/ :

In other words, we are thinking solutions of the four-dimensional d’Alembert equation as products of solutions of the two-dimensional d’Alembert equation and of the twodimensional Laplace equation. What is lacking in this case is, obviously, the square integrability of such solutions due to the presence of the harmonic function solution of the two-dimensional Laplace equation. To be more precise, the global square integrability is lacking, but there exist singular solutions of this form which far away from the singularities are perfectly well-behaved. Now the question is, can these solutions be considered “unphysical”? Put in another way, can reasonable sources be found to smooth out the singularities? The answer is positive as we will see in more detail in the next sections. Now we will show a simple and interesting example of such solutions. Comparison with the linearised theory. The exact gravitational field g D dx 2 C dy 2 C dz 2 dt 2 C d.w/d.ln jz t j/;

(25)

given by equation (5) for D 1, f D 1=2 , has the physically interesting form of a perturbed Minkowski metric with h D dwd ln jz t j . Moreover, besides being an exact solution of the Einstein equations, it is a solution of the linearised Einstein equations on a flat background too: (

@ @ h D 0; (26)

.2h ; h; / D 0: Then, to study its energy and polarization, the standard tools of the linearised theory and in particular the canonical energy-momentum pseudo-tensor, could be used. Nevertheless, with h D d.w/d.ln jz tj/ the 00 component of the canonical energymomentum tensor vanishes. This is due to the fact that the components of the tensor h cannot be expressed in the transverse-traceless gauge since h has only one index in the plane transversal to the propagation direction. It has been shown that, for square integrable perturbations, one can always choose the transverse-traceless gauge. For these reasons, the canonical energy-momentum pseudo-tensor, which is gauge invariant in the sense of the linearised gravity, “cannot see” the energy and momentum of gravitational fields given by equation (25) which

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have one index in the propagation direction2 (but they are not square integrable so they cannot be gauged away). The fact that square integrable perturbations with one index in the propagation direction are always pure gauge is equivalent to the fact that, for such perturbations, the canonical energy-momentum pseudo-tensor identically vanishes. Landau–Lifshitz’s and Bel’s energy-momentum pseudo-tensors. Besides the canonical energy-momentum pseudo-tensor, a deep physical significance can be given to the Landau–Lifshitz energy-momentum pseudo-tensor [26] defined by D

1 ˚ /.g g g g / .2 16k

C g g . C / C g g . C /

C g g . / :

(27)

There are strong evidences that, in some cases, it gives the correct definition of energy [36]. In fact, the energy flux radiated at infinity for an asymptotically flat spacetime, evaluated with the Landau–Lifshitz energy-momentum pseudo-tensor, has been seen to agree with the Bondi flux [9] that is with the energy flux evaluated in the exact theory. It is easy to check that the components p 0 of the 4-momentum density are (

p0 D

4 .tz/2 2

C1 .w;xx /2 C C2 .w;xy /2 C

p 1 D p D 0;

p3 D p0;

4 C3 r .tz/4

jrwj2 rw ;

where Ci are some positive numerical constants, r D .@x ; @y / and the harmonicity condition for w has been used. The use of the Bel’s superenergy tensor [6] T ˛ˇ D

1 ˛ ˇ ˛ ˇ R C R R ; R 2

where the symbol denotes the volume dual, leads to the same result. Indeed, in adapted coordinates the metric has the form g D dx 2 C dy 2 C .w .x; y/ 2q/dp 2 C 2dpdq and the only non-vanishing independent components of the covariant Riemann tensor R˛ˇı D g˛ R ˇı are Rxpxp D w;xx I

Rxpyp D w;xy I

Rypyp D w;yy :

Of course, it is possible to find a coordinate system in which the perturbation h has non-vanishing components only in the transverse plane. However such a coordinate system will be not harmonic. 2


517

It follows that the density energy represented by the Bel’s scalar W D T˛ˇ U ˛ U ˇ U U ; the U ˛ ’s denoting the components of a time-like unit vector field, depends on the squares of w;ij . Thus, both the Landau–Lifshitz pseudo tensor and the Bel superenergy tensor single out the same physical degrees of freedom. In particular, we can take the components h tx and h ty as fundamental degrees of freedom for the gravitational wave (25). Concerning the definition of the polarization, the above form for 0 is particularly appealing because, apart from a physically irrelevant total derivative that does not contribute to the total energy flux, the component 00 representing the energy density is expressed as the sum of square amplitudes. The momentum p i D 0i is non-vanishing only in the z-direction and it is proportional to the energy with proportionality constant c D 1; that is these waves move with light velocity along the z-axis. Moreover, this result is perfectly consistent with the one obtained with the Pirani criterion. Spin. The definition of spin or polarization for a theory, such as general relativity, which is non-linear and possesses a much bigger invariance than just the Poincaré one, deserves a careful analysis. It is well known that the concept of particle together with its degrees of freedom like the spin may be only introduced for linear theories (for example for the Yang– Mills theories, which are non-linear, it is necessary to perform a perturbative expansion around the linearized theory). In these theories, when Poincaré invariant, the particles are classified in terms of the eigenvalues of two Casimir operators of the Poincaré group, P 2 and W 2 , where P are the translation generators and W D 12 P M is the Pauli–Ljubanski polarization vector with M the Lorentz generators. Then, the total angular momentum J D L C S is defined in terms of the generators M as J i D 12 0ij k Mj k . The generators P and M span the Poincaré algebra O.3; 1/:

‚ M

; M

D i. M M M C M /; M ; P D i. P P /; P ; P D 0:

(28)

Let us briefly recall a few details about the representation theory of this algebra. The Pauli–Ljubanski operator is a translational invariant Lorentz vector, that is ŒP ; W D 0, M ; W D i. W W /. In addition it satisfies the equation W P D 0:

(29)

The unitary (infinite-dimensional) representations of the Poincaré group fall mainly into three different classes: • P 2 D m2 > 0, W 2 D m2 s.s C 1/, where s D 0; 12 ; 1; : : : denotes the spin. From equation (29) we deduce that in the rest frame the zero component

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of the Pauli–Ljubanski vector vanishes and its space components are given by Wi D 12 i0j k P 0 S j k so that W 2 D m2 S 2 . This representation is labelled by the mass m and the spin s. • P 2 D 0, W 2 D 0. In this case W and P are linearly dependent: W D P I the constant of proportionality is called helicity and it is equal to ˙s . The time component of W is W 0 D PE JE, so that PE JE P0 which is the definition of helicity for massless particles like photons. • P 2 D 0, W 2 D 2 , where is a continuous parameter. This type of representation, which describes particles with zero rest mass and an infinite number of polarization states labeled by , does not seem to be realized in nature. D

As it has been shown, the gravitational fields (25) represent gravitational waves moving at the velocity of light, that is, in the would be quantised theory, particles with zero rest mass. Thus, if a classification in terms of Poincaré group invariants could be performed, these waves would belong to the class of unitary (infinite-dimensional) representations of the Poincaré group characterized by P 2 D 0, W 2 D 0. But, in order for such a classification to be meaningful, P 2 and W 2 have to be invariants of the theory. This is not the case for general relativity, unless we restrict to a subset of transformations selected for example by some physical criterion or by experimental constraints. For the solutions of the linearised vacuum Einstein equations the choice of the harmonic gauge does the job [47]. There, the residual gauge freedom corresponds to the sole Lorentz transformations. For these reasons, only gravitational fields represented by equation (25) will be considered, which, besides being exact solutions, solve the linearised vacuum Einstein equations as well. There exist several equivalent procedures to evaluate their polarization. For instance, one can look at the 00 component of the Landau–Lifshitz pseudo-tensor and see how the metric components that appear in 00 transform under an infinitesimal rotation R in the .x; y)-plane transverse3 to the propagation direction4 . The physical components of the metric are h tx and h ty and under the infinitesimal rotation R in the .x; y)-plane transform as a vector. Applied to any vector .v1 ; v2 / the infinitesimal rotation R, has the effect Rv1 D v2 ;

Rv2 D v1 ;

from which

R2 vi D vi ; so that iR has the eigenvalues ˙1.

i D 1; 2;

3 With respect to the Minkowskian background metric, this plane is orthogonal to the propagation direction. With respect to the full metric this plane is transversal to the propagation direction and orthogonal only in the limit jz t j 7! 1. 4 It has been said before that this transformation preserves the harmonicity condition.


519

Thus, the components of h that contribute to the energy correspond to spin-1 fields, provided that only Lorentz transformations are allowed. Eventually, we can say that gravitational fields (25) represent spin-1 gravitational waves and that the reason why it is commonly believed that spin-1 gravitational waves do not exist is that, in dealing with the linearised Einstein theory, all authors implicitly assume a square integrable perturbation. In other words, square integrable spin-1 gravitational waves are always pure gauge. However, as we will now see in more detail, there exist interesting non square integrable wave-like solutions of linearised (as well as of exact) Einstein equations that have spin-1. These solutions are very interesting at least for two reasons. Firstly, they are asymptotically flat (with at least a ı-like singularity) in the plane orthogonal to the propagation direction. Secondly, they are solutions of the exact equations too, so that the spin-1 cannot be considered as an “artifact” of the linearised theory. In the next section realistic sources able to smooth out the mentioned singularities will be analyzed. 7.2 A simple example. Thus, more generally, it is natural to consider perturbations of the form h D dw.x; y/ df .z t / which are not square integrable and cannot be Fourier expanded. Nevertheless, in the next section it will be shown that the metric g D C dw.x; y/ df .u/;

u D z t;

@2x C @y2 w D 0

(30)

with f an arbitrary function, is asymptotically flat for a wide choice of harmonic functions w; thus, it represents a physically interesting gravitational field which, besides to be a solution of the linearized Einstein equations on flat background, it is also an exact solution of Einstein equations too. It is trivial to verify that the above perturbation h is written in harmonic coordinates and moreover has an off-diagonal form, that is, this perturbation has only one index in the .x; y/-plane orthogonal to the propagation direction z: for this reason the above gravitational wave has spin equal to 1 and obviously is not a pure gauge [11], that is, its Riemann tensor does not vanish. One could think that with a gauge transformation it is possible to bring the above gravitational wave in the standard transverse-traceless form. Indeed, it is possible [43] to find a diffeomorphism which gives to the metric (30) the standard transverse-traceless form but one can check that the new coordinates are not harmonic anymore. 7.2.1 Asymptotic flatness. From the physical point of view, it is important to understand under which conditions metrics (30) are asymptotically flat. In the vacuum case, the coordinates .x; y; z; t / are harmonic. Being z the propagation direction, the physical effects manifest themselves in the .x; y/-planes orthogonal to the propagation direction. This suggests to call (spatially) asymptotically flat a metric approaching, for x 2 C y 2 ! 1, the Minkowski metric. This intuitive definition of asymptotic flatness allows to obtain qualitative results by using the standard theory of partial differential equations.

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In terms of the function w the asymptotic flatness condition reads lim

x 2 Cy 2 !1

.w c1 x c2 y c3 / D 0;

where c1 ; c2 and c3 are suitable arbitrary constants and the behaviour of w can be easily recognized by looking at the Riemann tensor of the metrics (30): Ruiuj D f;u w;ij

(31)

which depends on the second derivatives of the two-dimensional harmonic function w. Therefore, to have an asymptotically flat metric, the function w must be asymptotically close to a linear functions. But, due to standard results in the theory of linear partial differential equations, this is impossible unless w is a linear function everywhere and this would imply the flatness of the metrics (30). However, if we admit ı-like singularities in the .x; y)-planes, non-trivial spatially asymptotically flat vacuum solutions with w ¤ const can exist [10]. Of course, it is not necessary to consider ı-like singularities: it is enough to take into account matter sources. For example, in the presence of an electromagnetic wave propagating along the z axis with energy density equal to , the exact non vacuum Einstein equations for metrics (30) read (see, for example, [10]) f;u @2x C @y2 w D ; where is the gravitational coupling constant. Thus, one can have non-singular spin-1 gravitational waves by considering suitable matter sources which smooth out the singularities: in the above case one can, for example, consider an energy density which vanishes outside a compact region of the .x; y)-planes. From the phenomenological point of view, it is worth to note that these kind of wave-like gravitational fields, unlike standard spin-2 gravitational waves which can be singularities free even in the vacuum case, have to be coupled to matter sources in order to represent reasonable gravitational fields. The observational consequence of this fact is that spin-1 gravitational waves are naturally less strong than spin-2 gravitational waves: typically, if the characteristic velocity of the matter source is v, the spin-1 wave is suppressed by factors .v=c/n with respect to a spin-2 wave. 7.2.2 Wave character of the field. Up to now, it has been assumed that metrics (30) indeed represent wave-like gravitational fields. Even if from a “linearized” perspective this is obvious, being the above metrics solutions of the exact Einstein equations too, one should try to use covariant criteria in order to establish their wave character. Here the following gravitational fields, a little more general than the ones expressed by equation (4), will be considered: g D dx 2 C dy 2 C 2dudv C 2.';x dx C ';y dy/du;

(32)

where the vacuum Einstein equations and the harmonicity conditions read, respectively, 2 @x C @y2 ' D 0: (33) @u @2x C @y2 ' D 0;


521

The wave character and the polarization of these gravitational fields can be analyzed in many ways. For example, we could use the Zel’manov criterion [48] to show that these are gravitational waves and the Landau–Lifshitz pseudo-tensor to find the propagation direction of the waves [11], [12]. However, the algebraic Pirani’s criterion is easier to handle since it determines the wave character of the solutions and the propagation direction both at once. Moreover, it has been shown that, in the vacuum case, the two methods agree [12]. To use this criterion the Weyl scalars must be evaluated according to the Petrov–Penrose classification [37], [34] (see Appendix). To perform the Petrov–Penrose classification, one has to choose a tetrad basis with two real null vector fields and two real spacelike (or two complex null) vector fields. Then, according to the Pirani’s criterion, if the metric belongs to type N [15], [48] of the Petrov classification, it is a gravitational wave propagating along one of the two real null vector fields. Since @u and @v are null real vector fields, and @x and @y are spacelike real vector fields, the above set of coordinates is the right one to apply for the Pirani’s criterion. Since the only non-vanishing components of the Riemann tensor, corresponding to the metric (32), are Riuj u D @2ij @u ';

i; j D x; y;

this gravitational fields belong to Petrov type N. Then, according to the Pirani’s criterion, the metric (32) does indeed represent a gravitational wave propagating along the null vector field @u .

Appendix: The Petrov classification In the study of the electrodynamics, the algebraic properties of the Maxwell and Faraday tensors play a key role. In fact, by looking at the eigenvalues and eigenvectors of the Maxwell stress-energy tensor, it is possible to understand if a given electromagnetic field is of wave type, Coulomb type, magnetostatic type, etc. Thus, since many properties of electrodynamics are shared by general relativity, it is a useful job to try to characterize the different algebraic types of the Riemann tensor. In the fifties, Petrov was able to classify the different algebraic types of the Riemann tensor of Lorentzian manifold. This classification is a good tool to understand the physical nature of a given gravitational field. Of course, the situation in general relativity is by far more complicated then in electrodynamics because in general relativity there is not a natural stress-energy tensor for the gravitational field. Nevertheless, the Petrov classification allows us to find reasonable intrinsic definitions of wave-like gravitational field that do not refer to the linearized theory. Moreover, the black-hole solutions also fit very well in this picture. In the following, we will adopt the Debever–Sachs formulation of the Petrov classification because, in the physical problems we will be interested in, it is easier to handle.

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Let us consider a Lorentzian manifold .M; gab / whose Weyl tensor is Cabcd : 1 Cabcd D Rabcd .gca Rbd gad Rbc C gbd Rac gbc Rad / 2 1 C R.gca gbd gad gbc /: 6 Then, when the vacuum Einstein equations are fulfilled, Cabcd D Rabcd . It is easy to see that the Weyl tensor has the same symmetries of the Riemann tensor and, moreover the following identity holds: g bd Cabcd D 0: Let l a be a null vector field: gca l c l a D 0. Let us consider the following equation for l a : lŒh C abcŒd l f l b l c D 0; (34) where square brackets mean antisymmetrization. It can be shown that, for general .M; gab /, there are at least one and at most four different solutions of the equation (34): a l.i/ , i D 1; : : : ; 4. The type I of the Petrov classification is characterized by the fact that all the four a solutions l.i/ of equation (34) are different; for type D they are identical in pairs (so there are two independent vectors); for type II there are three independent vectors (two out of four are identical); for type III there are two independent vectors (three out of four are identical); lastly, type N is characterized by the fact that all four vector are identical. In the following, these vector fields will be called Debever vectors. The mutual orientation of the Debever vectors is determined by equation (34). The general form of this equation characterizes the most general orientation of such vectors, that is, type I. For the other types equation (34) transforms into more stringent relations: N H) Cabcd l a D 0; III H) CabcŒd l c l g D 0;

(35) (36)

II; D H) CabcŒd l g l b l c D 0;

(37)

b c

I H) lŒh C abcŒd l f l l D 0:

(38)

It is easy to show that if a vector satisfies any one of the equations (35), (36), (37), (38) then automatically it will satisfy all the following as well. Thus, a gravitational field belongs to a given Petrov type if the Debever vector(s) satisfies the related equation and none of the preceding ones. Newman–Penrose formalism. Now we will give a brief description of the Newman– Penrose formalism and its relations with the Petrov classification. The physical idea of this formalism is to shed light in a direct way on the casual structure of the spacetime encoded in the null-cones. The Newman–Penrose formalism


523

is a tetrad formalism with a suitable choice of the basis vectors. This basis is made .a/ of four null vectors e (where a D 1; : : : ; 4 is a tetrad index, i.e. it counts the basis vectors): l, n, m and m z of which l and n are real, and m and m z are complex conjugates. The orthogonality conditions are l mDl m z DnmDnm z D 0: Usually, one also imposes on this null basis the following normalization conditions: l m D 1; mm z D 1: e.b/ has the following structure: Thus the tetrad metric .a/.b/ D e.a/

0

.a/.b/

0 B1 DB @0 0

1 1 0 0 0 0 0C C; 0 0 1A 0 1 0

with the correspondence e .1/ D l; e .2/ D n;

e .3/ D m;

e .4/ D m: z

(39)

The Weyl scalars. In the Newman–Penrose formalism the ten (in four spacetime dimension) independent components of the Weyl tensor are represented by the five complex Weyl scalars: ‰0 ‰1 ‰2 ‰3 ‰4

D C l m l m ; D C l n l m ; D C l m m z n ; D C l n m z n ; D C n m z n m z :

(40) (41) (42) (43) (44)

For a recent discussion see [31]. The Petrov classification and the Newman–Penrose formalism. Petrov worked out his classification before the formulation of the Newman–Penrose formalism. However, a very clear formulation of the Petrov classification is given by using the Newman– Penrose formalism and the Goldberg–Sachs theorem[32]. Now we will simply state the final results describing the various Petrov types, referring for details to [15] and [48]. Type I: this type is characterized by the vanishing of ‰0 and ‰4 . More exactly, ‰0 and ‰4 can be made to vanish with a rotation of the null tetrad basis that does not change the other scalars.

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Type II: this type is characterized by the vanishing of ‰0 , ‰1 and ‰4 . More exactly, ‰0 , ‰1 and ‰4 can be made to vanish with a rotation of the null tetrad basis that does not change the other scalars. Thus only ‰2 and ‰3 will be left non-vanishing. Type D: this type is characterized by the vanishing of ‰0 , ‰1 , ‰3 and ‰4 . Thus ‰2 is the only non-vanishing Weyl scalar for the type D. Type III: this type is characterized by the vanishing of ‰0 , ‰1 , ‰2 and ‰4 . Thus ‰3 is the only non-vanishing Weyl scalar for the type III. Type N: this type is characterized by the vanishing of ‰0 , ‰1 , ‰2 and ‰3 . Thus ‰4 is the only non-vanishing Weyl scalar for the type N. A very interesting fact (namely a corollary of the Goldberg–Sachs theorem) is that all the black-hole solutions of general relativity are of type D (see, for example, [15]). Thus, since the black-hole solutions represent gravitational fields of isolated body, we could say that the type D represents Coulomb-like gravitational fields. Acknowledgment. The results exposed here have been obtained in collaboration with F. Canfora, G. Sparano, A. Vinogradov and P. Vitale. The author wishes to thank Professors Dimitri Alekseevsky and Helga Baum for the kind invitation to participate in the ESI-Program Geometry of Pseudo-Riemannian Manifolds with Applications in Physics and for interesting discussions. The author is grateful to the Erwin Schrödinger International Institute for Mathematical Physics in Vienna for hospitality during the writing of this article.

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[28] Law, P. R., Neutral Einstein metrics in four dimensions. J. Math. Phys. 32 (1991), 3039–3042. 499 [29] Matsushita,Y., On Euler characteristics of compact Einstein 4-manifolds of metric signature .C C /. J. Math. Phys. 22 (1981), 979–982. 499 [30] Matsushita, Y., Thorpe-Hitchin inequality for compact Einstein 4-manifolds of metric signature .C C / and the generalized Hirzebruch index formula. J. Math. Phys. 24 (1983), 36–40. 499 [31] Nerozzi,A., Beetle, C., Bruni, M., Burko, L. M., and Pollney, D., Towards a Wave-Extraction Method for Numerical Relativity: II. The quasi-Kinnersley Frame. Phys. Rev. D 72, (2005) 024014. 523 [32] Newman, E., and Penrose, R. J., An Approach to Gravitational Radiation by a Method of Spin Coefficients. J. Math. Phys. 3 (1962), 566–578; Errata, ibid. 4 (1963), 998. 523 [33] Ooguri, H., and Vafa, C., N D 2 heterotic strings. Nucl. Phys. B 367 (1991), 83–104. 510 [34] Penrose, R. J., A spinor approach to general relativity. Ann. Physics 10 (1960), 171–201. 521 [35] Peres, A., Some Gravitational Waves. Phys. Rev. Lett. 3 (1959), 571–572. 500 [36] Persides, S., and Papadopoulos, D., A covariant formulation of the Landau-Lifchitz complex. Gen. Relativity Gravitation 11 (1979), 233–243. 516 [37] Petrov, A. Z., Einstein spaces. Pergamon Press, New York 1969. 496, 521 [38] Sakharov, A. D., Cosmological transitions with changes in the signature of the metric. Soviet Physics JETP 60 (1984), 214–218. 510 [39] Sparano, G., and Vilasi, G., Noncommutative integrability and recursion operators. J. Geom. Phys. 36 (2000), 270–284. 499 [40] Sparano, G., Vilasi, G., and Vinogradov A. M., Gravitational fields with a non-Abelian, bidimensional Lie algebra of symmetries. Phys. Lett. B 513, (2001) 142–146. 496, 497, 498 [41] Sparano, G., Vilasi, G., and Vinogradov A. M., Vacuum Einstein metrics with bidimensional Killing leaves. I. Local aspects. Differential Geom. Appl. 16 (2002), 95–120. 497, 498, 499, 504, 506 [42] Sparano, G., Vilasi, G., and Vinogradov A. M., Vacuum Einstein metrics with bidimensional Killing leaves. II. Global aspects. Diff. Geom. Appl. 17 (2002), 15–35. 496, 497, 499, 504, 505, 506 [43] Stephani, H., General Relativity. Cambridge University Press, Cambridge 1990. 519 [44] Stephani, H., Kramer, D., MacCallum, M., Honselaers C., and Herlt E., Exact solutions of Einstein field equations. 2nd., Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge 2003. 496 [45] Verdaguer, E., Soliton solutions in spacetimes with two spacelike killing fields. Phys. Rep. 229 (1–2) (1993), 3–80. 496 [46] Wald, R. M., General Relativity. University of Chicago Press, Chicago 1984. 507 [47] Weinberg, S., Gravitation and Cosmology. J. Wiley & Sons, N.Y. 1972. 512, 515, 518 [48] Zakharov, V. D., Gravitational waves in Einstein’s theory. Israel Program for Scientific Translations, Jerusalem and London; Halsted Press, New York 1973. 521, 523

List of Contributors Thierry Barbot, CNRS, UMR 5669, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France; e-mail: [email protected] Anna Maria Candela, Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy; e-mail: [email protected] Virginie Charette, Département de mathématiques, Université de Sherbrooke, Sherbrooke, Quebec, Canada; e-mail: [email protected] Andrew Dancer, Jesus College, Oxford OX1 3DW, United Kingdom; e-mail: dancer@ maths.ox.ac.uk Todd A. Drumm Department of Mathematics, University of Pennsylvania, Philadelphia, PA U.S.A.; e-mail: [email protected] Maciej Dunajski, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom; e-mail: [email protected] José Figueroa-O’Farrill, School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom; e-mail: [email protected] Charles Frances, Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay, France; e-mail: [email protected] Anton Galaev, Ústav matematiky a statistiky, Masarykova Univerzita, Janáˇckovo nám. 2a, 60200 Brno, Czech Republic; e-mail: [email protected] William M. Goldman, Department of Mathematics, University of Maryland, College Park, MD 20742 U.S.A.; e-mail: [email protected] Brendan Guilfoyle, Department of Computing and Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland; e-mail: [email protected] Ines Kath, Institut für Mathematik und Informatik, Jahnstraße 15a, 17487 Greifswald, Germany; e-mail: [email protected] Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom; e-mail: [email protected] Wolfgang Kühnel, Universität Stuttgart, Institut für Geometrie und Topologie, 70550 Stuttgart, Germany; e-mail: [email protected] Thomas Leistner, Department Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany; e-mail: [email protected] Karin Melnick, Department of Mathematics, Yale University, New Haven, CT 06520 U.S.A.; e-mail: [email protected]

528

List of Contributors

Ettore Minguzzi, Department of Applied Mathematics, Florence University, Via S. Marta 3, 50139 Florence, Italy; e-mail: [email protected] Martin Olbrich, University of Luxembourg, Mathematics Laboratory, 162 A, avenue de la Faïencerie, 1511 Luxembourg, Grand-Duchy of Luxembourg; e-mail: martin. [email protected] Hans-Bert Rademacher, Universität Leipzig, Mathematisches Institut, 04081 Leipzig, Germany; e-mail: [email protected] Miguel Sánchez, Departamento de Geometría y Topología Facultad de Ciencias, Avda. Fuentenueva s/n, 18071 Granada, Spain; e-mail: [email protected] Andrew Swann, Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark; e-mail: swann@imada. sdu.dk Gaetano Vilasi, Dipartimento di Fisica, Università di Salerno, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, GCSalerno, Italy; e-mail: [email protected] Simon West, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom; e-mail: [email protected] Frederik Witt, NWF I - Mathematik, Universität Regensburg, 93040 Regensburg, Germany; e-mail: [email protected]

Index acausal subset, 341 achronal subset, 341 action discontinuous, 373 free, 373 proper, 233 properly discontinuous, 373 adapted coordinates, 498 admissible measures, 328 affine lightcone, 184 Alekseevskii’s theorem, 269 Alexandrov’s topology, 325 ˛-plane, 118 ˛-surface, 120 Ambrose–Singer holonomy theorem, 57 angular momentum, 156 anti-de Sitter space, 186, 270, 396, 421, 422 anti-holomorphic involution, 132 anti-self-dual conformal structure, 114 form, 113 anti-symplectic involution, 201 associated graph, 281 asymptotically flat, 519 attracting fixed point, 211 Avez–Seifert result, 348, 349 Bach equation, 275 Bach tensor, 274 Bach-flat, 275 Beem and Buseman counterexample, 382 Bel’s superenergy tensor, 516 Berger algebra, 64, 68 Berger list, 59, 97 ˇ-plane, 118 B-field transformation, 457, 458, 474, 479 Bianchi-identity, 57, 63

Bondi flux, 516 Borel–Lichnerowicz property, 59, 64 bosonic particle, 463 Boubel coordinates, 76 Brinkmann coordinates, 76 Brinkmann space, 282 Brinkmann wave, 84 Brinkmann’s theorem, 271 C-complete, 280 C-space, 273 Cahen–Wallach space, 43, 86, 421 Calabi–Yau manifold, 143 neutral, 100 canonical bundle, 135 canonical isotropic ideal, 10 Cartan connection, 237 curvature of, 237 Cartan decomposition, 212 Cartan geometries, 237 Carter’s “virtuosity”, 337 category Ljusternik–Schnirelman, 392 relative, 405 Cauchy function, 342 Cauchy horizon, 341 Cauchy hypersurface, 341, 364 Cauchy temporal function, 341 Cauchy time function, 342 causal boundary, 310 cones, 301 curve, 196, 301 diamonds, 341 future, 305 ladder, 316, 317 mapping, 351 relations, 305 spacetimes, 319 structure, 304 vector, 182, 301

530 causality theory, 316 causally continuous spacetimes, 337 causally convex neighborhood, 306 causally related points, 196, 197 causally related events, 305 causally simple spacetimes, 338 charge conjugation operator, 467 chirality, 465 chronological future, 305 chronological spacetimes, 318 chronologically related events, 305 Clifford algebra, 464–466 Clifford multiplication, 464, 469, 476, 477, 485 Clifton–Pohl torus, 378 closed curve, 316 closed timelike geodesic, 350 closed vector field, 264, 277 coercivity, 399 cohomology, 11 Lie algebra, 12 quadratic, 13 cohomology class admissible, 20 balanced, 14 combinatorial interior, 106 compact spacetime, 318, 350 compactification, 109 complete vector field, 269, 280 completeness (spacelike, lightlike and timelike), 368 complex product structure, 99 complex structure, 99, 123, 202 complex symplectic structure, 100 concircular vector field, 265 condition .C /, 391 condition of Palais–Smale or (PS), 391 conformal class of metrics, 236, 305 conformal compactification, 116, 268 conformal development, 267, 269 conformal Einstein equation, 273 conformal gradient field, 271, 277 conformal group, 282, 283

Index

conformal immersion, 266 conformal inversion, 268 Euclidean, 190 Lorentzian, 191 conformal Killing vector field, 127, 317, 371 conformal mapping, 263 conformal Ricci collineation, 286 conformal spacetimes, 304 conformal structure, 231, 304 conformal transformation infinitesimal, 263 conformal vector field, 263 conformally closed vector field, 265 conformally Einstein metric, 272 conformally flat, 240, 265–267, 280 conjugate point, 363 multiplicity, 363 conjugated events, 312 connecting curve, 305 connection, 462, 463 contact 1-form, 204 plane field, 204 structure, 204 convex cone, 105 polyhedron, 106 set, 105 neighborhood, 306 coordinates adapted, 498 Boubel, 76 Brinkmann, 76 Rosen, 271 Schimming, 76 semiadapted, 498 Walker, 76 Cotton tensor, 240, 266 covariant derivative, 462, 463 crooked plane, 217 crooked surface, 219

Index

curvature Gauss, 498 Ricci, 265 scalar, 265 Weyl, 265 curvature endomorphism, 63, 64 curvature tensor, 265, 279, 313, 363 curve imprisoned, 369 length, 363 partially imprisoned, 369 cut hypersymplectic, 108 symplectic, 108 cut point, 365 de Rham–Wu decomposition, 2, 59, 60 de Sitter space, 270, 388, 421, 422 developable manifold, 268 developping map, 240 dilatation, 263 dilatino equation, 456, 470, 472, 490, 491 dilaton field, 456, 458, 459, 470, 472, 486 2-dimensional spacetime, 314, 335 Dirac current, 89, 290 Dirac operator, 288, 469, 472, 488, 491 Dirac spinor, 464 discrete subgroup of the first kind, 215 of the second kind, 216 distance, 393 distinguishes p in U (future, past), 320, 321 distinguishing spacetime, 320 distinguishing subsequence (limit of curves), 327 distribution generated by a Killing field, 495, 496 divergence, 263 dKP equation, 130 domain of dependence, 341 double fibration, 134

531

dynamical quadruple, 211 Einstein hypersphere, 186, 202 Einstein manifold, 90 Einstein metric, 35 Einstein space, 270, 283 Einstein’s universe, 243 Einstein–Weyl equations, 128 Einstein–Weyl structure, 101 Ejiri’s example, 270 endpoint future, past, 305 energy-momentum pseudo-tensors, 513 equivariant representation, 460 equivariant map, 101, 460 essential conformal transformation, 232 essential conformal vector field, 266, 269 essential structure, 232 Euler characteristic, 302, 335 event, 303 exponential map, 363 extension T -, 7 admissible quadratic, 20 balanced quadratic, 11 canonical quadratic, 10 double, 5 quadratic, 3, 9 extremal metric, 276 fermionic particle, 463 fibration canonical, 24, 27, 34, 40 special affine, 22 fierzing, 467, 477 fine topology on the metric space, 334 first prolongation, 69, 70 flat space, 266 focal set, 164 frame bundle, 99 free edge, 281 Frobenius distribution, 498 Frobenius-integrable, 495 functional action, 361, 391, 392, 404

532 functional energy, 361, 395 future causal, of a point, 196 -conjugate point, 198 lightcone, 195 -oriented causal curve, 196 -oriented tangent vector, 195 -oriented vector, 194 future-directed vector, 302 future reflecting spacetimes, 332 future set, 308 gauge transformations, 512 Gauss curvature, 498 Gauss Lemma, 363 generalized G2 -structure, 479, 486, 492 generalised monopole equation, 128 generalized Spin.7/-structure, 479, 486, 492 generalized SU.m/-structure, 478, 479 generalized tangent bundle, 458, 480, 481, 483 generalized time function, 333 geodesic, 363 causal character, 363 closed, 374 co-spacelike, 364 extremizing properties, 364 lightlike, 244, 312 maximizing causal, 364 minimizing Riemannian, 364 periodic, 374 geodesic connectedness ortogonal splitting spacetime, 407 Riemannian manifold, 395 static spacetime, 399 stationary spacetime, 400, 402 geodesic flow, 252, 499 geodesic ray, 327 geodesically complete, 504 geographic coordinates, 499 Geroch’s Cauchy time function, 344 globally hyperbolic neighborhood, 307 globally hyperbolic spacetimes, 340

Index

Goldberg–Sachs theorem, 523 gradient field, conformal, 277 grandfather’s paradox, 316, 341 gravitino equation, 456, 470–472 group action, 101 discontinuous, 373 free, 102, 106, 373 proper, 102, 233 properly discontinuous, 373 group of affine maps, 374 isometries, 373 rigid motions, 374 G-structure, 460 linear, 460 G2 -structure, 479, 482 harmonic gauge, 518 harmonic polynomial, 284 harmonic Weyl tensor, 273 harmonicity condition, 511 Hausdorff (topology), 326 Heavenly equations, 127 heterotic supergravity, 456, 457, 459, 469–472 hierarchy of spacetimes, 316 holomorphic sections, 135 holonomy group, 1, 2, 56, 100, 433 Abelian, 37–39, 62, 86 connected, 57 non-closed, 67, 84 of a linear connection, 57 of a Lorentzian manifold, 60, 65 of a semi-Riemannian manifold, 58 of space-times, 88 solvable, 85, 86 holonomy algebra, 57 abelian, 62, 86 homothetic vector field, 263 homothety, 127, 263, 282 homothety group, 283 homotopy classes causal, 366 timelike, 365

Index

horismos, 305 horismotically related events, 305, 313 horocyclic flow, 252 hyper-CR equation, 131 hyper-heavenly metric, 125 hyperkähler manifold, 30–35, 40, 97 modification, 108 hypersymplectic cut, 108 manifold, 30–35, 40, 97, 99 moment map, 101 quotient, 101 structure, 30, 35–40, 97, 126 ideal sphere, 192 imaginary Killing spinor, 289 imprisoned curve, 326 improper point, 191 spatial, 199 timelike, 199 indecomposable, 58 index form, 364 quotient, 364 inessential conformal structure, 232 conformal vector field, 269 inextendible curve, 305 infinitesimal conformal transformation, 263 Info-holes, 507 inner continuity, 329 integrable generalized G-structure, 486, 490–492 integrable system, 100, 114 interval topology on the metric space, 334 invariant under conformal transformations, 315 Inverse Scattering Transform, 495 isocausal ladder, 351 isocausal spacetimes, 351 isocausal structure, 353 isometric vector field, 263

533

isometry, 263 isometry group, 283 isotropic flag, 189 Jacobi class, 314, Jacobi equation, 313, 363 Jacobi field, 313, 363 Jones–Tod construction, 128 K-relation, 309 Kähler potential, 127 Kähler structure, 100 Killing algebra, 497 Killing fibering, 504 Killing leaf, 496, 497 Killing spinor, 289, 432 Killing vector field, 102, 107, 127, 263, 317, 371, 495, 496 Kleinian metrics, 499 Kodaira deformation theory, 135 Kruskal–Szekeres type solutions, 507 Kuiper’s theorem, 268 Kulkarni–Nomizu product, 240, 265, 438 Lagrangian Grassmannian, 199, 205 plane, 200 Landau–Lifshitz energy-momentum pseudo-tensor, 516 Lax pair, 119, 496 LeBrun–Mason construction, 139 Levi-Civita connection, 58, 362, 363, 389 Lichnerowicz’s conjecture, 232, 236, 239 Lie algebra, 496 admissible, 17 double extension, 427 indecomposable, 428 Lorentzian, 427 metric, 2 orthogonal, 2 quadratic, 2 Lie bialgebra, 45 Lie derivative, 263 light-like hypersurface curvature, 86

534 lightcone, 185, 244 lightlike curve, 301 lightlike foliations, 376 lightlike geodesic, 244, 312 lightlike vector, 182, 301 line congruence, 157 Liouville theorem, 180, 244, 282, 496 Lipschitzianity of causal curves, 323 Lorentz surfaces, 354 Lorentz–Minkowski spacetime, 307 Lorentzian distance, 310 Lorentzian length, 310 Lorentzian manifold, 281, 282, 301 3362 Lorentzian metrics, 510 Lorentzian symmetric space anti de Sitter, 421 Cahen–Wallach, 26, 43, 86, 421 de Sitter, 421 isometric embedding, 422 parallelisable, 424 Lorentzian vector space, 181 lower semi-continuous (Lorentzian distance), 311 manifold affine, 56, 379 conformally flat, 376 conformally homogeneous, 372 developable, 268 extendible semi-Riemannian, 369 globally symmetric, 369 homogeneous, 372 indefinite, 362 locally symmetric, 369 Lorentzian, 281, 282, 301, 362 model space, 380 negative definite, 362 pseudo-Riemannian, 263 Riemannian, 362 semi-Riemannian, 362 smooth, 362 spaceform, 380 warped product, 382

Index

Manin pair, 46 Manin triple, 45 Markus conjecture, 379 Maslov cycle, 205 Maslov index, 205 maximizing lightlike curve, 312 metric conformally Einstein, 272 extremal, 276 neutral, 99, 156 pointwise conformal, 367 weakly generic, 274 metric Lie algebra, 2 .h; K/-equivariant, 7 Lorentzian, 6, 18 nilpotent, 18 Minkowski patch, 185 Minkowski space, 183 Misner’s cylinder, 374 Misner’s group, 375 model solution, 504 modification hyperkähler, 108 moduli space, 501 moment map, 101 hypersymplectic, 103–105 symplectic, 101 -deformed Laplace equation, 500 multiplicity of Jacobi fields, 315 naked singularities, 341 neighborhood convex, 363 normal, 306, 363 starshaped, 363 neutral Calabi–Yau, 100 metric, 99, 156 signature, 99 neutral metric, 156 Neveu–Schwarz, 472 Newman–Penrose formalism, 523 Nijenhuis torsion, 496, 502 nilindex, 35, 39

Index

nilmanifold, 100 non-degenerate structure, 102, 106 non-degnerate Weyl tensor, 275 non-imprisoning spacetime, 326 non-totally vicious spacetimes, 317 nonspacelike vector, 301 normal coordinates, 499 NS-NS field, 472 null conformal Killing vector, 131 null frame, 154 null tetrad, 117 null vector, 301 null-Kähler structures, 125 nullcone, 182 Obata’s theorem, 269 Ooguri–Vafa metrics, 144 operator Dirac, 288, 469, 472, 488, 491 Penrose, 289 twistor, 289 optical scalars, 157, 159 orientable manifold, 303 orientation, 195 orthogonal leaf, 497 orthogonal part, 62, 65, 66 orthonormal basis, 301, 363 oscillator algebra, 7, 18 outer continuity, 329 Palais–Smale condition, 391 para-quaternionic Kähler manifold, 30, 35–40, 101 parabolic subalgebra, 210 subgroup, 209, 210 parabolic geometry, 238 parabolic group, 61 parallel displacement, 57 parallel vector field, 271, 282 partially imprisoned curve, 326 past -oriented causal curve, 196 -oriented tangent vector, 195

535

-oriented vector, 194 causal, of a point, 196 lightcone, 195 of a point, 196 past reflecting spacetimes, 332 past set, 308 past-directed vector, 302 Pauli–Ljubanski polarization vector, 517, 518 Penrose operator, 289 Penrose limit, 287 periodic curve, 316 periodic timelike geodesic, 350 Petrov–Penrose classification, 119, 521 Petrov type, 285 photon, 185, 210 Pirani’s criterion, 521 plane wave, 283 Poincaré group, 517 Poincaré transformations, 512 pointed photon, 189 pointwise conformal metrics, 304 pointwise conformal spacetimes, 304 Poisson–Lie group, 45 polar decomposition, 212 polarization, 517 pp-wave, 84, 143, 283 pr-wave, 85 precompactness, 401 pregeodesic, 312, 367 primary Kodaira surface, 100 product structure, 99 projective quadric, 267 projective structure, 131 proper action, 233 pseudo-coerciveness, 401 pseudo-Euclidean space, 267 pseudo-hyper-Kähler metrics, 126 pseudo-hyperhermitian metrics, 120 pseudo-Riemannian manifold, 263 pseudo-Riemannian metric, 234

536 pseudoconvex nonspacelike geodesic system, 388 pseudosphere, 380 quasi-limit, 327 quaternions split, 98 quotient Jacobi equation, 314 quotient space of vector fields, 314 Ramond–Ramond field, 456, 472, 485 real Killing spinor, 289 real type Lie algebra, 68, 71, 74 recurrent vector field, 60 recursion operator, 496 reduction, 460 geometrical, 462, 463 topological, 462, 463 reductive geometry, 238 reflecting spacetimes, 332 reflection of congruences, 162 regular measures, 329 relative category theory, 404 repelling fixed point, 211 representation, 460 representation of real type, 70, 74 reversed triangle inequality, 311 Ricci collineation, 286 Ricci identity, 277 Ricci tensor, 265, 277, 282 Ricci-flat metric, 90, 100 Ricci-isotropic, 85 root diagram sp.4; R/, 207 root space, 71 Rosen coordinates, 271, 287 R-R field, 472 R-R sector, 472 scalar curvature, 265, 270 scalar product, 301 scalar-flat Kähler metrics, 122 Schimming coordinates, 76 Schottky groups, 248

Index

Schouten tensor, 240, 265 Schur’s Lemma, 460, 471 Schwachhöfer–Merkulov list, 58 Schwarzian tensor, 273 screen holonomy, 61, 66 screw dilations, 63 screw isometries, 63 semiadapted coordinates, 498 semi-Riemannian manifold, 362 sequence balanced distortion, 214 bounded distortion, 213 mixed distortion, 214 no-distortion, 213 unbalanced, 214 unbounded, 213 set of incomplete directions, 371 set of parts P .M /, 320 Siegel upper-half space, 203 signature Lorentz, 101 neutral, 99 split, 97 similarity transformation, 62, 282 Euclidean, 190 Minkowski, 184 simple Lie group, 257 simple neighborhood, 306 smooth quotient, 102, 106 smoothability problems, 343 Sobolev space of causal curves, 323 solid cone, 104 solid convex cone, 105 space of causal curves, 347 spaceform, 380 spacelike circle, 188 curve, 301 hypersphere, 186 vector, 182, 301 spacetime, 282, 303, 364 anti-de Sitter, 396 de Sitter, 388

Index

Generalized Robertson–Walker or GRW, 385 globally hyperbolic, 364 Lorentz–Minkowski, 363, 373 orthogonal splitting, 406 standard static, 388, 397 standard stationary, 388 static, 388 stationary, 388 spanning triple, 73 spectral parameter, 120 spin, 517 spin connection, 119 spin representation, 463, 466, 467 spin structure, 460, 463, 468, 469, 472, 482 spinor, 117 imaginary Killing, 289 Killing, 289, 432 parallel, 89, 288 real Killing, 289 twistor, 289 split quaternions, 98 split signature, 97 stability of completeness, 376 stable dynamics, 253 stable isocausality, 354 stable spacetime property, 335 stably causal spacetimes, 334 stably chronological spacetimes, 335 standard conformal vector field, 284 star “outside” the universe, 507 static spacetime, 349 stationary spacetime, 317 stem, 217 stem configuration, 187, 220, 224 stereographic projection, 245 strictly causally related events, 305 Strong Cosmic Censorship Hypothesis, 316 strong essentiality, 236 strongly causal spacetimes, 324 supergravity

537

d D6 .1; 0/ background, 435 .1; 0/ supersymmetric background, 435 .2; 0/ background, 436 .2; 0/ supersymmetric background, 436 R-symmetry groups, 435 d D10 type II background, 446 parallelisable background, 446 supersymmetric background, 447 d D10 IIB background, 434 duality group, 435 Killing spinor, 434 d D11 background, 431 Killing spinor, 432 supersymmetric background, 432 type I, 469 type II, 456, 469, 471, 472, 480, 485 type IIA, 471, 472 type IIB, 471, 472 type N, 119 supersymmetry, 100, 456 supersymmetry equations, 456–458 supersymmetry parameters, 456, 470, 471 SU.1/-Toda equation, 129 symmetric pair, 22 symmetric R-space, 42 symmetric space, 2, 19, 369, affine, 22 extrinsic, 40–45 hyper-Kähler, 30–35, 40 hypersymplectic, 30–35, 40 Lorentzian, 25, 43 non-reductive, 2 para-Hermitian, 30, 33–35 para-quaternionic Kähler, 30, 35–40 pseudo-Hermitian, 30, 32–35 pseudo-Riemannian, 2, 19–30 quaternionic Kähler, 30, 35–40

538 reductive, 2 symmetric triple, 3, 19 extrinsic, 41 symplectic basis, 206 cut, 108 geometry, 97 involution, 202 plane, 202, 209 vector space, 199 Taub-NUT metric, 107 temporal function, 333 tensor Cotton, 240, 266 harmonic Weyl, 273 Bach, 274 curvature, 265 Ricci, 240, 265 Schouten, 240, 265 Schwarzian, 273 Weyl, 240, 265 Theorem Alekseevskii, 269 Ambrose–Singer, 57 Berger, 59 Brinkmann, 271 Cartan, 369 de Rham–Wu, 59 Gauss–Bonnet, 379 Goldberg–Sachs, 523 Hawking’s singularity, 381 Hopf–Rinow, 370, 395 Kuiper, 268 Liouville, 180, 244, 282, 496 Nash, 393, 395 Obata, 269 Saddle Point, 404 Uniformization, 376 time function, 333 time orientation, 194, 302, 364 time-orientable double covering, 302 time-orientable manifold, 302, 303

Index

time-orientation, 302, 364 time-separation, 310 time-separation (or Lorentzian distance), 310, 364 timelike circle, 188 curve, 196, 301 vector, 182, 301 toric fibration, 144 toric variety, 103 tortoise equation, 498 torus, 376 totally vicious, 318 transvection group, 2, 19, 22, 42 twisted product, 384 twistor construction, 99 twistor distribution, 134 twistor operator, 289 twistor space, 99, 132 twistor spinor, 289 type I, 469 type II, 456, 469, 471, 472, 480, 485 type IIA, 471, 472 type IIB, 471, 472 type N, 119 unitary type Lie algebra, 69, 70 vacuum spacetime, 283 variational principle classical, 393 in the static case, 397 natural constraint, 401 vector causal, 362 lightlike, 362 null, 362 timelike, 362 vector field closed, 264, 277 complete, 269, 280, 388 concircular, 265 conformal, 263 conformal Killing, 127, 317, 371

Index

conformally closed, 265 essential conformal, 266, 269 homothetic, 263 isometric, 263 Killing, 102, 107, 127, 263, 371 vector representation, 463, 467 volume functions, 329

Weyl scalars, 521, 523 Weyl spinor, 465 Weyl tensor, 114, 240, 265, 522 non-degenerate, 275 wing, 217

Walker coordinates, 76 warped complete triple, 383 warped geodesic projection, 383 warped product, 277, 279 wave profile, 288 weak curvature endomorphism, 63 weak-Berger algebra, 64, 66, 68–70, 74 weakly generic metric, 274 weight space, 71 Weyl chamber, 211 Weyl group, 210

Yang–Mills theories, 517

X-ray transform, 114

Zel’manov criterion, 521 -complex numbers, 502 -complex curve, 502 -complex structure, 502 -harmonic function, 503 -holomorphic function, 502 -holomorphic map, 503 -Laplace equation, 502

539

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