Lectures on spaces of nonpositive curvature

GEOMETRIA 3

Lectures on Spaces of Nonpositive Curvature with an appendix by Misha Brin Ergodicity of Geodesic Flows

Werner Ballmann

Birkhauser Verlag Basel· Boston· Berlin

Author: Werner BaUmann Mathematisches Institut der Universitiit Bonn Wegelerstr. 10 D-53ll5 Bonn

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data BaUmann, Werner: Lectures on spaces of nonpositive curvature / Werner Ballmann. With an appendix Ergodicity of geodesic flows / by Misha Brin. - Basel; Boston; Berlin: Birkhiiuser, 1995 (DMV-Seminar; Bd. 25) ISBN 3-7643-5242-6 (Basel ... ) ISBN 0-8176-5242-6 (Boston) NE: Brin, Misha: Ergodicity of geodesic flows; Deutsche MathematikerVereinigung: DMV-Seminar

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 the permission of the copyright owner must be obtained. © 1995 Birkhiiuser Verlag, P.O. Box 133, CHAOlO Basel, Switzerland Camera-ready copy prepared by the author Printed on acid-free paper produced from chlorine-free pulp 00 Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN 3-7643-5242-6 ISBN 0-8176-5242-6

987654321

Contents Introduction

1

Chapter I. On the interior geometry of metric spaces 1. Preliminaries 2. The Hopf-Rinow Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Spaces with curvature bounded from above 4. The Hadamard-Cartan Theorem 5. Hadamard spaces

11 13 15 19 22

Chapter II. The boundary at infinity 1. Closure of X via Busemann functions 2. Closure of X via rays 3. Classification of isometrics 4. The cone at infinity and the Tits metric

27 28 31 32

Chapter III. Weak hyperbolicity 1. The duality condition 2. Geodesic flows on Hadamard spaces...... .. .. 3. The flat half plane condition 4. Harmonic functions and random walks on r

..

Chapter IV. Rank rigidity 1. Preliminaries on geodesic flows 2. Jacobi fields and curvature 3. Busemann functions and horospheres 4. Rank, regular vectors and flats 5. An invariant set at infinity 6. Proof of the rank rigidity Appendix. Ergodicity of geodesic flows By MISHA BRIN 1. Introductory remarks 2. Measure and ergodic theory preliminaries 3. Absolutely continuous foliations 4. Anosov flows and the Holder continuity of invariant distributions 5. Proof of absolute continuity and ergodicity Bibliography

43 47 50 55 61 67 70 73 76 79 81 82 85 90 93 97

Index.................................................... v

111

Introduction These notes grew out of lectures which I gave at a DMV-seminar in Blaubeuren, Bavaria. My main aim is to present a proof of the rank rigidity for manifolds of nonpositive sectional curvature and finite volume. Since my interest in the last couple of years has shifted to singular spaces of nonpositive curvature, I take the opportunity to include a short introduction into the theory of these spaces. An appendix on the ergodicity of geodesic flows has been contributed by Misha Brin. Let X be a metric space with metric d. A geodesic in X is a curve of constant speed which is locally minimizing. We say that X has nonpositive Alexandrov curvature if every point p E X has a neighborhood U with the following properties: (i) for any two points x, y E U there is a geodesic ax,y : [0,1] -+ U from x to y of length d(x, y); (ii) for any three points x, y, z E U we have

where ax,y is as in (i) and Tn = a x ,y(1/2) is in the middle between x and y. The second assumption requires that triangles in U are not fatter than the corresponding triangles in the Euclidean plane ]R2: if U is in the Euclidean plane then we have equality in (ii). It follows that the geodesic ax,y in (i) is unique. Modifying the above definition by comparing with triangles in the model surface l\J~ of constant Gauss curvature K instead of ]R2 = M;}, one obtains the definition of Alexandrov cnrvature Kx ~ K. It is possible, but not very helpful, to define K x and not only the latter inequality. Reversing the inequality in the triangle comparison, that is, requiring that (small) triangles are at least as fat as their counterparts in the model surface M~, leads to the concept of spaces with lower curvature bounds. The theory of spaces with lower curvature bounds is completely different from the theory of those with upper curvature bounds. Triangle comparisons are a standard tool in global Riemannian geometry. In the Riemannian context our requirement (ii) is equivalent to nonpositive sectional curvature and an upper bound K for the sectional curvature respectively. We say that X is a Hadamard space if X is complete and the assertions (i) and (ii) above hold for all points x, y, Z EX. This corresponds to the terminology in the

2

Introduction

Riemannian case. If X is a Hadamard space, then for any two points in X there is a unique geodesic between them. It follows that Hadamard spaces are contractible, one of the reasons for the interest in these spaces. There arc many examples of Hadamard spaces, and one of the sources is the following result, Gromov's version of the Hadamard-Cartan theorem for singular spaces.

Theorem A (Hadamard-Cartan Theorem). Let Y be a complete connected metric space of nonpositive Alexandrov curvature. Then the universal covering space X of Y, with the induced interior metric, is a Hadamard space. For X and Y as in Theorem A, let r be the group of covering transformations of the projection X ----+ ~nce X is contractible, Y is the classifying space for r. This implies for example that the homology of r is equal to the homology of Y. The contractibility of X also implies that the higher homotopy groups of Y arc trivial and that indeed the homotopy type of Y is determined by r. This is the reason why spaces of nonpositive curvature are interesting in topology. Since r acts isometrically on X, the algebraic structure of r and the homotopy type of Yare tied to the geometry of X. The action of r on X is properly discontinuous and free, but for various reasons it is also interesting to study more general actions. This is the reason that we formulate our results below for groups r of isometrics acting on Hadamard spaces instead of discussing spaces covered by Hadamard spaces. Here are several examples of Hadamard spaces and metric spaces of nonpositive and respectively bounded Alexandrov curvature. (1) Riemannian manifolds of nonpositive sectional curvature: the main examples arc symmetric spaces of noncompact type, sec [Hel, ChEb]. One particular such space is X = SI(n,ra)/SO(n) endowed with the metric which is induced by the Killing form. Every discrete (or not discrete) linear group acts on this space (where n has to be chosen appropriately). Many Riemannian manifolds of nonpositive curvature are obtained by using warped products, see [BiON]. For Riemannian manifolds with boundary there are conditions on the second fundamental form of the boundary - in addition to the bound on the sectional curvature _. which are equivalent to the condition that the Alexandrov curvature is bounded from above, see [ABB]. (2) Graphs: Let X be a graph and d an interior metric on X. Then for an arbitrary K E ra, the Alexandrov curvature of d is bounded from above by K if and only if for every vertex v of X there is a positive lower bound on the length of the edges adjacent to v. This example shows very well one of the main technical problems in the theory of metric spaces with Alexandrov curvature bounded from above - namely the possibility that geodesics branch. (3) Euclidean buildings of Bruhat and Tits: these are higher dimensional versions of homogeneous trees, and they are endowed with a natural metric (determined up to a scaling factor) with respect to which they are Hadamard spaces.

Introduction

Similarly, Tits buildings of spherical type are spaces of Alexandrov curvature :S l. See [BruT, Ti2, Bro]. (4) (p, q )-spaces with p, q ::::: 3 and 2pq ::::: p+q: by definition, a two dimensional CIV-space X is a (p, q)-space if the attaching maps of the cells of X are local homeomorphisms and if (i) every face of X has at least p edges in its boundary (when counted with multiplicity); (ii) for every vertex v of X, every simple loop in the link of v consisits of at least q edges. If X is a (p, q)-space, then the interior metric d, which turns every face of X into a regular Euclidean polygon of side length 1, has nonpositive Alexandrov curvature. In combinatorial group theory, (p, q)-spaces arise as Seifert-van Kampen diagrams or Cayley complexes of small cancellation groups, see [LySe, GeSh, BB2]. The angle measurement with respect to the Euclidean metric on the faces of X induces an interior metric on the links of the vertices of X, and the requirement 2pq ::::: p + q implies that simple loops in the links have length at least 2'if. This turns the claim that d has nonpositive Alexandrov curvature into a special case of the corresponding claim in the following example. If 2pq > p + q, then X admits a metric of Alexandrov curvature :S -l. (5) Cones: Let X be a metric space. Define the Euclidean cone C over X to be the set [0, (0) x X, where we collapse {O} x X to a point, endowed with the metric dc((a, x), (b, y)) := a 2 + b2 - 2abcos(min{ dx(x, y), 'if}) . Then C has nonpositive Alexandrov curvature if and only if X has Alexandrov curvature K x :S 1 and injectivity radius::::: 'if. By the latter we mean that for every two points x, y E X of distance < 'if there is one and only one geodesic from x to y of length d(x, y). In a similar way one can define the spherical and respectively hyperbolic cone over X. The condition on X that they have Alexandrov curvature :S 1 and respectively :S -1 remains the same as in the case of the Euclidean cone, see [Ber2, Gr5, Ba5, BrHa]. (6) Glueing: Let Xl and X 2 be complete spaces of nonpositive Alexandrov curvature, let Yl C Xl and Y2 C X 2 be closed and locally convex and let f : Yl ---> Y2 be an isometry. Le X be the disjoint union of Xl and X 2 , except that we identify Yl and Y2 with respect to f. Then X, with the induced interior metric, is a complete space of nonpositive curvature. For Xl, Yl given as above one may take X 2 = Yl X [0,1]' Y2 = Yl X {O} and f the identity, that is, the map that forgets the se~d coordinate O. One obtains a porcupine. Under the usual circumstances it is best to avoid such and similar exotic animals. One way of doing this is to require that X is geodesically complete, meaning that every geodesic in X is a subarc of a geodesic which is parameterized on the whole real line. Although the geodesic completeness does not exclude porcupines with spines of infinite length, it is a rather convenient regularity assumption and should be sufficient for most purposes, at least in the locally compact case. Of

Introduction

4

course one might wonder how serious such a restriction is and how many interesting examples are excluded by it. A rich source of examples are locally finite polyhedra with piecewise smoothrnetrics of nonpositive Alexandrov curvature and, in dimension 2, such a polyhedron always contains a homotopy equivalent subpolyhedron which is geodesically complete and of nonpositive Alexandrov curvature with respect to the induced length metric, see [BB3]. Whether this or something similar holds in higher dimensions is open (as far as I know). In Riemannian geometry, the difference between strict and weak curvature bounds is very important and has attracted much attention. It is the contents of rigidity theorems that certain properties, known to be true under the assumption of a (certain) strict curvature bound, do not hold under the weak curvature bound, but fail to be true only in a very specific way. Well known examples are the Maximal Diameter Theorem of Toponogov and the Minimal Diameter Theorem of Berger (see [ChEb]). In the realm of nonpositive curvature, the above mentioned rank rigidity is one of the examples. Let X be a Hadamard space. A k-fiat in X is a convex subset of X which is isometric to Euclidean space JR.k. Rigidity phenomena in spaces of nonpositive curvature are often caused by the existence of flats. To state results connected to rank and rank rigidity, assume now that !v! is a smooth complete Riemannian manifold of nonpositive sectional curvature. Let '"Y : JR. ----+ M be a unit speed geodesic. Let JP("j) be the space of parallel Jacobi fields along '"Y and set rank("j) = dimJP("j)

and

rankM = minrank("j),

where the minimum is taken over all unit speed geodesics in M. Then 1

< rankM < rank("j)

:s:

dimM

and rank(Ml x M 2 )

=

rank M 1

+ rank M2 .

The space JP ('"Y) can be thought of as the maximal infinitesimal fiat containing the unit speed geodesic '"Y. In general, JP("j) is not tangent to a flat in M. However, if M is a Hadamard manifold of rank k, then every geodesic of Al is contained in a k-flat [B.Kleiner, unpublished]. (We prove this in the case that the isometry group of Al satisfies the duality condition, see below.) If Al is a symmetric space of noncompact type, then rank M coincides with the usual rank of M and is given as rank M min{ k I every geodesic of M is contained in a k-flat} max{ kiM contains a k-flat} . The second property is false for Hadamard manifolds. Counterexamples are easy to construct. Let X be a geodesically complete Hadamard space. The geodesic fiow gt, t E JR., of X acts by reparameterization on the space GX of complete unit speed geodesics on X,

Introduction

5

We say that a unit speed geodesic I : lR. ----> X is nonwandering mod r if there are sequences of unit speed geodesics (Tn : lR. ----> X), real numbers (t n ) and isometries ('Pn) of X such that In ----> I, t n ----> 00 and 'Pn 0 gt n (Tn) ----> r. This corresponds to the usual definition of nonwandering in x/r in the case where r is a group of covering transformations and where geodesics in x/r are considered. In a similar way we translate other definitions which refer, in their standard versions, to objects in a covered space. Following Chen and Eberlein [CEll we say that a group r of isometries of X satisfies the duality condition if any unit speed geodesic of X is nonwandering mod r. If M is a Hadamard manifold and r is a properly discontinuous group of isometries of M such that vol(MIn < 00, then r satisfies the duality condition. This is an immediate consequence of the Poincare Recurrence Theorem. The duality condition is discussed in Section I of Chapter III. The technical advantages of this notion are that it is invariant under passage to subgroups of finite index and to supergroups and that it behaves well under product decompositions of X. Theorem B. Let M be a Hadamard manifold and r a group of isometries of M satisfying the duality condition. Assume that the rank of M is one. Then we have: (i) [Bal} r contains a free non-abelian subgroup; (ii) [Bal} the geodesic flow of M is topologically transitive mod r on the unit tangent bundle SM of M; (iii) [Bal} the tangent vectors to r -closed geodesics of rank one are dense in SM. Moreover, if r is properly discontinuous and cocompact, then we have: (iv) [Kn} the exponential growth rate of the number of r -closed geodesics of rank one, where we count according to the period, is positive and equal to the topological entropy of the geodesic flow; (v) [BBl}, [Bu} the geodesic flow is ergodic mod r on the set n of unit vectors which are tangent to geodesics of rank one; (vi) [Ba3}, [BaLl} the Dirichlet problem at M(oo) is solvable and M(oo) together with the corresponding family of harmonic measures is the Poisson boundary ofM. Assertions (i)-(iii) are proved in Section 3 of Chapter III. The version in Chapter III is actually more general than the statements above inasmuch as we consider locally compact, geodesically complete Hadamard spaces instead of Hadamard manifolds. Assertions (iv) and (v) are not proved in these notes. The main reason is the unclear relation between proper discontinuity and cocompactness on the one hand and the duality condition on the other in the case of general Hadamard spaces. The ergodicity of the geodesic flow in S M Ir for compact rank one manifolds, claimed in [BBI, Bu] (and for compact surfaces of negative Euler characteristic in [Pe2]) , is based on an argument in [Pel, 2] which contains a gap, and therefore remains an open problem (even for surfaces). The main unresolved issue is whether the set has full measure in S/>'1. In case of a surface of nonpositive curvature, consists of all unit vectors which are tangent to geodesics passing through a point

n

n

6

Introduction

where the Gauss curvature of the surface is negative. Observe that R docs have full measure if the metric on .M is analytic. For Iv! a Hadamard manifold with a cocompact group of isometries, the relation of the rank one condition to the Anosov condition can be expressed in the following way: (1) M has rank one if M has a unit speed geodesic I such that dim JP(') = 1; (2) the geodesic flow of M is of Anosov type if dim JP b) = 1 for every unit speed gf,odesic I of ]v!. Sec [Eb4, .5] for this and other conditions implying the Anosov conditions for the geodesic flow. The proofs of the various assertions in (vi) in [Ba.5] and [BaLl] use.a discretization procedure of Lyons and Sullivan (see [LySu, Anc3, BaL2, Kai2]) to reduce the claims to corresponding claims about random walks on f. In Section 4 of Chapter III we show that the Dirichlet problem at 111 (00) is solvable for harmonic functions on f if f is countable and satisfies the duality condition. We also discuss briefly random walks and Poisson boundary of f. Theorem B summarizes more or less what is known about rank one manifolds (as opposed to more special cases). Rank one manifolds behave in many ways like manifolds of negative sectional curvature. Rank rigidity shows that higher rank is an exceptional case.

Theorem C (Rank Rigidity). Let Iv! be a Hadamard manifold and assume that the gmup of isometries of ]v! satisfies the duality condition. Let N be an irreducible factor in the deRham decomposition of A! with rank N = k :::: 2. Then N is a symmetric space of noncompact type and rank k. In particular, either Iv! is of rank one or else a Riemannian product or a 8ymmetric 8pace. The breakthrough in the direction of thIS theorem was obtained in the papers [BBE] and [BBS]. Under the stronger assumptions that 1'v! admits a properly discontinuous group f of isometrics with vol AI jf < 00 and that the sectional curvature of Iv! has a uniform lower bound, Theorem C was proved in [Ba2] and, somewhat later by a different method, in [BuSp]. The result as stated was proved by Eberlein (and is published in [EbHe]). The proofs in [Ba2], [BuSp] and [EbHe] build on the results in [BBE, BBS] and other papers and it may be somewhat difficult for a beginner to collect all the arguments. For that rcason it seems useful to present a streamlined and complete proof in one piece. This is achieved (I hope) in Chapter IV, where the details of the proof arc given. That is, we show that the holonomy group of Iv! is not transitive on the unit sphere (at some point of M), and then the Holonomy Theorem of Berger and Simons applies, sec [Be, Si]. Theorems Band C, when combined with other results, have some striking applications. We discuss some of them.

Theorem D. Let]v! be a Hadamard manifold and f a properly discontinuou8 group ofisometrie8 of M with vol (Mjf) < 00. Then eitherf contain8 a free non-abelian 8ubgroup or else ]v! i8 compact and fiat and f a Bieberbach group.

Introduction

7

This follows from Theorem A if the deRham decomposition of NI contains a factor of rank one (since the duality condition is preserved under products, see Section I in Chapter III). If there is no such factor, then NI is a symmetric space by Theorem C. In this case, f is a linear group and the Free Subgroup Theorem of Tits applies, see [Til]. The following application is proved in [BaEb].

Theorem E. Let !VI be a Hadamard manifold and let f be an irreducible properly discontinuous group of isometries of NI such that vol (!VIIf) < 00. Then the property that ivi is a symmetric space of noncompact type and rank k 2: 2 depends only on f. The result in [BaEb] is more precise in the sense that explicit conditions on f are given which are equivalent to NI being a symmetric space of rank k 2: 2. The proof in [BaEb] relies on results of Prasad-Raghunathan [PrRa] and Eberlein [EbI3] and will not be discussed here. For the sake of completeness we mention the following immediate consequence of Theorem E and the Strong Rigidity Theorem of Mostow and Margulis [Mos, Marl, 2].

Theorem F. Let !VI, NI* be Hadamard manifolds and f, f* be properly discontinuous groups of isometries of NI and !VI* respectively such that vol(M If), vol (!VI* If*) < 00. Assume that NI* is a symmetric space of noncompact type and rank k 2: 2, that f* is irreducible and that f is isomorphic to f*. Then NI is a symmetric space of noncompact type and, up to scaling factors, isometric to N1*. In the cocompact case this was proved earlier by Gromov [BGS] and (under the additional assumption that !VI* is reducible) Eberlein [EbI2]. Today, Theorem F does not represent the state of the art anymore, at least in the cocompact case, where superrigidity has been established in the geometric setting. This development was initiated by the striking work of Corlette [Col, 2]' and the furthest reaching results can be found in [MSY]. Another recent and exciting development is the proof of the non-equivariant Strong Rigidity Theorem by Kleiner and Leeb [KILe]. Theorem C has the following important companion, proved by Heber [Heb]. Theorem G (Rank Rigidity for homogeneous spaces). Let M be an irreducible homogeneous Hadamard manifold. If the rank of NI is at least 2. then !VI is a symmetric space of noncompact type. Note that the isometry group of a homogeneous Hadamard manifold !VI satisfies the duality condition if and only if M is a symmetric space, see [EbIO, Wotl,2]. In this sense, the assumptions of Theorems C and G are complementary. In both cases, the limit set of the group of isometrics is equal to NI ((X) ), and one might wonder whether the rank rigidity holds under this weaker assumption. In fact, it might even hold without any assumption on the group of isometries.

Introduction

8

Another interesting problem is the question of extendability of Theorems B and C to metric spaces of nonpositive Alexandrov curvature. As mentioned above already, part of that is achieved for the results in Theorem B. However, we use the flat half plane condition as in [Ba1,2] because there is, so far, no good definition of rank in the singular case. To state some of the main problems involved, assume that X is a geodesically complete and locally compact Hadamard space.

Problem 1. Assume that the isometry group of X satisfies the duality condition. Define the rank of X in such a way that (i) rank X = k ;::: 2 if and only if every geodesic of X is contained in a k-flat; (ii) rank X = 1 implies some (non-uniform) hyperbolicity of the geodesic flow. Problem 2. Assume that X is irreducible and that the group of isometries of X satisfies the duality condition. Show that X is a symmetric space or a Euclidean building if every geodesic of X is contained in a k-flat, k ;::: 2. Problem 3. Assume that r is a cocompact and properly discontinuous group of isometries of X. Show that r satisfies the duality condition. Problems 1-3 have been resolved in the case of two dimensional simplicial complexes with piecewise smooth metric and cocompact group of automorphisms, see [BB3]. Kleiner has solved Problem 2 in the case rank X = dim X, cf. [KI] (see also [BB3, Bar] for the case dimX = 2). Recalling the possible definitions of rank in the case of symmetric spaces we get a different notion of rank, Rank X = max {k I X contains a k- flat} Clearly rank X ::; Rank X ::; dim X . If X is locally compact and the isometry group of X cocompact, then the following assertions are equivalent: (i) X satisfies the Visibility Axiom of Eberlein-O'Neill; (ii) X is hyperbolic in the sense of Gromov; (iii) Rank X = 1, see [Eb1, Gr5]. If X admits a properly discontinuous and cocompact group of isometries r, then

Rank X ;::: max{ k

Ir

contains a free abelian subgroup of rank k}

by the Flat Torus Theorem of Gromoll-Wolf and Lawson-Yau, see [ChEb, GroW, LaYa, Bri2]. On the other hand, Bangert and Schroeder proved that RankM is the maximal integer k such r contains a free abelian subgroup of rank k if !vI is an analytic Hadamard manifold and r a properly discontinuous and cocompact group of isometries of M, see [BanS]. It is rather unclear whether this result can be extended to the smooth or the singular case. For more on Rank we refer to the discussion in [Gr6]. In these Lecture Notes, only topics close to the area of research of the author are discussed. References to other topics can be found in the Bibliography. It would

Introduction

9

be desirable to have complete lists of publications on the subjects of singular spaces and nonpositive curvature. The Bibliography here is a first and still incomplete step in this direction. For obvious reasons, papers about spaces of negative curvature are not included unless they are closely related to topics discussed in the text. I am very greatful to M. Brin for contributing the Appendix on the ergodicity of geodesic flows and drawing the figures. The assistance and criticism of several people, among them the participants of the DMV-seminar in Blaubeuren and S. Alexander, R. Bishop, M. Brin, S. Buyalo, U. Lang, B. Leeb, J. Previte and P. Thurston, was very helpful when preparing the text. I thank them very much. I am also grateful to Ms. G. Goetz, who TEXed the original manuscript and its various revisions. The work on these notes was supported by the SFB256 at the University of Bonn. A good part of the manuscript was written during a stay of the author at the IRES in Bures-sur-Yvette in the fall of '94.

CHAPTER I

On the Interior Geometry of Metric Spaces We discuss some aspects of the interior geometry of a metric space X. The metric on X is denoted d, the open respectively closed metric ball about a point x E X is denoted B(x,r) or Br(x) respectively B(x,r) or Br(x). Good references for this chapter are [Ri, AlBe, AlBN, Gr4, BrHa]. 1. Preliminaries

A curve in X is a continuous map IJ" : I ----> X, where I is some interval. The length L(IJ") of a curve IJ" : [a, b] ----> X is defined as k

(1.1)

L(IJ") = sup Ld(lJ"(ti-d,lJ"(ti)) i=l

where the supremum is taken over all subdivisions

a = to < t 1 < ... < tk = b of [a, b]. If rp : [a', b'] ----> [a, b] is a homeomorphism, then L(IJ" 0 rp) = L(IJ"). We say that IJ" is rectifiable if L(IJ") < 00. If IJ" : [a, b] ----> X is a rectifiable curve, then IJ"1[t, t'] is rectifiable for any subinterval [t, t'] c [a, b]. The arc length

5: [a, b]

---->

[0, L(IJ")],

s(t) = L(IJ" I [a, t])

is a non-decreasing continuous surjective map and

a: [0, L(IJ")]

---->

X,

a(s(t)):= IJ"(t)

is well defined, continuous and of unit speed, that is,

L(al [5,5']) More generally, we say that a curve

IJ" :

Is - 5'1 .

=

I

---->

L(IJ"I[t, t']) =

X has speed v ::;,. 0 if

vIt -

t'l

for all t, t' E I. A curve IJ" : I ----> X is called a geodesic if IJ" has constant speed v ::;,. 0 and if any tEl has a neighborhood U in I such that (1.2)

d( IJ"( t'), IJ"( til))

=

for all t', til in U. We say that a geodesic c: I all t', til E I.

vlt' - till ---->

X is minimizing if (1.2) holds for

Chapter I: On the interior geometry of metric spaces

12

The interior metric di associated to d is given by (1.3)

di(x,y) := inf{L(o-) I

0-

is a curve from x to y}.

One can show that di is a metric, except that di may assume the value pairs of points x, y E X. We have

00

at some

We say that X is an interior metric space if d = di . An interior metric space is pathwise connected. It is easy to see that the usual distance function on a connected Riemannian manifold is an interior metric. More generally we have the following result. 1.4 Proposition. If X is complete and for any pair x, y of points in X and there is a z E X such that 1 d(x,z),d(y,z)::; 2d(x,y)

E

>

°

+E,

then X is interior.

The proof of Proposition 1.4 is elementary, see [Gr4, p.4j. We omit the proof since we will not need Proposition 1.4. We say that X is a geodesic space if for any pair x, y of points in X there is a minimizing geodesic from x to y. 1.5 Proposition. If X is complete and any pair x, y of points in X has a midpoint, that is, a point m E X such that 1 d(x, m) = d(y, m) = 2d(x, y),

°

then X is geodesic. More generally, if there is a constant R > such that any pair of points x, y E X with d(x, y) ::; R has a midpoint, then any such pair can be connected by a minimizing geodesic. Proof. Given x, y in X with d(x, y) ::; R, we define a geodesic 0- : [0,1] ---> X from x to y of length d(x, y) in the following way: choose a midpoint m between x and y and set 0-(1/2) = m. Now d(x, m), d(y, m) ::; R and hence there are midpoints ml and m2 between x and 0-(1/2) respectively 0-(1/2) and y. Set 0-(1/4) = ml, 0-(3/4) = m2. Proceeding in this way we obtain, by recursion, a map 0- from the dyadic numbers in [0, 1j to X such that d(o-(s),o-(t))

= Is - tld(x,y)

for all dyadic s, t in [0,1]. Since X is complete, 0- extends to a minimal geodesic from x to y as asserted. D

The Hopf-Rinow Theorem

13

2. The Hopf-Rinow Theorem We present Cohn-Vossen's generalization of the Theorem of Hopf-Rinow to locally compact interior metric spaces, see [Coh]. 2.1 Lemma. Let X be locally compact and interior. Then for any x E X there is

°

an r > such that (i) if d(x, y) ::; r, then there is a minimizing geodesic from x to y; (ii) if d(x, y) > r, then there is a point z E X with d(x, z) = rand d(x, y) = r

+ d(z, y).

°

Proof. Since X is locally compact, there is an r > such that B(x, 2r) is compact. For any y E X there is a sequence O"n : [0,1] -+ X of curves from x to y such that L(O"n) -+ d(x, y) since X is interior. Without loss of generality we can assume that O"n has constant speed L(O"n). If d(x, y) ::; r, then L(O"r,) ::; 2r for n sufficiently large. Then the image of O"n is contained in B(x, 2r) and O"n has Lipschitz constant 2r. Hence the sequence (O"n) is equicontinuous and has a convergent subsequence by the theorem of Arzela-Ascoli. The limit is a minimizing geodesic from x to y, hence (i). If d(x, y) > r, then there is a point t n E (0,1) such that d(O"n(t n ), x) = r. A limit z of a subsequence of (O"n(t n )) will satisfy (ii). D

°

We say that a geodesic 0" : [0, w) -+ X, < w ::; 00 is a ray if 0" is minimizing and if limt--->wO"(t) does not exist. The most important step in Cohn-Vossen's argument is the following result. 2.2 Theorem. Let X be locally compact and interior. Then for x, y in X there is either a minimizing geodesic from x to y or else a unit speed ray 0" : [0, w) -+ X with 0"(0) = x, 0< w < d(x, y), such that the points in the image of 0" are between x and y; that is, if z is in the image of 0", then

d(x, z)

+ d(z, y)

=

d(x, y).

Proof. Let x, y E X and assume that there is no minimizing geodesic from x to y. Let rl be the supremum of all r such that there is a point z between x = Xo and y with d(xo, z) = r and there is a minimizing geodesic from Xo to z. Then

°

rl > by Lemma 2.1. We let Xl be such a point with 01 = d(xo, xd ~ r1l2 and 0"1 a minimizing unit speed geodesic from Xo to Xl. Since there is no minimizing geodesic from Xo to y we have Xl =I y. Since Xl is between Xo and y, any point z in the image of 0"1 is between Xo and y. There is no minimizing geodesic 0" from Xl to y since otherwise the concatenation 0"1 * 0" would be a minimizing geodesic from x = Xo to y. Hence we can apply the same procedure to Xl and obtain r2 > 0, X2 between Xl and y with 02 = d(Xl' X2) ~ r2/2 and a minimizing unit speed geodesic 0"2 from Xl to :);2. Since .1:2 is between Xl and y, any point in the image of 0"1 * 0"2

Chapter I: On the interior geometry of metric spaces

14

is between Xo and y. In particular, 171 * 172 is a minimizing unit speed geodesic and X2 -1= y. Proceeding inductively, we obtain a minimizing unit speed geodesic

such that all points on a are between x and y. In particular, a is minimizing. By the definition of T1 we have (T1

+ T2 + ... )/2

~ w := 61

+ t5 2 + ...

~ T1

It remains to show that a is a ray. If this is not the case, the limit x := limt--->w a(t) exists. Since there is no minimizing geodesic from x to y we have x -1= y. But then there is a (short) minimizing unit speed geodesic 0' : [0, r] ---+ X with r > such that 0'(0) = x and such that all the points on 0' are between x and y. Then all points on 17*0' are between x: and y and, in particular, r ~ T n /2 for all n by the definition of Tn. This contradicts r > and Tn ---+ 0. D

°

°

2.3 Theorem of Hopf-Rinow (local version). Let X be locally compact and inteTioT, and let x E X and R > 0. Then the following aTe equivalent: (i) any geodesic a : [0,1) ---+ X with 17(0) = x and L(a) < R can be e:r:tended to the closed interval [0,1]; (ii) any minimizing geodesic a : [0,1) ---+ X with 17(0) = x and L(a) < R can be extended to the closed inteTval [0, 1]; (iii) B(x, T) is compact fOT ~ T < R. Each of these implies that fOT any paiT y, z of points in B(x, R) with d(x, y) d(x, z) < R theTe is a minimizing geodesic from y to z.

°

+

Proof. The conclusions (iii) =} (i) and (i) =} (ii) are clear. We prove (ii) =} (iii). Let TO E (0, R] be the supremum over all T E (0, R) such that B(x:, T) is compact. We assume TO < R. Let (x n ) be a sequence in B(.T, TO). From (ii) and Theorem 2.2 we conclude that there is a minimizing geodesic an : [0,1] ---+ X from x to x n . Then d( x, an (t)) ~ tTo for ~ t ~ 1, and by a diagonal argument we conclude that an I [0,1) has a subsequence converging to a minimizing geodesic (J : [0,1) ---+ X. By (ii), a can be extended to 1 and clearly (J(1) is the limit of the (corresponding) subsequence of (a n (l)) = (x n ). Hence B(x, TO) is compact. Since X is locally compact, there is an E > such that B(y, E) is compact for any y E B(x, TO). But then B(x, TO + 15) is compact for 15 > sufficiently small, a contradiction to the definition of TO. D

°

°

°

2.4 Theorem of Hopf-Rinow (global version). Let X be locally compact and inteTioT. (i) (ii) (iii)

Then the following aTe equivalent: X is complete; any geodesic (J : [0,1) ---+ X can be extended to [0,1]; fOT some point x E X, any minimizing geodesic a: [0,1) can be extended to [0, 1];

---+

X with 17(0) =:r

Spaces with curvature bounded from above

15

(iv) bounded subsets of X are relatively compact. Each of these implies that X is a geodesic space, that is, for any pair x, y of points in X there is a minimizing geodesic from x to y. 0

The local compactness is necessary. If X is the space consisting of two vertices y and a sequence of edges an of length 1 + lin, n ?: 1, then X (with the obvious interior metric) is a complete interior metric space; but X is not a geodesic space since d(x,y) = 1. :I:,

3. Spaces with curvature bounded from above For K E lR, we denote by M~ the model surface of constant curvature K. Motivated by corresponding results in Riemannian geometry, we will define upper curvature bounds for X by comparing triangles in X with triangles in 1vl~. A triangle in X consists of three geodesic segments aI, a2, a3 in X, called the edges or sides of the triangle, whose endpoints match (in the usual way). If II = (al,a2,a:l) is a triangle in X, a triangle II = (al,a2,a3) in M~ is called an Alexandmv triangle or comparison triangle for II if L(ai) = L(ai), 1 ::; i ::; 3. A comparison triangle exists and is unique (up to congruence) if the sides satisfy the triangle inequality, that is (3.1a) for any permutation (i, j, k) of (1. 2, 3), and if the perimeter (3.1b) Here and below we usc the convention 21f I ~

=

00

for

K ::;

O.

3.2 Definition. We say that a triangle II has the C AT",-property, or simply: is CAT"" if its sides satisfy the inequalities (3.1a), (3.1b) and if d(x, y) ::; d(x, y)

for all points x, yon the edges of II and the corresponding points X, yon the edges of the comparison triangle II in 1vl~. In short, a triangle is CAT", if it is not too fat in Comparison to the Alexandmv Triangle in M~.

3.3 Lemma. Let II = (aI, a2, (3) and ll' = (a~, af, a_~) be triangles in X and assume that a:l = a(J and that a2 * af is a geodesic. If lll! = (aI, a2 * .:rf, a~) has perimeter < 21f/~ and ll, ll' are CAT", then lll! -is CAT",. Moreover, if the length of a3 = a~ is strictly smaller than the distance of the pair of points on the comparison triangle lll! in 1vl~ corresponding to the endpoints of a3 = a~, then the distance of any pair x, y of points on the edges of lll!, such

Chapter I: On the interior geometry of metric spaces

16

that x, yare not on the same edge of 1::1", is strictly smaller than the distance of the corresponding pair of points on 1::1". Proof. Match the comparison triangles 1::1 and 1::1 ' in M~ along the sides 0'3 and O'~. Let x be the vertex of (J3 = (J~ opposite to (J1 and (J~ respectively. Since (J2 * (J& is a geodesic, we have d(y, y') = d(y, x) + d(x, y') for y E (J2 and y' E (J& sufficiently close to x. We claim that the interior angle of -I 1::1 U 1::1 at x is at least 7r. If not, there is a point z on (J3 = (J~ different from x such that the minimizing geodesic (in M~) from y to y' passes through z. But then

-

d(y, y') = d(y, x)

+ d(x, y')

d(y, x)

+ d(x, y')

> d(y, z) + d(z, y') > d(y, z) + d(z, y') :?: d(y, y'), where we use, in the second line, that 1::1 and 1::1 ' are CAT",. Hence we arrive at a contradiction and the interior angle at x is at least 7r. In particular, the sides of 1::1" satisfy the triangle inequality (3.1a). -I If the angle is equal to 7r, then 1::1 U 1::1 is the comparison triangle of 1::1". In this case it is clear that 1::1" is CAT",. If the angle is strictly bigger than 7r, the comparison triangle is obtained by straightening the broken geodesic 0'2 * 0'& of -I 1::1 U 1::1 , keeping the length of 0' 1 , O'~ , 0'2 and O'~ fixed. Now let Q be the union of the -I two triangular surfaces bounded by 1::1 and 1::1 and denote by Qt the corresponding surface obtained during the process of straightening at time t. The interior distance dt of two points on the boundary of Qt, that is, the distance determined by taking the infimum of the lengths of curves in Qt connecting the given points, is strictly smaller than the distance d s of the corresponding points at a later time s. The proof of this assertion is an exercise in the trigonometry of M~. At the final time t f of the deformation, Qtj is a triangle in M~ of perimeter < 27r / /K, and, therefore, the interior distance d t j is equal to the distance in M~. Hence the lemma follows. D 3.4 Lemma. Let x E X and assume that any two points y, z E Br(x) can be connected by a minimizing geodesic in X. If r < 7r /2/K, and if all triangles in B 2r (x) with minimizing sides and of perimeter < 27r / /K, are CAT"" then B r (x) is convex; more precisely, for all y, z E Br(x) there is a unique geodesic (Jyz : [0,1] ----+ B r (x) from y to z and (Jyz is minimizing and depends continuously on y and z.

Proof. We prove first that any geodesic (J : [0,1] ----+ Br(x) is minimizing. Otherwise there would exist, by the definition of geodesics, a maximal t a E (0,1) such that (J1 = (JI[O, tal is minimizing and a 8> a with 8 < t a , l~ta such that (J1[t a -8, t a +8] is minimizing. Let (J2 = (J1[ta,t a +8],y = (J(t a ),Zl = (J(t a -8) and Z2 = (J(t a +8). We have

Spaces with curvature bounded from above

17

On the other hand, al [0, t(J + 8] is not minimizing by the definition of t(J' Let a3 be a minimizing geodesic from a(O) to Z2 = a(t(J + 8). Then

and a3 is contained in B 2r (x) since d(x,a(O)) + d(X,Z2) < 2r. Now the triangle ~ = (al' a2, a3) has minimizing sides and is contained in B 2r (x). In the comparison triangle ~ of ~ we get from (**) and (*),

d(Zl,Z2) < d(Zl'y) +d(y,Z2) =d(Zl,y)+d(y,Z2) = d(Zl,Z2), a contradiction to ~ being CAT". Hence a is minimizing. Now let Y, Z E Br(x) and a be a minimizing geodesic from Y to z. Then a is contained in B 2r (x) and a together with minimizing geodesics from x to Y and x to Z forms a triangle ~ in B 2r (x) with minimizing sides. Since ~ is CAT" and r < 1f/2/fi, any point on a has distance < r to Xj that is, a is in Br.(x). Now Lemma 3.4 follows easily. D 3.5 Remark. The assumptions and assertions in Lemma 3.4 are not optimal; compare also Corollary 4.2 below. 3.6 Definition. For K, E IR, an open subset U of X is called a CAT,,-domain if and only if (i) for all x, y E U, there is a geodesic a xy : [0,1] ---> U of length d(x, y); (ii) all triangles in U are CAT". We say that X has Alexandrov curvature at most K" in symbols: K x :::; K" if every point x E X is contained in a C AT,,-domain.

For K, = 1, the standard example of a C AT,,-domain is the open hemisphere in the unit sphere. Euclidean space respectively (real) hyperbolic space are examples of C AT,,-domains for K, = 0 respectively K, = -l. If U is a C AT,,-domain, then, by the definition of CAT", triangles in U have perimeter < 21f/ /fi. In particular, any geodesic in U has length < 1f/ /fi. Otherwise we could construct (degenerate) triangles of perimeter 2: 21f/ /fi. Since triangles in U are CAT", we conclude that the geodesic a xy as in (i) is the unique geodesic in U from x to y and that a xy depends continuously on x and y. By the same reason, if x E U and if Br(x) C U for some r < 1f/2/fi, then Br(x) is a C AT,,-domain. 3.7 Exercises and remarks. (a) If X is a Riemannian manifold, then the Alexandrov curvature of X is at most K, iff the sectional curvature of X is bounded from above by K,. (b) Define a function K x on X by

Kx(;x;) = inf{K,

E IRlx is contained in a CAT,,-domain

U

C

X}.

Chapter I: On the interior geometry of metric spaces

18

Show that this function is bounded from above by K, if the Alexandrov curvature of X is at most K,. (This is a reconciliation with the notation.) (c) Let X be a geodesic space. Show that X has Alexandrov curvature at most K, iff every point :1: in X has a neighborhood U such that for any triangle in X with vertices in U and minimizing sides the distance of the vertices to the midpoints of the opposite sides of the triangle is bounded from above by the distance of the corresponding points in the comparison triangle in lvl~. In [Ri] , the interested reader finds a thorough discussion of the above and other definitions of upper curvature bounds. Now assume that U

c

X is a C AT,,-domain. Let e

> 0 and let (}1, 0"2 :

[O,e] ----+ U be two unit speed geodesics with (}1(0) = (}2(0) =: x. For o5,t E (O,e) let 6. st be the triangle spanned by 0"11 [0,05] and 0"21 [0, t]. Let ,(05, t) be the angle at x of the comparison triangle 6. st in lvl~. Then ,(8, t) is monotonically decreasing as 8, t decrease and hence (3.8) exists and is called the angle subtended by 0"1 and the triangle 'inequality

0"2.

The angle function satisfies

(3.9) The triangle inequality is very useful in combination with the fact that

(3.10) Here 0"1 1 is defined by (}1 1(t) = (}l(-t), t:s. O. The trigonometric formulas for spaces of constant curvature show that we can use comparison triangles in lvlJ as well, where ,\ E lR is arbitrary (but fixed), and obtain the same angle measure. In particular, we have the following formula:

-e:s.

(3.11) If y, Z E U \ {;r} and 0"1,0"2 are the unit speed geodesics from ;r to y and z respectively, we set Lx(y, z) = L((}l' (}2).

3.12 Exercise. In U as above let X n ----+ 'X, Yn ----+ y, Zn ----+ Z with x E U and y, z E U \ {x}. Then Lx n (Yn' zn) and Lx (Yn' zn) are defined for n sufficiently large and

In a CAT,,-domain U, we can also speak of the triangle 6.(X1, X2, X3) spanned by three points Xl, :1:2, X3 since there are unique geodesic connections between these points.

The Hadamard-Cartan Theorem

19

3.13 Proposition. Let U be a CAT,,-domain, let 6. = .6.(X1,X2,X3) be a triangle in U, and let .6. be the comparison triangle in (i) Let OOi be the angle of 6. at Xi, 1 :::; i :::; 3. Then OOi :::; ai, where ai is the corresponding angle in 6.. (ii) If d(x, y) = d(x, y) for one pair of points on the boundary of 6. such that X, y do not lie on the same edge, or if OOi = ai for one i, then 6. bounds a convex region in U isometric to the triangular region in bounded by .6..

M;.

M;

Proof. The first assertion follows immediately from the definition of angles since triangles in U are CAT". The equality in (ii) follows easily from the last &'lsertion

in Lemma 3.3.

0 4. The Hadamard-Cartan Theorem

In this section, we present the proof of the Hadamard-Cartan Theorem for simply connected, complete metric spaces with non-positive Alexandrov curvature. Our version of the theorem includes the ones given in [AB1] and [Ba5]; it was motivated by a discussion with Bruce Kleiner. We start with a general result of Alexander and Bishop [AB1] about geodesics in spaces with upper curvature bounds. 4.1 Theorem. If X is a complete metric space with K x :::; "'", and if 0" : [0,1] -+ X is a geodesic segment of length < 7f /,j"K, then 0" does not have conjugate points in the following sense: for any point X close enough to 0"(0) and any point y close enough to 0"(1) theT'e is a unique geodesic 0"xy : [0, 1] -+ X from X to y close to 0". Moreover, any triangle (O"xy,O"xz,O"yz), where x is close enough to 0"(0), y and z are close enough to 0"(1) and O"yz is minimizing from y to z, is CAT". Proof. For 0 :::; L :::; L(O") consider the following assertion A(L): Given c > 0 small there is 15 > 0 such that for any subsegment 0' = XoYo of 0" of length at most L and any two points x, y with d( x, xo), d(y, Yo) < 15 there is a un'ique geodesic 0"x'y from :1: to y whose distance from 0' is less than c. M oreover, IL(O";ry) - L(O') c and any triangle (O";ry, O"xz, O"yz), where d(x, xo), d(y, Yo), d(z, Yo) < 15 and where O"yz is minimizing from y to z, 'is CAT". 1

:::;

Let l' > 0 be a uniform radius such that BT(z) is a CAT,,-domain for any point z on 0". Then A(L) holds for L :::; r. We show now that A(3L/2) holds if A(L) holds and 3L/2 :::; L(O") < 7f/,j"K. To that end choose 0 > 0 with L(O") + 300 < 7f /,j"K. Then L + 200 < 27f /3,j"K and therefore there is a constant ,\ < 1 such that for any two geodesics 0'1, 0'2 [0,1] -+ 1\:1; with 0'1(0) = 0'2(0) and with length at most L + 20 we have

Now let 0' = XoYo be a subsegment of 0" of length 31/2 :::; 3L/2 and let c > 0 be given. Choose c' > 0 with c' < min{c/3,oo}, and let 15 < 15'(1- ,\)/,\, where 15' is the value guaranteed by A (L) for c'.

20

Chapter 1: On the interior geometry of metric spaces

Subdivide (j into thirds by points Po and qo. Let x, y be points such that d(x, xo), d(y, Yo) < 8. By A(L) there are unique geodesics O"xqO from x to qo and O"pOY from Po to Y of distance at most e' to xoqo respectively PoYo. Furthermore, their length is in [l - e', l + e'l and the triangles (xoqo, 0"xqo' 0" XXQ) and (Po Yo , O"poY' O"yyo) are CATK. Hence we can apply (*) to the midpoints PI of O"xqO and qI of O"poY and obtain d(po,pd,d(qo,qd:O:; A max{d(x, xo), d(y, Yo)} < A8 < 8'. By A(L) there are unique geodesics O"xq, from x to qI and O"PIY from PI to y of distance at most e' to xoqo respectively PoYo. Furthermore, their length is in [l - e', l + e'] and the triangles (O"xqO' O"xq" O"qOq,) and (O"pOY' O"PIY' O"POPl) are CATK· Hence we can apply (*) to the midpoints P2 of O"xq, and q2 of O"PIY and obtain

Hence by the triangle inequality

Recursively we obtain geodesics O"xqn from x to the midpoint qn of O"pn_lY and O"pnY from the midpoint Pn of O"xqn_l to y of distance at most e' to xoqo respectively PoYo. Their length is in [l - e', l + e'] and the triangles (0"xqn-l , 0"xqn , 0"qn-l qn) and (O"pn_lY' O"pnY' O"pn-lpJ are CATK· Furthermore, we have the estimates

and

d(Po,Pn), d(qo, qn) < (A

+ ... + .\")8 < 8'.

In particular, the sequences (Pn), (qn) are Cauchy. Since X is complete, they converge. If P = limpn and q = lim qn, then

and hence, by A(L), the geodesics O"xqn and O"pnY converge to O"~,q and O"py. By construction, P E O"xq and q E O"py. Therefore, by the uniqueness of O"pq, the geodesics O"xq and O"py overlap in O"pq and combine to give a geodesic O";r;y from x to y. The length of O"xy is given by L(O"xq)+L(O"py)-L(O"pq), hence IL(O"xy)-L(O")1 < e by A(L) and since e' < e/3. The last assertion follows from Lemma 3.3 by subdividing the triangle suitably. 0 4.2 Lemma. Let X be a complete metric space with K x :0:; "'", and let 0"0 : [0,1] -> X be a geodesic segment of length L o = L(O"o) < 7f /,j""K. For e > a with L+2e < 7f /,j""K assume that the balls BE(xo) and BE(yo) are CATK-domains, where :1:0 = 0"0(0) and Yo = 0"0 (1). Then there is a continuous map

The Hadamard-Cartan Theorem

21

with the following properties: (i) er xy = 'L.(x, y,.) is a geodesic from x to y of length L(er xy ) S; La + d(x, xo) + d(y,yo) and erxoYo = era; (ii) any triangle (er xy , er xz , er), where er is the geodesic in BE(yo) from y to z, is CAT". Moreover, 'L. is unique in the following sense: if er" OS; 8 S; 1, is a homotopy of era by geodesics with ers(O) = Xs E BE(xo) and er s (l) = Ys E BE(xd, then er s = er xsYs ' Proof. The assertion about the uniqueness is immediate from the uniqueness assertion in Theorem 4.1. Now let x E BE(xo), y E BE(yo) and let

be the geodesics from Xo to x respectively Yo to y. Let ro 2': over all r E [0,1] such that there is a continuous map

Fr : [O,r] x [O,r] x [0,1]

-+

°

be the supremum

X

with Fr(O,O,.) = era and (i')erst = F r (8, t, .) is a geodesic from 10 (8) to 11 (t) of length L(ersd S; La + d(xo"o(8)) + d(Yo"l(t)); (ii') triangles (,0 I [81,82]), ers,t, er s2 t) and (erst I ,erst2' 11 I [t l , t2]) are CAT". The uniqueness assertion in Theorem 4.1 implies that Frl and Fr2 agree on their common domain of definition for all rl, r2 E [0, ro). Now L + 2E < 7r /,j"K, and hence the circumference of triangles as in (ii'), for 81,82 respectively t l , t2 small, is uniformly bounded away from 27r /,j"K. Comparison with triangles in M~ shows that limr--->ro Fr =: Fro exists. It is clear that Fro (0, 0, .) = era and that Fro satisfies (i') and (ii'). This shows that the set J of r, for which a map F r as above exists, is closed in [0,1]. On the other hand, J is open by Theorem 4.1. Hence ro = 1. We set Fxy = F l and er xy = F l (l, 1, .). It follows from Theorem 4.1 and uniqueness that Fry, and hence er xy , depends continuously on x and y. Assertion (ii) follows from Lemma 3.3. D 4.3 Corollary. Let X be a complete interior metric space with Kx S; K. Let R S; 7r /,j"K and assume that for any two points x, y with d( x, y) < R there is a unique geodesic er xy : [0,1] -+ X from x to y of length < R. Then (i) L(er xy ) = d(x, y) for all such x, y; (ii) each triangle in X of perimeter < 2R is CAT". In particular, B R / 2(:r) is a CAT,,-domain for any x E X.

Proof. Since X is interior, there is a curve c : [0,1]

-+

X from x to y of length

°

< R. By Lemma 4.2, such a curve c gives rise to a homotopy er s , S; 8 S; 1, such that er s (t) = c( t) for 8 S; t S; 1 and er s (t), S; t S; 8, is a geodesic from x to c( 8)

°

°

which is not longer than c(t), S; t S; 8. In particular, erl is a geodesic from x to y of length S; L(c) < R. Hence erl = er xy , independently of the choice of c. Since

Chapter I: On the interior geometry of metric spaces

22

X is interior we obtain L(a xy ) Corollary 4.3 follow easily. D

=

d(x, y), hence (i). The remaining assertions of

We now apply the above argument in the case of a family of curves. To avoid asumptions on the lengths of the curves, we assume that the Alexandrov curvature is nonpositive. 4.4 Lemma. Let X be a complete metric space with K x ::; o. Let f : K x I ----+ X be a continuous map, where K is a compact space and I = [0,1]. Then there is a homotopy F : I x K x I ----+ X of f such that (i) F(s, k, t) = f(k, t) for all k E K and t 2: s; (ii) F(s,k,t),O::; t::; s, is a geodesic from f(k,O) to f(k,s). (iii) F(s,k,t),O::; t::; s, is not longer than f(k,t),O::; t::; s; (iv) d(F(s,k,t),F(s,k',t))::; rd(f(k,O),f(k',O)) + (1- r)d(f(k,s),f(k',s)), r = t/ s, if k and k' are sufficiently close.

Proof. The existence of a map F satisfying (i) and (ii) is immediate from Lemma 4.2. Since the Alexandrov curvature of X is nonpositive, (iii) and (iv) follows from (ii) in Lemma 4.2. D 4.5 Theorem of Hadamard-Cartan. Let X be a simply connected complete metric space with K x ::; o. Then for any pair x, y of points in X there is a unique geodesic a xy : [0,1] ----+ X from 2; to y. Moreover, a xy is continuous in 2; and y and L(a xy ) = di(x,y), where d i denotes the interior metric associated to d.

Proof. Let f : [0,1] ----+ X be a path from x to y. By Lemma 4.4 there is a homotopy F of f such that a = F(l,.) is a geodesic from x to y. This proves existence. Note also that L(a) ::; L(f). As for uniqueness, if ao and al are geodesics from x to y, then there is a homotopy f between them fixing the endpoints x and y. Consider the associated map F as in Lemma 4.4. The uniqueness assertion in Lemma 4.2 implies F(l, 0, .) = ao and F(l, 1,.) = al. Now F(l, 0, 0)

=

F(l, 1,0)

=

x

and

and therefore F(l, 0,.) = F(l, 1,.) by (iv) We have proved that, for any pair geodesic a xy : [0,1] ----+ X from 2; to y. By y. The uniqueness of a = a xy implies that x to y. Hence L(a) = di(x, y). D

F(l, 0,1)

=

F(l, 1, 1)

=

y

in Lemma 4.4. Hence ao = al. x, y of points in X, there is a unique Lemma 4.2, a xy is continuous in x and L(a) ::; L(f), where f is any curve from

5. Hadamard spaces

A Hadamard manifold is a simply connected, complete smooth Riemannian manifold without boundary and with nonpositive sectional curvature. Following this terminology, we say that a metric space X is a Hadamard space if X is simply connected, complete, geodesic with K x ::; o. A Hadamard spcl,Ce need not be

Hadamard spaces

23

geodesically complete, that is, we do not require that every geodesic segment of X is contained in a geodesic which is defined on the whole real line. The following elegant characterization of Hadamard spaces can be found in the paper of Bruhat and Tits [BruT], see also [Bro,p.155].

5.1 Proposition. Let X be a complete metric space. Then X is a Hadamard space if and only if for any pair x, y of points in X there is a point m E X (the midpoint between x and y) such that

Using Karcher's modified distance function [Kar] one obtains a similar formula characterizing complete geodesic spaces X with K x :S '" such that for any pair of points x, y E X with d(x, y) < R (for some fixed R> 0, R:S 7f /"fii) there is a unique geodesic from x to y of length < R. Proof of Proposition 5.1. Substituting x and y for z we see that m is in the middle between x and y, d(x,m) = d(y,m) = ~d(x,y), hence X is geodesic, see Proposition 1.5. It is also immediate that m is unique and varies continuously with the endpoints, hence the geodesic connecting x and y is unique and varies continuously with x and y. Therefore X is contractible and, in particular, simply connected. Now if x. y, z are the vertices of a geodesic triangle in X, then the inequality in Proposition 5.1 asserts that the distance of z to m is bounded by the distance of z to in in the comparison triangle in the Euclidean plane. Therefore, K x :S 0, compare Exercise 3.7. 0 We now collect consequences of our discussion in the previous sections, in particular of Proposition 3.13, Corollary 4.3 and the Theorem of Hadamard-Cartan. The main property is that for any pair x, y of points in a Hadamard space X, there is a unique geodesic a xy : [0,1] -+ X connecting x and y.

5.2 Proposition. Let X be a Hadamard space and let l:l be a triangle in X with edges of length a, band c and angles a, {i and, at the opposite vertices respectively. Then (i) a+p+,:S7f; (ii) c2 ?: 0,2 + b2 - 2ab cos, (First Cosine Inequality) bcosa+acos{3 (Second Cosine Inequality). (iii) In each case, equality holds if and only if l:l is fiat, that is, l:l bounds a convex region in X isometric to the triangular region bounded by the comparison tr-iangle in the fiat plane. 0

c:s

The following formula is useful in connection with the Second Cosine Inequality, see [BGS] or respectively the corresponding argument in the proof of Theorem

II.4.3.

Chapter I: On the interior geometry of metric spaces

24

5.3 Proposition. Let X be a Hadamard space and let .0. be a triangle in X with edges of length a, band c and angles 0', f3 and I in the opposite vertices. Let O'E and f3E be the angle in the Euclidean triangle .0.E spanned by two edges of length a and b which subtend an angle of measure IE = 7f - 0' - f3. Then bcos 0'

+ a cos f3 = C E cos (0' - 0' E) = c E cos (f3 -

f3E)

where CE is the length of the third edge of .0.E. In particular, bcos 0' + a cos f3 = CE if and only if .0. is flat. Proof. Consider the function

f (s)

=

b cos s

+ a cos (0' + f3 -

8).

Then

l' =

-bsin8 + asin(O'

+ f3 -

8)

and

1" = - f.

By the Law of Sines, f assumes its maximum CE in O'E and hence we have f(8) = CE cos(s - O'E), where CE is the length of the third edge in .0.E. D 5.4 Proposition. Let I be an interval, and let 0"1,0"2 : I ---> X be two geodesics in a Hadamard space X. Then d(0"1(t),0"2(t)) is convex in t. Proof. Let to < t1 be points in I, and let 0" : [to, t 1] ---> X be the geodesic from O"l(tO) to 0"2(td. Since triangles in X are CATo , we have, for t = ~(to + td, d(0"1(t),0"2(t)) :::; d(O"l(t),O"(t)) + d(0"(t),0"2(t)) 1 1 :::; "2 d (0"1(td'0"2(td) + "2 d (0"1(tO)'0"2(tO)).

D

5.5 Remark. The convexity of d( 0"1 (t), 0"2 (t)) in t, for any pair of geodesics 0"1, 0"2 in X, is not equivalent to X being a Hadamard space. For example, if X is a Banach space with strictly convex unit ball, then geodesics are straight lines and hence the above 'convexity property holds. On the other hand, a Banach space has an upper curvature bound in the sense of Definition 3.6 if and only if it is Euclidean, see [Ri]. We say that a function f on a geodesic space X is convex if f for any geodesic 0" in X. 5.6 Corollary. Let X be a Hadamard space and C de : X

--->

c

0

0" is convex

X a convex subset. Then

JR., dc(z) = d(z, C),

is a convex function. If C is closed, then for any z E X there is a unique point 7fc(z) E C with d(z, 7fe(z)) = dc(z). The map 7fe is called the projection onto C; it has Lipschitz constant 1. Proof. Everything except for the existence of the point 7fc(z) is clear. Now let (x n ) be a sequence in C with d(z, x n ) ---> dc(z). Since C is convex, the midpoint

Hadamard spaces

25

m = mnl between X n and Xl is also in C. Its distance to z satisfies the estimate in Proposition 5.1, where we subsitute X n and Xl for X and y. It follows that (x n ) is a Cauchy sequence. Now X is complete and C is closed, hence the sequence has a limit and the limit is in C. By continuity, it realizes the distance from z to C. 0 5.7 Proposition. Let X be a Hadamard space and let 0 = (CT1' CT2, CT3, CT4) be a quadrangle of four consecutive geodesic segments in X with interior angle (Xi subtended by CTi and CTi+1 at their common vertex (where we count indices mod 4). Then (Xl + (X2 + (X3 + (X4 ::; 27f, and equality holds if and only if 0 is fiat, that is, o bounds a convex region isometric to a convex region in the fiat plane bounded by four line segments.

Proof. Subdivide 0 into two triangles by the geodesic from the initial point of CT1 to the endpoint of CT2 and apply (i) of Proposition 5.2. 0 We say that two unit speed rays CT1, CT2 : [0,(0) ---+ X (respectively unit speed geodesics CT1,CT2 : lR ---+ X) are asymptotic (respectively parallel) if d(CT1(t),CT2(t)) is uniformly bounded. 5.8 Corollary. Let X be a Hadamard space. (i) If CT1, CT2 : [0,(0) ---+ X are asymptotic unit speed rays, then

and equality holds if and only if CT1, CT2 and the geodesic CT from CTdO) to CT2 (0) bound a fiat half strip: a convex region isometric to the convex hull of two parallel rays in the fiat plane. (ii) (Flat Strip Theorem) If CT1, CT2 : lR ---+ X are parallel unit speed geodesics, then CT1 and CT2 bound a fiat strip: a convex region isometric to the convex hull of two parallel lines in the fiat plane. 0 5.9. Proposition. Let X be a Hadamard space and CT : lR ---+ X a unit speed geodesic. Then the set P = P a C X of all points in X which belong to geodesics parallel to CT is closed, convex and splits isometrically as P = Q X lR, where Q C X is closed and convex and {q} x lR is parallel to CT for any q E Q.

Proof. The convexity of P follows immediately from the Flat Strip Theorem 5.8(ii). The closedness of P follows from the completeness of X. Now set qo = CT(O) and let X E P. Then there is, up to parameterization, a unique unit speed geodesic CT x parallel to CT with X E CT x . Denote by qx the point on CT x closest to qo and choose the parameter on CT x such that CTx(O) = qx. Let t x E lR be the unique parameter value with x = CTx(t x ). It follows from the Flat Strip Theorem that qx (respectively qo) is the unique point on CTx (respectively CT) with d( er( t), qx) - t ---+ 0 (respectively d(CTx(t), qo) - t ---+ 0) as t ---+ 00. Now let yEP and define qy and t y accordingly. Then we have

t 2

d(CTy(t), qx) - t ::; d(CT,,(t), CT(t/2)) - J

+ d(CT(t/2), qx) -

t 2

-

Chapter I: On the interior geometry of metric spaces

26

and hence, since d(O"y(t), 0"(t/2)) - t/2

-+

0 as t

-+ 00,

lim sup d(O"y(t), qx) ~ t

< O.

t-----+oo

Similarly

limsupd(O"x(t),qy)

~t::;

O.

t->oo

Now O"x and O"y are parallel and bound a flat strip. From (*) and (**) we conclude that qy is the closest point to qx on O"y and qx is the closest point to qy on 0"x. Hence (don't forget the flat strip)

Hence the assertion with Q = {qx I x E P}.

D

Following the presentation in [Bro], we now discuss circumcenter and circumscribed balls for bounded subsets in Hadamard spaces. 5.10 Proposition. Let X be a Hadamard space and let A c X be a bounded subset. For x E X let r(x, A) = SUPyEA d(x, y) and set r(A) = inf xEx r(x, A). Then there is a unique:1: E X, the circumcenter of A, such that r(x,A) = r(A), that is, such that A c B(x, r(A)).

Proof. Let x, y E X and let

Tn

be the midpoint between them. Then 1 2 4d (x, y),

see Proposition 5.1, and hence

+ r 2 (y, A)) < 2(r 2 (x, A) + r 2 (y, A))

d 2 (x, y) < 2(r 2 (x, A)

- 4r 2 (m, A) - 4r 2 (A).

Now the uniqueness of the circumcenter is immediate. It also follows that a sequence (x n ) with r(x n , A) -+ r(A) is Cauchy. Now X is complete, hence we infer the existence of a circumcenter. D With similar arguments one obtains centers of gravity for measures on Hadamard spaces. 5.11 Exercise. Let X be a Hadamard space and let p, be a measure on X such that g(x) = I d 2 (x, Y)JL(dy) is finite for one (and hence any) x E X. Show that g assumes its infimum at precisely one point. This point is called the barycenter or center of gravity of p,. Barycenters and circumcenters can also be defined in CAT,,-domains; compare [BuKa] for the discussion in the Riemannian case.

CHAPTER II

The Boundary at Infinity In this chapter, we present a variation of §§ 3-4 of [BGS]. We assume throughout that X is a complete geodesic space. The various spaces of maps discussed are assumed to be endowed with the topology of uniform convergence on bounded subsets. 1. Closure of X via Busemann functions

On X we consider the space C(X) of continuous functions, endowed with the topology of uniform convergence on bounded subsets. For x, y, z E X we set

b(x, y, z) = d(x, z) - d(x, y).

(1.1 ) Then we have b(x,y,y) (1.2)

= 0 and Ib(x,y,z) - b(x,y,z')I:::; d(z,z'),

and hence b(x, y,.) E C(X). Furthermore (1.3)

Ib(x, y, z) - b(x', y, z)1 :::; 2d(x, :1:')

and (1.4)

b(x,y,z) - b(:I:,]}.y') = b(x,y',z).

For y E X fixed we have the map

by : X

--->

C(X),

by(x) = b(x, y, .).

It follows from (1.3) that by is continuous and from (1.2) that by(x) has Lipschitz constant 1 for all J; E X. If X, x' E X and, say, d(x', y) 2: d(x, y), then

by(x)(x') - by(x')(x') 2: d(x,x'), and hence by is injective. In fact, by is an embedding since by(x) and by(x') are strictly separated if d(x,x') is large. To sec this suppose d(x',y) 2: 2d(x,y) + 1 and let z E Br(x) be the point on the geodesic form x to x' with d(x, z) = r := d(J:, y) + 1. Then

by(x)(z) - by(.Y')(z) = 1 - d(x', z)

+ d(x', y)

2:1+d(x',x)-d(x,y)-d(x',z) Now r depends only on x and not on x'. Hence by is an embedding.

2.

Chapter II: The boundary at infinity

28

We say that a sequence (x n ) in X converges at infinity if d(x,,, y) ----+ 00 and by(x n ) converges in C(X) for some y E X. From (1.4) we conclude that this is independent of the choice of y. Two such sequences (x n ) and (x~) will be called equivalent if limn~CX) by(x n ) = limn~CX) by(x~) for one and hence any y E X. We denote by X (00) the set of equivalence classes. For any ~ E X (00) and y E X there is a well defined function f = b(~, y, .) E C(X), called the Busemann function at ~ based at y, namely f = limn~CX) by(x n ), where (x n ) represents ~. The sublevels f- 1(-00,a) of f are called horoballs and the levels horospheres (centered at O. Note that, by the above argument, X( 00) corresponds to the points in by(X) \ by (X). Hence the embedding by gives a topology on X (00) and X = X U X (00 ). Because of (1.4) this topology does not depend on the choice of y. With respect to this topology, the function b extends to a continuous function

b:XxXxX----+lR such that (1.2) and (1.4) still hold and b(x, y, y)

= 0 for

all x E X and y E X.

1.5 Remarks. (a) If X is locally compact, then X and X(oo) are compact. For this note that we can apply the Theorem of Arzela-Ascoli since by (x) is normalized by by (x) (y) = 0 and since by (x) has Lipschitz constant 1 for all x EX. (b) In potential theory, one uses Green's functions G(x,y) to define the Martin boundary in an analogous way. Instead of using differences b(x, y, z) = d(x,z) - d(x,y), one takes quotients K(x,y,z) = G(x,z)/G(x,y).

2. Closure of X via rays In this section, we add the assumption that X is a Hadamard space. We describe the construction of X (00) by asymptote classes of rays, introduced by Eberlein-O'Neill [EbON]. We recall that a ray is a geodesic (5 : [0, w) ----+ X, 0 < w ~ 00, such that (5 is minimizing and such that limt~w (5(t) does not exist. Since we assume that X is complete, we have w = 00 for any unit speed ray in X. As in Section 1.5 we say that two unit speed rays (51,(52 are asymptotic if d((51(t), (52 (t)) is bounded uniformly in t. This is an equivalence relation on the set of unit speed rays in X; the set of equivalence classes is denoted X (00 ). If ~ E X (00) and (5 is a unit speed ray belonging to ~, we write doo) =~. Recall that d((51(t),(52(t)) is convex in t. In particular, if (51 is asymptotic to (52 and (51(0) = (52(0), then (51 = (52. Hence for any ~ E X(oo) and any x E X, there is at most one unit speed ray (5 starting in x with (5 ( 00) = ~. Our first aim is to show that, in fact, for each x E X and each ~ E X (00) there is a unit speed ray (5 starting at x with d 00) = ~. (5 : [0,00) ----+ X be a unit speed ray and let ~ = d 00). Let x E X and for T > 0 let (5T : [0, d(x, (5(T))] ----+ X be the unit speed geodesic from x to (5(T). Then, for R > 0 and 10 > 0 given, we have, for 5, T sufficiently large,

2.1 Lemma. Let

d((5s(t), (5T(t))
0 and E > 0 given, we have, for T sufficiently large, d(CTx,~(t),

CTT(t)) < E,

0:::; t :::; R.

Proof. Let aT = La(T)(X, cr(0)). Then aT ----t 0 since d(CT(O), CT(T)) = T ----t 00. Hence, for a > 0 given, we have La(T) (x, CT(S)) 2': 7r - a for T large and S > T. Hence the assertion follows from a comparison with Euclidean geometry. D The above lemma together with our previous considerations shows that for any x E X and any ~ E X (00) there is a unique unit speed ray CT x,~ : [0,(0) ----t X with CTx,~(O) = x and CTx,~(oo) = ~. For y E X, y -I- x, we denote by CTx,y : [0, d(x, y)] ----t X the unique unit speed geodesic from x to y. On X = X u X(oo) we introduce a topology by using as a basis the open sets of X together with the sets U(x,~, R, E) =

where x E that

X,~ E

{z E Xlz tJ- B(x, R), d(CTx,z (R), CTx,dR)) < E},

X(oo). Recall that, by convexity,

d(CTx,AR),CTx,~(R))

< E implies

The following lemma is an immediate consequence of Lemma 2.1.

2.2 Lemma. Let x,y E X, ~,Tj E X(oo) and let R > 0, E > 0 be given. Assume Tj E U(x, C R, E). Then there exist S > 0 and 15 > 0 such that

U(y,Tj,S,I5)

C U(X,~,R,E).

D

This lemma shows that for a fixed x E X the sets U (x, ~, R, E) together with the open subsets of X form a basis for the topology of X. With respect to this topology, a sequence (x n ) in X converges to a point ~ E X (00) if and only if d(x,x n ) ----t 00 for some (and hence any) x E X and the geodesics CTx,x n converge to CTx,~. Note also that, for any x E X, the (relative) topology on X(oo) is defined by the family of pseudometrics dx,R, R > 0, where

(2.3) Our next aim is to show that X and X (00) are homeomorphic to the corresponding spaces in Section 1. To that end, let Xo E X and R > 1 be given and assume Xl,X2 E X satisfy d(xo,xd,d(xo,X2) > R. Let Yl = CT XQ ,Xl(R), Y2 = CTXQ ,X2(R) and assume d(Yl,Y2) 2': E. By comparison with the Euclidean triangle we get

Chapter II: The boundary at infinity

30

since YI is on the geodesic from Xo to Xl, hence

From the First Cosine Inequality we obtain (for

E

< 1)

and hence (2.4) where b(Xi,XO") is defined as in (1.1). From this we obtain the desired conclusion about X and X(oo). 2.5 Proposition. Let (x n ) be a sequence in X such that d(xo, x n ) -+ 00. Then b( X n , XO, .) converges to a Busemann function f if and only ifa xo .x" converges to a ray a = a xo,~' Furthermore, we have f = b", where b,,(x) := lim (d(a(t),x) - t). t--->oc

Proof. If b(x n , xo,.) -+ f, then b(J: n , xo,.) converges uniformly to f on B(xo, R) for any R > 1 (by the choice of topology on C(X)). By (2.4), the sequence (axo,x" (R)) converges; hence axo,x" converges to a ray a. For the proof of the converse, note that. for T > 0 and E > 0 given, there is an R > 0 such that Ib(y,xo,x) - b(z,xo,x)1

for all

X E

B(xo, r) if d(z, xo)


Rand y = axo,z(R).

E

0

2.6 Exercise. There is the following intrinsic characterization of Busemann functions in the case of Hadamard spaces: a function f : X -+ lR is a Busemann function based at Xo E X if and only if (i) f(xo) = 0; (ii) f is convex; (iii) f has Lipschitz constant 1; (iv) for any X E X and T > 0 there is a z E X with d(x,z) = rand f(x) - f(z) = r. This characterization of Busemann functions is a variation of the one given in Lemma 3.4 of [BGS]. It has been suggested that a proof of Exercise 2.6 is included. The interested reader finds it in Section 3 of Chapter IV. Exercise 2.6 will be used in the discussion of fixed points of isometries.

Classification of isometries

31

3. Classification of isometries Let X be a Hadamard space and let

is called the displacement function of is convex.

ip.

ip :

X

Since

----+

ip

X be an isometry. The function

maps geodesics to geodesics, d",

Definition 3.1. We say that ip is semisimple if d", achieves its mimimum in X. If is semisimple and min d", = 0, we say that ip is elliptic. If ip is semisimple and min d", > 0, we say that ip is axial. We say that ip is parabolic if d", does not achieve a minimum in X. ip

Proposition 3.2. An isometry alent conditions holds: (i) ip has a fixed point;

ip

of X is elliptic iff one of the following two equiv(ii)

ip

has a bounded orbit.

Proof. It is obvious that ip is elliptic iff (i) holds and that (i) :=} (ii). For the proof of (ii) :=} (i), let B be the unique geodesic ball of smallest radius containing the bounded orbit Y = {ipn(x) In E Z}, see Proposition 1.5.10. Now ip(Y) = Y, hence ip(B) = B by the uniqueness of B. Therefore ip fixes the center of B. D Proposition 3.3. An isometry ip of X is axial iff there is a unit speed geodesic 0" : lK ----+ X and a number to > 0 such that ip(O"(t)) = O"(t + to) for all t E K Such a geodesic 0" will be called an axis of ip. Now let ip be axial, let m", = mind", > 0 and set A = A", = {x E X I d",(x) = m",}. Then A is closed, convex and isometric to C x lK, where C c A is closed and convex. Moreover, the axes of ip correspond precisely (except for the parameterization) to the geodes'ics {c} x lK, c E C.

Proof. Suppose ip is axial and let x E A. Consider the geodesic segment p from x to ip(x) and let y be the midpoint of p. Then d(y, x) = d(y, ip(x)) = d(x, ip(x))j2. Since d(ip(y), ip(x)) = d(y, x), we get d(y, ip(y))

S d(y, ip(x)) + d(ip(x), ip(y)) S d(x, ip(x)).

Now x E A, hence d(y, ip(y)) = d(x, ip(x)). Hence the concatenation of p with ip(p) is a geodesic segment. Therefore 0"

=

UnEZipn (p)

is an axis of ip. Hence there is an axis of ip through any x E A. Clearly, any two axes of ip are parallel. Hence A is closed, convex and consists of a family of parallel geodesics. Therefore A is isometric to C x lK as claimed, where C is a closed and convex subset of X, sec Proposition 1.5.9. The other assertions are clear. D

Chapter II: The boundary at infinity

32

Propostion 3.4. If X is locally compact and if rp is a parabolic isometry of X, then there is a Busemann function f = b(C y,.) invariant under rp, that is, rp fixes ~ E X (00) and all horospheres centered at ~ are invariant under 'P.

Proof. Let m

=

inf d'{J 2':

o. For 8 > m set X 8 = {x E X

I d'{J(x)

~

8}.

Then X 8 is closed and convex since d'{J is convex. Moreover,

n8>mX8

=0

since rp is parabolic. Let Xo E X and set

1o(x)

=

d(x,X 8) - d(xo,X 8).

Then (i) f8(xo) = 0; (ii) 18 is convex; (iii) 10 has Lipschitz constant 1; (iv) for any x E X\X 8 and r > 0 with r ~ d(x,X 8) there is a z E X with d(x,z) = rand 1o(x) - 1o(z) = r. Now apply the Theorem of Arzela-Ascoli and Exercise 2.6. D

4. The cone at infinity and the Tits metric Throughout this section we assume that X is a Hadamard space. For X (00) we define the angle by L(~, TJ)

= sup xEX

~,TJ E

Lx(~, TJ)·

Note that L is a metric on X(oo) with values in [0,7f]. The topology on X(oo) induced by this metric is, in general, different from the topology defined in the previous sections; to the latter we will refer as the standard topology.

4.1 Proposition. (X (00 ), L) is a complete metric space. Furthermore, (i) if ~n ----> ~ with respect to L, then ~n ----> ~ in the standard topology; (ii) if ~n ----> ~ and TJn ----> TJ in the standard topology, then L(~,TJ) ~ liminfL(~n,TJn)' n----+CX)

Proof. Let (~n) be a Cauchy sequence in X(oo) with respect to L. Let x E X and let Un = ux,E;n be the unit speed ray from x to ~n' Given R > 0 and E > 0, there is N = N(E) EN such that L(~n, ~m) < E for all m, n 2': N. Then L(J"n(R)(X,~m)

2': 7f

-'L(J"n(R)(~n,~m)

2': 7f -

L(~n,~m)

> 7f -

E.

By comparison with the Euclidean plane, dx,R(~n, ~m)

= d(un(R), um(R))

~ 2Rtan

E

2'

Therefore (un) converges to a unit speed ray U and ~n ----> ~ := u(oo) with respect to the standard topology. Since N = N (E) does not depend on x and R, we also get ~n ----> ~ with respect to L. This proves (i) and that L is complete. The proof of (ii) is similar. D

The cone at infinity and the Tits metric

33

4.2 Proposition. Let~, 7) E X(oo). For x E X, let (J = (Jx,t; be the ray from x to~. Then Lcr(t) (~, 7)) and Jr - Lcr(t) (x, 7)) are monotonically increasing with limit L(~, 7)) as t --+ 00. If L(~, 7)) < Jr and Lx(~, 7)) = L(~, 7)),

then (J and the ray from x to 7) bound a fiat convex region in X isometric to the convex hull of two rays in the fiat plane with the same initial point and angle equal to L(~, 7)). Proof. Let r(t) = Lcr(t)(~,7)) and oo r(t). This proves the assertion about r(t). Now Jr -

00 (Jr - ;3 as t ----> 00. We also have

---->

SUPxEX

Ix we get

----> L(~, r]) and ,6. We conclude that

at - f3t

7r -

a, ,6t,E

---->

where Pt : [0,1] ---->]R;.2 is a line of length c(t) and d(x,pt(O)) = at, d(x,pt(l)) = bt. Now d(x, Pt (s)) It ----> e as t ----> 00, therefore . 1 ll1nsup -d(x, Pt(s)) a t2 ~

c

(t)2 + S2 -2~ t

by the Fin.,t Cosine Inequality. Since at

----> 0

-

c(t) () 2as - t cos at.

and c(t)lt

---->

c we conclude that

and hence that d(x,Pt(.'))lt ----> e at t ----> 00. We now prove that the geodesics CT s,t converge to a ray as claimed. Given c > 0, there is T 2: 0 such that

C(t)2

It2 -

c

2

I,

1

c(t) . c 2as - - cos at - 2asccos al < '2 t

for all t 2: T. Then we obtain for t

d2(CTo(at'), Pt(s)) t2

>

t;

> t' 2: T and r

= t'lt,

(a 2(t - t')2 + s2c(t)2 - 2a(t - t')sc(t) cos at)

> a 2(1- r)2 + s2 C2 - 2a(1- r)sccosa -

c.

Chapter II: The boundary at infinity

36

Now denote by at the angle in the Euclidean triangle with side lengths a, b, c(t)lt corresponding to at. Then at 2: at and at ---> a. By enlarging T if necessary, we also have

c(t) E 12as- cos at - 2asccosal < t 2 for all t 2: T. Since triangles in X are CATo we obtain, for t > t l 2: T, d 2 ( 0'0 ( at l ), Pt (s ))

t2

t~ (a 2(t -


oo

ci

~d2((Jo(at),(JI(et)) 2

t

s2 c2 + E2 or

= lim ~d2((Jl(bt),(JI(et)) 2 (1- s)2 c2 + E2. t--->oo

t

However, by the first part of the proof we have

L(~, (') = a

2

2

2

\eae - Co ,

L(

2 ,>/) = b T), "

+e

2

2

- c1 .

2be

Hence (**) implies c6 = s2 c2 and ci = (1- s)2 c2. We conclude (= ('. As for the last claim, let

Then O'(t) + (3(t) 2 7r. By comparison with Euclidean geometry, applied to the triangles t:.(Pt (s), x, (Jo (at)) and t:.(Pt (s), x, (Jl (bt)) we get limsupO'(t):::;

0"

and

Now

0"

+ (3'

=

7r,

hence the claim.

lim sup (3(t) :::; (3'. t---+cx:::>

t---+CXJ

D

4.5 Exercise. Use the notation of Theorem 4.4 and assume that e i=- 0. Let F be a point on the edge of length a on the Euclidean triangle such that d( C, F) = ra for some r E (0,1). Prove: 1

d(E, F) = lim -d(pt(s), (Jo(rat)) t--->oo t and corresponding assertions about the angles at F. 4.6 Corollary. If~,T) E X(oo) are points with L(~,T)) < 7r, then there is a unique minimizing geodesic in X(oo) (with respect to L) from ~ to T).

Proof. The existence of a geodesic from ( to T) follows from Proposition 1.1.5 and Theorem 4.4, where we put a = b = 0, s = 1/2. The uniqueness follows from the uniqueness of the point ( in Theorem 4.4. D The cone at infinity, Coo X , is the set [0,(0) x X(oo)/ "', where {O} x X(oo) shrinks to one point, together with the metric d oo defined by

(4.7)

Chapter II: The boundary at infinity

38

That is, (Ccx;X, d oo ) is the Euclidean cone over X (00). We may also think of CooX as the set of equivalence classes of geodesic rays, where we do not require unit speed and where we say that geodesic rays (Jo and (J1 are equivalent if d((JO(t),(J1(t)) is uniformly bounded in t. In the notation of Theorem 4.4 we have

d oo ((a , 0, (b , ry))

=

1

lim -d((Jo(at), (J1 (bt)).

t~oo

t

In this sense, the geometry of X with respect to the rescaled metric d/t converges to the geometry of CooX as t ---+ 00. Given :r E X and a sequence of geodesics (In : [0, t n ] ---+ X with speed Un 2: 0 and (In(O) = X, we say that (In ---+ (u , O if t n ---+ 00 , if Un ---+ a and, if a i=- 0, if (In(t n )

---+~.

Theorem 4.8. (Coo X , d oo ) is a Hadamard space.

Proof. Note that CooX is simply connected since we can contract CooX onto the distinguished point Xoo = [{O} x X(oo)]. The metric doo is complete since L is a complete metric on X (00 ). We show now that doo is a geodesic metric. To that end , let (a, 0, (b , T)) E Coo X . We want to show that there is a minimizing geodesic between these two points. The only non-trivial case is U , b > 0 and 0 < L(~, T)) < 7f. Choose X E X and let (Jo, (J1 be the unit speed rays from :r with (Jo (00) = ~ and (J1 (00) = ry. Then, in the notation of Theorem 4.4, the geodesics (Js,t converge to geodesic rays (Js of speed e s > 0, 0 ::; s ::; 1, and, by Theorem 4.4 and the definition of doo1 the curve p(s) = (es,(JS(00)),0::; s::; 1, is a minimizing geodesic form (a,~) to (b,ry). It is immediate from Corollary 4.6 (or Theorem 4.4) that P is the unique minimizing geodesic from (a,~) to (b,ry). It remains to show that CooX is nonpositively curved. By Lemma 1.3.4, it suffices to consider triangles in CooX with minimizing sides. Let D. be such a triangle and let (Ui 1 ~i) 1 1 ::;i ::; 3, be the vertices of D.. The only non-trivial case is ai > 0 and 0 < L (~i 1 ~j) < 7r for i i=- j. Recall that the minimal geodesics Pi: [0 , 1] ---+ CooX from (ail~i) to (ai+11 (i+d are unique (indices are counted mod 3). Let (e1,(d = pi(sd and (e21 (2) = Pj(S2)' where i i=- j and 0 < 131,132 < l. Let (J1 and (J2 be the unit speed rays from a given point .r E X to (1 and (2 respectively. By Theorem 4.4,

Let Crk be the unit speed ray from x to ~k:, 1 ::; k ::; 3. If Pk.t : [0,1] ---+ X denotes the geodesic from Crk(Ukt) to Crk+1(ak+1t) 1 and if Cru respectively Cr2,t are the geodesics from x with Cr1,t(e1t) = Pi,t(sd respectively Cr2,t(e2t) = Pj,t(S2), then , by Theorem 4.4 ,

The cone at infinity and the Tits metric

39

For t large, the triangle (PI,t, P2,t, P3,t) has sides of length approximately tlk, where lk = L(Pk)' Now let (7h ,75 2 , (3) be the Euclidean comparison triangle of (PI, P2, P3) andlct dE = d(Pi (sd, Pj (S2))' Since X is a Hadamard space we conclude

where

E:

> 0 is given and t is sufficiently large. Hence 1 1 I/(uI(eIR), u2(e2 R )) = t~~ lid(o-l,t(eIR), o-2,t(e2 R )) 1 ::; lim sup -t d(pi,t(sd, Pj,t(S2)) ::; dE. t~'X;

It follows that triangles in CocX are CATo and hence that CocX is a Hadamard

space.

D

4.9 Corollary. (X (00), L) has curvature::; 1. More pr'ecisely, (i) geodesics in X (00) of length < 1T are minimal; (ii) triangles in X (00) of perimeter < 21T are C ATI .

PT'Oof. The relation between distances in C ocX and angles in X (00) is exactly as the relation between distances in Euclidean space E 3 and angles in the unit sphere 52. D 4.10 Lemma. Let X be a locally compact Hadamard space. In the notation of Theorem 4.4 let a = b = 1 and assume that there is no geodesic in X fT'Om ~ to T/. Let Pt(St) be the point on Pt closest to :r. Then d(x, pt(sd) --+ 00. If ( E X(oo) is an accumulation point of Pt (.'id for t --+00, then L(~,()

= L((,r/) = ,/2.

PT'Oof. Since X is locally compact and since there is no geodesic from ~ to r/ we have d(x,Pt) --+ 00 for t --+ 00. From Theorem 4.4 we conclude d(x,pt(l/2))/t --+ cos(,/2) < 1 and hence St # 0, 1. Therefore

Since (in the notation of Theorem 4.4)

Qt,

f3 t

--+

(1T -,)/2 we get

The triangle inequality implies that we must have equality in these inequalities.

D

Following Gromov, we denote by Td, the Tits metric, the interior metric on X (00) associated to the angle metric L. The name derives from its close relationship with the Tits building associated to symmetric spaces. By definition, we have Td(~,T/) 2: L(~,T/) for all ~,T/ E X(oo).

Chapter II: The boundary at infinity

40

4.11 Theorem. Assume that X is a locally compact Hadamard space. Then each connected component of (X(oo), Td) is a complete geodesic space with curvature :::; 1. Furthermore, for ~,7) E X (00) the following hold: (i) if there is no geodesic in X from ~ to 7), then Td(~, 7)) = L(~, 7)) :::; Jr; (ii) if L(~, 7)) < Jr, then there is no geodesic in X from ~ to 7) and there is a unique minimizing Td-geodesic in X(oo) from ~ to 7); Td-triangles in X of perimeter < 2Jr are CAT!; (iii) if there is a geodesic a in X from ~ to 7), then Td(~, T)) 2: Jr, and equality holds iff a bounds a fiat half plane; (iv) if (~n), (7)n) are sequences in X (00) such that ~n -+ ~ and 7)n -+ 7) in the standard topology, then Td(~, 7)) :::; lim infn-->CXJ Td(~n, T)n). Proof. The first assertion in (ii) and (iii) respectively is clear since L(~, 7)) = Jr if there is a geodesic in X from ~ to 7). If there is no geodesic in X from ~ to 7), then, by Lemma 4.10, there is a midpoint (in X(oo) between them with respect to the L-metric. Now the L-metric on X(oo) is complete, see Proposition 4.1. Hence we can construct an L-geodesic from ~ to 7) by choosing midpoints iteratively. Since Lgeodesics are Td-geodesics, we obtain (i). The uniqueness assertion in (ii) follows from the corresponding uniqueness assertion of midpoints in Theorem 4.4. The completeness of Td also follows since the L-metric is complete, see Proposition 4.1. We now prove (iv) and that each connected component of (X(oo), Td) is geodesic. By the corresponding semicontinuity of L, see Proposition 4.1, we conclude from (i) that (iv) holds if liminfn-->CXJ Td(~n, 7)n) < Jr. From (ii) we also get that for any pair of points ~,7) E X(oo) with Td(~,7)) < Jr there is a minimizing Td-geodesic from ~ to 7). By induction we assume that this also holds when we replace Jr by h, k 2: 1. Now suppose lim infn-->CXJ Td(~n, 7)n) < (k + l)Jr. By th definition of Td, there is a point (n almost half way between ~n and 7)n. Passing to a subsequence if necessary, we assume (n -+ ( in the standard topology and obtain, by induction,

+ Td((, 7)) :::; liminfTd(~n, (n) + liminfTd((n, 7)n)

Td(~, 7)) :::; Td(~, () n~~

In particular,

Td(~, 7))

n~~

:::; liminfTd(~n, 7)n). n~~

< (k + l)Jr. There exists (another) sequence ((n) such that

hence, by the semicontinuity, there is a point ( with Td(~, () = Td(7), () = Td(~, 7))/2.

By the inductive hypothesis, there are minimizing Td-geodesics from ~ to ( and ( to T). Their concatenation is a minimizing Td-geodesic from ~ to 7). Hence we conclude (iv) and that the components of (X(oo),Td) are geodesic spaces.

The cone at infinity and the Tits metric

41

As for the statement about flat half planes in (iii), let ( be a point with = Td((,T/) = 7f/2. Since

Td((,~)

for any x on the geodesic (in X) from

~

to T/ and Lx ::; Td, we have

for all such x. Now the rigidity part of Lemma I.5.8(i) applies. The claims about the curvature bound and about triangles in (ii) are immediate from Corollary 4.9. 0

CHAPTER III

Weak Hyperbolicity One of our main objectives in this chapter is the discussion of transitivity properties of the geodesic flow and related topics. The first two sections are greatly influenced by the work of Eberlein on geodesic flows in [Eb2, Eb3]. The third section deals with the main results of the author's thesis (published in [Bal]) , but here in the context of Hadamard spaces instead of Hadamard manifolds. The fourth section deals with applications to harmonic functions and random walks on countable groups, compare [Ba3, BaLl]. Many of the ideas in Section 4 go back to Furstenberg [Ful, Fu2, Fu3]. Let X be a Hadamard space. We say that X is geodesically complete if every geodesic segment of X is part of a complete geodesic of X, that is, a geodesic which is defined on the whole real line. For X geodesically complete, we denote by X the set of unit speed geodesics of X, equipped with the topology of uniform convergence on bounded subsets. The geodesic flow (gt) acts on ex by reparameterization, l(a)(s) = a(s + t).

e

If X is a smooth manifold, then

ex

----+

SX; a

----+

&(0)

defines a homeomorphism from ex to the unit tangent bundle SX of X. With respect to this homeomorphism, the geodesic flow above corresponds to the usual geodesic flow on SX (also denoted gt),

l(v) = 1v(t), v

E

SX,

where Iv denotes the geodesic in X with 1v(0) = v. 1. The duality condition

In our considerations it will be important to have a large group of isometries. Let X be a Hadamard space and r a group of isometries of X. Following Eberlein [Eb2]' we say that ~,T) E X (CXl) are r -dual if there is a sequence ('Pn) in r such that 'Pn(x) ----+ ~ and 'P~l(X) ----+ 77 as n ----+ CXl for some (and hence any) x E X.

Chapter III: Weak hyperbolicity

44

Duality is a symmetric relation. The set of points f -dual to a given point in X (00) is closed and f-invariant. We say that f satisfies the duality condition if for any unit speed geodesic a : lR -+ X the endpoints a( -00) and a( 00) are f-dual. The duality condition was introduced by Chen and Eberlein [CE1]. The condition is somewhat mysterious, at least as far as Hadamard spaces are concerned (as opposed to Hadamard manifolds). May be it is only reasonable for geodesically complete Hadamard spaces. 1.1 Lemma [Eb2J. Let X be a geodesically complete Hadamard space, and let a, p : lR -+ X be unit speed geodesics in X. Let (CPn) be a sequence of isometries of X such that cP;; 1(x) -+ a( 00) and CPn (x) -+ p( -00) as n -+ 00 for some (and hence any) x E X. Then there are sequences (an) in ex and (t n ) in lR such that an -+ a, t n -+ 00 and CPn 0 gt n (an) -+ p as n -+ 00. 1.2 Remarks. (a) The converse to Lemma 1.1 is clear: let a, p E ex and suppose there are sequences (an) in ex, (t n ) in lR and (CPn) of isometries of X such that an -+ a, t n -+ 00 and CPn 0 gtn(an ) -+ a as n -+ 00. Then cp;;l(X) -+ a(oo) and CPn(x) -+ p(-oo) as n -+ 00 for one (and hence any) x E X. (b) Note that the assumption on the sequence (CPn) in Lemma 1.1 only specifies a( 00) and p( -00). Thus we may replace a by any asymptotic a' E ex and p by any negatively asymptotic p' E ex, keeping the sequence (CPn) fixed. Since an -+ a and cp;;l 0 gtn(an ) -+ p we have

tn-d(a(O),cp;;,-l(p(O))) -+0

as

n-+oo.

For a' asymptotic to a we also have

d(a'(O),cp;;,-l(p(O))) - d(a(O),cp;;,-l(p(O))) -+ b(a(oo),a(O),a'(O)) as n

-+

00. Hence we may replace the sequence (t n ) by the sequence

(t~.)

given by

t~ = tn+b(a(oo),a(O),a'(O)), nEN,

when replacing a by a'. A similar remark applies to geodesics p' E asymptotic to p.

ex negatively

Proof of Lemma 1.1. For m ;::: 0 and n E N let an,m : lR -+ X be a unit speed geodesic with an,m( -m) = a( -m)

and

an,m(tn,m

+ m)

= cp;;,-l(p(m)) ,

where

tn,m = d(a(-m),cp;;,-l(p(m))) - 2m. Set N(O) 1 and define N(m), m > 2, recursively to be the smallest number > N(m - 1) such that 1 1 and d(CPn 0 an,m(tn,m - m), p( -m)) :::: m m for all n ;::: N(m). Note that such a number N(m) exists since cp;;l(p(m)) -+ a(oo) and CPn(a( -m)) -+ p( -00). Now set an := an,N(m) for N(m) :::: n < N(m+ 1). D

d(an,m(m), a(m))
X. Let ('Pn) be a sequence of isometries of X such that 'Pn (x) ---> ~ and 'P~ I (x) ---> ( E X (00) as n ---> 00. Then 'P~l (7)) ---> ( as n ---> 00.

Proof. Let x

= dO).

Then Lx(~, 'Pn(x))

--->

0 and hence

By comparison with Euclidean geometry we get, for the metric dx,R as in (11.2.3),

where R > 0 is arbitrary. Since 'P~I(X)

--->

(we conclude 'P~l(rl)

---> (.

D

1.9 Proposition. Suppose X is geodesically complete and f satisfies the duality condition. Let ~ E X(oo). Then f operates minimally on the closure f(~) of the f-orbit of~, that is, f(() = f(~) for any ( E [(0.

Proof. It suffices to show that ~ E [(() for any ( E f(O. Since X is geodesically complete, there are geodesics (J, P : ]R;. ---> X with (J( (0) = ~ and p( (0) = (. Let = (J( -(0) and (' = p( -(0). Since f satisfies the duality condition, is f-dual to ~ and (' is f-dual to (. Now ( is f-dual to because the set of points f-dual to is f-invariant and closed. By Lemma 1.8 this implies that E [(('). Choose a sequence ('Pn) in f with 'Pn ((') ---> For rn > 0, let (J m be a complete geodesic with (Jrn(O) = (J(rn) and (Jrn(~oo) = 'Pn(('), where n = n(rn) 2: rn is chosen so large that d ((J m, (J (0)) :s; 1. Then (Jrn ((0) ---> ~ and (J m ( - (0) is f-dual to (Jm(oo). Therefore (Jm(oo) is f-dual to (' = 'P~I((Jm(-oo)). Hence (' is f-dual to ~ = lim m---> 00 (Jm(oo). Now Lemma 1.8 implies ~ E [((). D

e

e

e

e

e

e.

1.10 Remark. The duality condition has other useful technical properties. (1) Let Xl and X 2 be Hadamard spaces with metrics d l and d2 respectively and let X be the Hadamard space Xl x X 2 with the metric d = Jdr + d~. If f is a group of isometries of X satisfying the duality condition such that any 'P E f is of the form 'P = ('PI, 'P2), where 'Pi is an isometry of Xi, i = 1,2, then

and the corresponding group f 2 satisfy the duality condition. This is clear since geodesics in X are pairs ((JI, (J2), where (JI is a geodesic of Xl and (J2 is a geodesic

Geodesic flows on Hadamard spaces

47

of X 2 . See [Ebl0,16] for applications of this property in the case of Hadamard manifolds. Note that r 1 and/or r 2 need not act properly discontinuously even if r does. Thus a condition like cocompactness + proper discontinuity does not share the above property of the duality condition. (2) Clearly the duality condition passes from a group r of isometries of a Hadamard space X to a larger group of isometries of X. Vice versa, if X is geodesically complete and separable and if t. is a subgroup of finite index in r, then t. satisfies the duality condition if r does [EblO]. To see this, let a E GX be a rrecurrent geodesic. Choose sequences (t n ) in lR and ('Pn) in r such that t n ---> 00 and 'Pn 0 gt n (a) ---> a. Now t. has finite index in r. Hence we may assume, after passing to a subsequence, that 'Pn = 'l/Jn'P, where 'P E r is fixed and 'l/Jn E t. for all n E N. Then and

arn,n = ('l/Jrn'I/Jr~l)ogtm-tn(a~) = 'Prnogtm(a) --->a. Choose rn = rn(n) convenient to get that am(n),n ---> a. This shows that a is nonwandering mod t.. Now the set of geodesics in GX which are nonwandering mod t. is closed, and the set of r-recurrent geodesics in GX is dense in GX, see Corollary 1.4. Hence t. satisfies the duality condition. We come back to the beginning of this section. We said there that we will be in need of a large group of isometries, and we made this precise by introducing the duality condition. One might think that the isometry group of a homogeneous Hadamard manifold X is large; however, it is not in our sense, except if X is a symmetric space (sec [Ebl0], Proposition 4.9).

1.11 Definition/Exercise/Question. Let X be a Hadamard space and r a group of isometrics of X. The limit set L(r) c X (00) of r is defined to be the set of all ~ E X (00) such that there is a sequence ('Pn) in r with lim n -+= 'Pn (x) = ~ for one (and hence any) x E X. Show L(r) = X(oo) (i) if there is a bounded subset B in X with UcpEl'P(B) = X or (ii) if X is geodesically complete and r satisfies the duality condition. In many results below we assume that X is geodesically complete and that r satisfies the duality condition. Can one (you) replace the duality condition by the assumption L(r) = X (00). 2. Geodesic flows on Hadamard spaces 2.1 Definition. Let X be a geodesically complete Hadamard space and r a group of isometrics of X. We say the geodesic flow of X is topologically transitive mod r if for any two open non-empty subsets U, V in G X there is t E lR and 'P E r such that gt(U) n 'P(V) -# 0.

Chapter III: Weak hypcrbolicity

48

2.2 Remark. If X is separable, then the topology of X, and therefore the topology of ex, has a countable base. Hence the geodesic flow is topologically transitive mod f if and only if it has a dense orbit mod f: there is a a in ex such that for any a ' in ex there are sequences (t n ) in JR and If!n in f with If!nlna ---+ a ' . After all our preparations we are ready for the following result from [Eb3]. 2.3 Theorem. Let X be a geodesically complete, separable Hadamard space, and let f be a group of isometries of X satisfying the duality condition. Then the following are equivalent: (i) gt is topologically transitive mod f; (ii) for some ~ E X(oo), the orbit r(~) is dense in X(oo); (iii) for every ~ E X(oo), the orbit f(O is dense in X(oo); (iv) X (00) does not contain a non-empty f -invariant proper closed subset. Proof. Clearly (iii) {::} (iv) and (iii) =} (ii). The implication (i) =} (ii) is immediate from Remark 2.2 above. The implication (ii) =} (iii) follows from Proposition 1.9. It remains to prove (ii) =} (i). We let U(oo) respectively V(oo) be the set of points a(oo) in X(oo) with a E U respectively V. Then U(oo) and V(oo) are open and non-empty. Applying (ii) we can assume U(oo) C V(oo). Let a E U. Then there is a geodesic p E V with p(oo) = a(oo). Now Corollary 1.3 applies. D 2.4 Theorem. Let X be a geodesically complete and locally compact Hadamard space, and let f be a group of isometries of X satisfying the duality condition. Then the following are equivalent: (i) the geodesic flow (gt) of X is not topologically transitive mod f; (ii) X (00) contains a non-empty f -invariant proper compact subset; (iii) the geodesic flow (gt) of X admits a non-constant f -invariant continuous first integral f : ex ---+ JR such that f (a) = f (p) for all a, pEeX with a(oo) = p(oo) or a( -(0) = p( -(0). Each of these conditions implies that every geodesic of X is contained in a flat plane, that is, a convex subset of X isometric to the Euclidean plane. Proof. X is separable because X is locally compact. Hence (i) {::} (ii) by Theorem 2.3. The implication (iii) =} (ii) is clear since we assume the integral to be continuous, non-constant and f-invariant and since X(oo) is compact. It remains to show (ii) =} (iii) and the existence of the flat planes. To that end, let C C X(oo) be a non-empty f-invariant proper compact subset. For a E ex we set (2.5)

f(a) = minL,,(o)(a(oo),(). (EC

We claim that the function f is an integral of the geodesic flow as asserted. Note that f is f-invariant since Cis.

Geodesic flows on Hadamard spaces

49

The continuity of angle measurement at a fixed point, see 1.3.2, implies that for any a E GX there is ( E G with f(a) = Lo-(O)(a(oo),(). Obviously, or by Proposition 11.4.2, we have that f(gt(a)) is monotonically not decreasing in t. Suppose first that a E GX is r-recurrent. Choose sequences (t n ) in lH. and «Pn) in r such that t n ----+ 00 and 'Pn 0 gtn (a) ----+ a. By the r -invariance of f and the monotonicity we have

The semicontinuity of angle measurement, see 1.3.2, implies liminf f('Pn n--+oo

0

in(a)) ~ f(a).

We conclude that f (gt (a) = f (a) for all t ;:::: 0, and hence for all t E lH. since gS (a) is r-recurrent for all S E R We also conclude that there is a point ((a) E G such that

f(i(a)) = Lo-(t)(a(oo),((a)) = L(a(oo),((a)) = minL(a(oo),(). (EC

rt

In particular, a bounds a flat half plane Ho- with ((a) E Ho-(oo) if a(oo) G. Now let a E GX be arbitrary. Let (an) be a sequence of r-recurrent unit speed geodesics converging to a. Passing to a subsequence if necessary, the corresponding half planes Ho- n and points ((an) converge to a flat half plane Ho- along a respectively a point ((a) E Ho-(oo). We conclude

f (a) = lim f (an) = L (a (00 ), (( a)) . n--+=

Therefore f is continuous; now Lemma 1.1 implies that f(a) = f(p) if a(oo) p( 00) respectively if a( -00) = p( -00); f is non-constant since G is proper. Let a E GX and let Ho-, ((a) be as above. Let Pt : lH. ----+ X be a unit speed geodesic with Pt(O) = a(t) and Pt(oo) = ((a). For S E lH., let as : lH. ----+ X be a unit speed geodesic as(t) = Pt(s) and as(oo) = a(oo). Then clearly

f(a)

=

f(a s )

=

Lo-s(t)(((a), a 8 (00)).

Therefore ptl [s, 00) and as I [t, 00) span a flat cone Gs,t. This cone contains aUt, 00)) if s < O. Now X is locally compact. Hence we obtain a flat plane containing a as a limit of the cones GSn,t n for some appropriate sequences Sn ----+ -00 and t n ----+ -00. 0 2.6 Problem. Suppose X is geodesically complete and irreducible with a group r of isometries satisfying the duality condition. Assume furthermore that every geodesic of X is contained in a flat plane. Claim: X is a symmetric space or a Euclidean building. This is known in the smooth case, which excludes the case of Euclidean buildings. In Chapter IV we will discuss the proof. In the singular case, if dim X = n and every geodesic of X is contained in an n-f1at, that is, a convex subset of X isometric to n-dimensional Euclidean space, then X is a Euclidean building [KI].

Chapter III: Weak hypcrbolicity

50

3. The flat half plane condition In this section we assume that X is a locally compact Hadamard space. Then, if a : III --> X is a geodesic which does not bound a flat half plane, there is a constant R > 0 such that a does not bound a flat strip of width R.

3.1 Lennna. Let a : III --> X be a unit speed geodesic which does not bound a fiat strip of width R > O. Then there are neighborhoods U of a( -(X)) and V of a( (0) in X such that for any ~ E U and T) E V there is a geodesic from ~ to T), and for any ,mch geodesic a' we have d( a', a(O)) < R. In particular, a' does not bound a fiat strip of width 2R. Proof. If the assertion of the lemma docs not hold, then there are sequences (x n ) and (Yn) in X with X n --> a( -(X)) and Yn --> a( (0) such that the unit speed geodesic segment an from X n to Yn, parameterized such that an(O) is the point on an closest to z:= a(O), satisfies d(an(O),z) :::: R for all n E N. Let Zn be the point on the geodesic from z to an(O) with d(z, zn) = R. By the choice of an(O), 7f

"2'

Lan(O)(z, x n ), Lan(O)(z, Yn) ::::

Hence also

7f

Lzn(z,xn ), L zn (Z,Yn)::::"2'

Passing to a subsequence if necessary we assume Zn and

-->

z=. Then d(z, z=)

=

R

We also have Lz(a( -(0), z=)

+ Lz(a(oo), zoe) >

7f.

Applying Proposition I.5.8(i) we conclude Lz(a(-oo), zoe)

7f

= Lz=(a(-oo),z) = Lz(a(oo), zoe) = Lz=(a(oo),z) ="2'

Hence the unit speed rays w_ respectively w+ from z= to a( -(X)) and a( (0) respectively span flat half strips with al (-00,0] respectively al [0, (0), sec Proposition I.5.8(i). In particular, a( -t) respectively a(t) is the point on a closest to w_(t) respectively w+(t), t > O. Hence d(w_(t),w+(t)) :::: 2t, and hence w_ and w+ combine to give a geodesic w from a( -(X)) to a( (0) through z=. Therefore w is parallel to a and wand a span a flat strip of width R. This is a contradiction. 0

3.2 Lemma. Let a : III

--> X be a unit speed geodesic which does not bound a fiat half plane. Let ('Pn) be a sequence of isometries of X such that 'Pn (x) --> a( (0) and 'P~1 (x) --> a( -ex)) for one (and hence any) x EX. Then fOT n sufficiently large, 'Pn has an a.Tis an such that an(oo) --> a(oo), an(-oo) --> a(-oo) as n --> 00.

Proof. There is an R> 0 such that a docs not bound a flat strip of width R. Let x = a(O) and choose c > 0, r > 0 such that U

=

U(x,a(-oo),r,c) and V

=

U(x,a(oo),T,c).

The flat half plane condition

51

are neighborhoods as in Lemma 3.1. By comparison with Euclidean geometry we find that there is an R' > r such that for any y E X we have

d(CJ(r), CJx,y(r)) < c or ifd(y,CJ)

d(CJ( -r), CJx,y(r)) < c

~Randd(y,x)

'2R'.

By the definition of U and V this means that y E U U V if d(y, CJ) ~ Rand

d(y, x) '2 R'. Assume that l{!n, for any n in a subsequence, fixes a point X n E X outside U U V with d(x,x n ) '2 2R'. Let CJ n = CJx,x n be the unit speed geodesic (or ray respectively if X n E X (00 )) from x to x n . Then the displacement function d( l{!n (z), z) is monotonically decreasing along CJ n . Let Yn = CJn(R') and y~ = CJ n (2R'). Then

L Yn (y~, l{!n (Yn)) + L Yn (y~, l{!;; 1 (Yn)) L Yn (y~, l{!n(Yn))+L'Pn(Yn) (l{!n(Y;,), Yn) ~

7r

since

(consider the quadrangle spanned by Yn,Y;"l{!n(Yn),l{!n(Y~)).Now

and therefore

We also have

and hence rp,,(Yn) ----+ CJ(oo) and l{!;;l(Yn) ----+ CJ(-oo) (for the given subsequence). After passing to a further subsequence if necessary, we can assume Yn ----+ y. Then d(y,CJ) '2 R by (*) since Y tt U U V and d(y,x) = R'. By (**), the rays from Y to CJ( -(0) respectively CJ( (0) combine to give a geodesic parallel to CJ. Therefore CJ bounds a fiat strip of width R, a contradiction. Hence we obtain that, for n sufficiently large, any fixed point of l{!n is contained in U U V U B(x, 2R'). Using Lemma 3.1 it is now easy to exclude that l{!n is elliptic or parabolic (for this compare Proposition 11.3.4) for all n sufficiently large. For large n, the endpoints of an axis CJ n of l{!n have to be in U U V by what we said above. It is clear from Lemma 3.1 that not both endpoints of CJ n are in one of the sets, U or V. Hence the lemma. 0

Chapter III: Weak hyperbolicity

52

3.3 Lemma. Let rp be an isometry of X with an axis u : lR - t X, where u is a unit speed geodesic which does not bound a fiat half plane. Then (i) for any neighborhood U of u( -(0) and any neighborhood V of u( (0) in X there exists N E N such that rpn(x \ U)

c

V,

rp~n(x

\ V)

c

U

for all n 2: N ;

(ii) for any ~ E X (00) \ {u( oo)} there is a geodesic UE, from ~ to u((0), and any such geodesic does not bound a fiat half plane. For K C X (00) \ {u( 00)} compact, the set of these geodesics is compact (modulo parameterization).

= u(O). We can assume that

Proof. Let x

U

= U(x,u(-oo),R,c) and V = U(x,u(oo),R,c).

By comparison with Euclidean geometry we find that rp-l(U)CU

and

rp(V)cV.

If (i) is not true, there is a sequence nk - t 00 and, say, a sequence (Xk) in X \ U such that rpn k (Xk) ¢ V. Passing to a subsequence, we may assume Xk - t ~ E X \ U. From (*) we conclude rpm(Xk) ¢ V for O:S m:S nk, and hence we get

In particular, ~ E X (00). Now the set K of all ~ E X ((0) \ U satisfying (**) is compact and invariant under rpm, m 2: 1. By our assumption K i= 0. Now choose ~ E K such that La(o) (u(oo),O is minimal. Since rp(u(oo)) = u(oo),

where a

>

°

is the period of rp. By the choice of

~

we conclude

hence the rays from u(O) and u(ma) to rpm(o together with ul[O, ma] bound a flat convex region. Since m is arbitrary and since rp shifts u, we conclude that u bounds a flat half plane, a contradiction. Thus (i) follows. Now (ii) is an immediate consequence of (i) and Lemma 3.1. 0 Let f be a group of isometrics of the Hadamard space X. We say that a non-constant geodesic u in X is f-closed if it is an axis of an axial isometry rp E f.

The flat half plane condition

53

3.4 Theorem. Assume that X is a locally compact Hadamard space containing a unit speed geodesic a : ~ --+ X which does not bound a flat half plane and that r is a group of isometries of X satisfying the duality condition. Then we have: If X (=) contains at least three points, then X (=) is a perfect set: for any ~ E X (= ), there is a sequence (~n) in X (= ) \ {O such that ~n --+ ~. Furthermore, for any two non-empty open subsets U, V of X(=) there is a 'P E r with

'P(X(=) \ U) c V, 'P-I(X(=) \ V) c U. More precisely, there is a r -closed geodesic a with a( -=) E U and a( =) E V such that a does not bound a flat half plane. Proof. By Lemma 3.2, there is an isometry 'Po in r with an axis ao which does not bound a flat half plane. Since X (=) contains more than two points, there is a point ~ E X(=) \ {ao(±=)}. By Lemma 3.3, 'P~n(~) --+ ao(±=), hence the first assertion holds for ao (±=). Again by Lemma 3.3, there is a geodesic al from ao( -=) to ~ and this geodesic does not bound a flat half plane. By Lemma 3.2, any two neighborhoods of the endpoints of al contain endpoints of an axis a~ of an isometry 'PI in r and a~ does not bound a flat half plane. Then a sufficient high power of 'PI maps ao (+=) into the given neighborhood, hence the first assertion also holds for ~. If U, V are non-empty and open in X (=), there is, by the first step, a point ~ E U - ao (=). By Lemma 3.3, there is a geodesic half plane. By Lemma 3.2 there is a f -closed geodesic a2 with a2 (-=) E U such that a2 does not bound a flat half plane. Again by the first part, there is a point 7] E V - a2 ( -=) and, by Lemma 3.3, a geodesic a3 from a2 (-=) to 7] which does not bound a flat half plane. By Lemma 3.2, there is a f-closed geodesic a as claimed. D 3.5 Theorem. Let X and f be as in Theorem 3.4. Suppose X (=) contains at least three points. (i) If X is geodesically complete, then the geodesic flow is topologically transitive mod f. (ii) f contains free non-abelian subgroups.

Proof. (i) This is immediate from Theorem 2.4. Alternatively we can argue as follows: Theorem 3.4 and Lemma 3.3 imply that the f-orbit of any point ~ E X(=) is dense in X (=). Hence the geodesic flow of X is topologically transitive mod f by Theorem 2.3. (ii) By the first assertion of Theorem 3.4, X (=) contains non-empty open subsets WI, W 2 with

By the second assertion of Theorem 3.4 and Lemma 3.3, f contains isometries

'PI, 'P2 with axis aI, a2 which do not bound flat half planes such that ai (±=) E Wi and

Chapter III: Weak hyperbolicity

54

for all k =1= o. Let ~ E X(oo) \ (WI U W 2 ). It follows from (*) and (**) that a non-trivial word w in 0 small enough. Now there is a k 2: 1 such that J1k ('P) = a > O. We split r into three disjoint parts G,L and {'P}, where G is finite and J1k(L) sup IfI < a8/2. Since h f is J1k-harmonic we get lim sup Ihf('Pnk)1 k---'ooo

= limsup!L: hf('Pnk 'l/J)J1k('l/J)J k---'ooo

r

:::; limsuplL: hf('Pnk 'l/J)J1k('l/J)!+ lim sup Ihf('Pnk'P)la + a8/2 k---'ooo

G

k---'ooo

< J1(G) ·8+ a8/2 + a8/2 :::; (1 - a)8 + a8 = 8. This is a contradiction.

D

We now consider the left-invariant random walk on r defined by J1; that is, the transition probability from 'P E r to 'l/J E r is given by J1('P- 1 'l/J). For 'l/Jl, ... ,'l/Jk E r, the probability P that a sequence ('Pn) satisfies 'Pi = 'l/Ji, 1 :::; i :::; k, is by definition equal to

Since the support of J1 generates r and since r is not amenable (r contains a free subgroup), random walk on r is transient, see [Fu3, p.212], that is, d(x, 'Pnx) -7 00 for P-almost any sequence ('Pn) cr. An immediate consequence of Theorem 4.10 and the Martingale Convergence Theorem is the following result. 4.11 Theorem. Let X, rand J1 be as in Theorem 4.10. Then for almost any sequence ('Pn) in r, the sequence ('Pnx), x EX, tends to a limit in X (00). The hitting probability is given by v. In particular, the stationary measure l/ is uniquely determined.

Proof. If h is a bounded J1-harmonic function, then Md('Pn)) := h('Pk), k 2: 1, defines a martingale and hence M k(('Pn)) converges P -almost surely by the Martingale Convergence Theorem, see [Fu3]. Let ~ i= 'rJ E X(oo) and let U, V be neighborhoods of U and V in X(oo). Then there is a continuous function f on X (00) which is negative at ~, positive at 'rJ and 0 outside UuV. Applying the Martingale Convergence Theorem in the case h = h f we conclude that the set of sequences ('Pn) in r such that ('Pn (x)), x EX, has accumulation points close to both ~ and 'rJ has P-measure O. Now X(oo) is compact, hence the first assertion. Let f : X(oo) -7 JR be continuous and set Mk(('Pn)) = hf('Pk), k 2: 1. If 7r denotes the hitting probability at X (00), then

Harmonic functions and random walks

59

On the other hand, since (Mk ) is a martingale,

J

(lim Mk)dP =

J

MkdP

hf(e) =

Hence v =

7r

and the proof is complete.

J

fdv

0

We may ask whether (X (00), v) is the Poisson boundary of (f, /1,). By that we mean that the assertions of Theorem 4.11 hold and that every bounded pharmonic function h on f is given as h = hf, where f is an appropriate bounded measurable function on X (00). This is true if X, f and p satisfy the assumptions of Theorem 4.10 and (i) f is finitely generated and p has finite first moment with respect to the word norm of a finite system of generators of f; (ii) f is properly discontinuous and cocompact on X. For the proof (in the smooth case) we refer to [BaLl]; see alsox [Kai3, Theorem 3] for a simplification of the argument in [BaLl]. We omit a more elaborate discussion of the Poisson boundary since the relation between cocompactness on the one hand and the duality condition on the other is unclear (as of now). For the convenience of the reader, the Bibliography contains many references in which topics related to our random walks are discussed: Martin boundary, Brownian motion, potential theory ... for Hadamard spaces and groups acting isometrically on them.

CHAPTER IV

Rank Rigidity In this chapter we prove the Rank Rigidity Theorem. The proof proper is in Sections 4-7, whereas Sections 1-3 are of a preliminary nature. The initiated geometer should skip these first sections and start with Section 4. In Section 1 we discuss geodesic flows on Riemannian manifolds, in Section 2 estimates on Jacobi fields in terms of sectional curvature (Rauch comparison theorem) and in Section 3 the regularity of Busemann functions. 1. Preliminaries on geodesic flows

Let M be a smooth n-dimensional manifold with a connection D. Let T M be the tangent bundle of M and denote by 7r : T M ----+ M the projection. Recall that the charts of M define an atlas for T M in a natural way, turning T Minto a smooth manifold of dimension 2n. Namely, if x: U ----+ U' C jRn is a chart of M and

are the basic vector fields over U determined by x, then we define

(1.1 )

x: 7r- 1 (U)

----+

U' x

jRn;

x(v) = (x(p), dx Ip(v)) ,

where p = 7r(v). If we write vasa linear combination of the Xi(p),

then (~1, ...

,C) -'

~(v)

and hence

where x = x 0 7r. It is easy to see that the family of maps X, where x is a chart of M, is a COO-atlas for TM.

Chapter IV: Rank rigidity

62

The connection D allows for a convenient description of TTM, the tangent bundle of the tangent bundle. To that end, we introduce the vector bundle [; ---+ TM, [; = 7r*(TM) EB 7r*(TM). The fibre [;v of [; at v E TM is (1.2) where p = 7r( v) is the foot point of v. We define a map I : TT M (1.3)

I(Z) = (c(O),

---+ [;

as follows:

~~ (0)) ,

where V is a smooth curve in TM with V(O) = Z and c = 7r 0 V. To see that (1.3) defines a map TTM ---+ [;, we express it in a local coordinate chart i: as above. Then i: 0 V = (0",0, where 0" = x 0 c and V = L ~i Xi. Hence

is represented by

(1.4) where r denotes the Christoffel symbol with respect to the chart x and where we use i: EB i: as a trivialization for [;. We conclude that I is well-defined, smooth and linear on the fibres. It is easy to see that I is surjective on the fibres. Now I is bijective since the fibres of TT M and [; have the same dimension 2n. Therefore: 1.5 Lemma. The map I : TTM

---+ [;

is a vector bundle isomorphism.

1.6 Exercise. For a coordinate chart x and a tangent vector v = the beginning of this section show that

0

LeXk (p)

as in

and

e

where V = L X k is the "constant extension" of v to a vector field on the domain of the coordinate chart x. Henceforth we will identify TTM and [; via I without further reference to the identification map I.

Geodesic flows

63

For v E TTM and p

= 1f(v),

the splitting

singles out two n-dimensional subspaces. The first one is the vertical space

(1.7)

Vv

= ker1f*v = {(a, Y) lYE TpM}.

The vertical distribution V is smooth and integrable. The integral manifolds of V are the fibres of 1f, that is, the tangent spaces of M. Here we visualize the tangent spaces of M as vertical to the manifold ~ a convenient picture when thinking of TM as a bundle over M. The second n-dimensional subspace is the horizontal space

(1.8)

Hv

=

{(X,D) I X E TpM}.

The horizontal distribution H of T M is smooth but, in general, not integrable. By definition, a curve V in T M is horizontal, that is, V belongs to H, if and only if V' == 0, where V' denotes the covariant derivative of V along 1f0 V; in other words, V is horizontal if and only if V is parallel along c = 1f 0 V. The map, which associates to two smooth horizontal vector fields X, Yon T M the vertical component of [X, YJ, is bilinear and, in each variable, linear over the smooth functions on TM. Hence the vertical component of [X, YJ at v E TM only depends on X(v) and Y(v). To measure the non-integrability of H, it is therefore sufficient to consider only special horizontal vector fields. Now let X be a smooth vector field on M. The horizontal lift XH of X is defined by

(1.9)

XH(v) = (X(1f(v)),O),

V

E

TM.

By definition, X is 1f-related to XH, that is, 1f* 0 XH = X 0 1f. This allows us to express the Lie bracket of horizontal lifts of vector fields X, Y on M in a simple way. 1.10 Lemma. Let X, Y be smooth vector fields on M. Then

where p = 1f(v) and the index p indicates evaluation at p. Proof. Since X and Yare 1f-related to XH and yH respectively, we have

This shows the claim about the horizontal component.

Chapter IV: Rank rigidity

64

Since the vertical component only depends on X p and Yp , it suffices to consider the case that [X, Y] = 0 in a neighborhood of p. Denote by 'Pt the flow of X and by 'ljJs the flow of Y. Then the flows O

t

The RHS is precisely the formula for R(Yp, Xp)v using parallel translation.

0

1.11 Exercise. Show that [XH, Zj = 0, where XH is as above and Z is a smooth vertical field of T M.

Recall that the geodesic flow (l) acts on T M by (1.12) where D v denotes the maximal interval of definition for the geodesic "'Iv determined by 1'v(O) = v. By definition, the connection is complete if D v = ~ for all v E TM. To compute the differential of the geodesic flow, let vETM and (X, Y) E TvTM. Represent (X, Y) by a smooth curve V through v, that is, C(O) = X for c = 7r 0 V and DV (0)

dt

=

Y. Let J be the Jacobi field of the geodesic variation "'IV(s) of "'Iv.

By the definition of the splitting of TTM we have

g;v(X, Y) =

:sgt(V(S))ls=o (:s "'IV(s) (t), (J(t),

~

:s

~ 1'V(s) (t)) Is=o

"'IV(s) (t)) Is=o

(J (t), J' (t)) , where the prime indicates covariant differentiation along "'Iv. Hence: 1.13 Proposition. The differential of the geodesic flow is given by

g;v(X, Y) = (J(t), J'(t)), where J is the Jacobi field along "'Iv with J(O) = X and J'(O) = Y.

0

We now turn our attention to the case that D is the Levi-Civita connection of a Riemannian metric on M. By using the corresponding splitting of TT M, we obtain a Riemannian metric on TM, the Sasaki metric,

(1.14)

Geodesic flows

65

There is also a natural I-form

0:

on TM,

O:v((X, Y)) = < v, X>

(1.15)

Using Lemma 1.10 it is easy to see that the differential w = do: is given by (1.16) The Jacobi equation tells us that w is invariant under the geodesic flow. The I-form 0: is not invariant under the geodesic flow on T M.

1.17 Exercise. Compute the expressions for 0: and w with respect to local coordinates i = (x,~) as in the beginning of this section. Note that w is a nondegenerate closed 2-form and hence a symplectic form on TM. The measure on TM defined by Iwnl is proportional to the volume form of the Sasaki metric. It is called the Liouville measure. The Liouville measure is invariant under the geodesic flow since wis. Denote by SM c TM the unit tangent bundle. We have (1.18)

TvSM

=

{(X,Y) I X,Y E TpM, Y 1.. v},

where v E SM and p = 1f(v). The geodesic flow leaves SM invariant. The restrictions of 0: and w to S M, also denoted by 0: and w respectively, are invariant under the geodesic flow since < 1v, J >= const if J is a Jacobi field along '"'Iv with J' 1.. 1v. The (2n - I)-form 0: 1\ w n - 1 is proportional to the volume form of the Sasaki metric (1.14) on SM. In particular, 0: is a contact form. The measure on SM defined by 0: 1\ w n - 1 is also called the Liouville measure. It is invariant under the geodesic flow on SM since 0: and w = do: are. A first and rough estimate of the differential of the geodesic flow with respect to the Sasaki metric is as follows, see [BBB1].

1.19 Proposition. For v E SM set RvX

IIg~ (v) I

:::;

exp

(~

R(X, v)v. Then we have

=

I Ilid t

Rgs(v) lidS)

Proo]. Let J be a Jacobi field along '"'Iv. Then

((1, J)

+ (J', 1'))'

2 (1', J 2

+ JII)

\1', (id -

Rgs(v))

J)

< 211JII ·IIJ'II·llid - RgS(v)11

< ((J, J) + (1', J')) Ilid - Rgs(v) II

o We finish this section with some remarks on the geometry of the unit tangent bundle.

Chapter IV: Rank rigidity

66

1.20 Proposition [Sas]. A smooth curve V in SM is a geodesic in SM if and only if V" = ~ < V', V' > V and e' = R(V', V)e,

where c =

7f 0

V and prime denotes covariant differentiation along c.

Proof. The energy of a piecewise smooth curve V: [a, b]

E(V) =

~

l

b

IIel1 2 dt + ~

l

----+

SM is given by

b

11V'11 2 dt.

The second integral is the energy of the curve V in the unit sphere at c(O) obtained by parallel translating the vectors V(t) along c to V(O). Assume V is a critical point of E. By considering variations of V which leave fixed the curve c of foot points and V(a) and V(b), we see that V is a geodesic segment in the unit sphere. Hence V satisfies the first equation. Now consider an arbitrary variation V(s,·) of V with foot point c(s,·) which keeps V(-, a) and V(·, b) fixed. Then

o ~~ =

l l

b

l

=

[(

~ e, c) + ( ~ V', V')] dt

~ ~: c) + (~ ~ V, V') + ( R ( ~:' ~~) V, V') ] dt [~ (~:' c))- (~:' e') + :t ((~ V, V') ) '

[( b

b

(

Note that

l

b

:t

(~:' c) dt

(~ V, V" ) = 0 =

l

+ ( R(V', V)e,

b

:t

~:) ] dt .

(~ V, V' ) dt

since c(s,a),c(s,b), V(s,a), and V(s,b) are kept fixed. Also ((D/ds)V, V") = 0 since V" and V are collinear by the first equation and (( D / ds) V, V) = 0 because V is a curve of unit vectors. Hence the second equation is satisfied. Conversely, our computations show that V is a critical point of the energy if it satisfies the equations in Proposition 1.20. 0 1.21 Remarks. (a) The equations in Proposition 1.20 imply

(V', V')' = 0 = (e, c)' and therefore

IIV'II

= const.

and

Ilell

= const.

(b) The great circle arcs in the unit spheres SpM, p EM, are geodesics in SM. In particular, we have that the Riemannian submersion 7f : SM ----+ M has totally geodesic fibres. (c) If V is horizontal, then V is a geodesic if and only if c = 7f 0 V is a geodesic in M.

Jacobi fields and curvature

67

1.22 Exercise. Let M be a surface and V a geodesic in the unit tangent bundle of M. Assume that V is neither horizontal nor vertical. Then V' =/=- 0 and c =/=- 0, where c = 7f 0 V, and

-IIV'11 2 . V(t) IIV'II WK(c(t))

VI/ (t)

k(t)

where k is the geodesic curvature of c with respect to the normal n such that (c, n) and (V, V') have the same orientation and K is the Gauss curvature of M. Use this to discuss geodesics in the unit tangent bundles of surfaces of constant Gauss curvature. 2. Jacobi fields and curvature

In Proposition 1.13 we computed the differential of the geodesic flow (gt) on TM in terms of the canonical splitting of TTM. We obtained that

g:,JX, Y) = (J(t), J'(t)),

(2.1)

where J is the Jacobi field along the geodesic Iv with J(O) = X and J'(O) = Y. In order to study the qualitative behaviour of (gi) it is therefore useful to get estimates on Jacobi fields. 2.2 Lemma. Let I : lR ----> M be a unit speed geodesic, and suppose that the sectional curvature of M along I is bounded from above by a constant f'. If J is a Jacobi field along I which is perpendicular to 'Y, then IIJIII/(t) 2: -kIIJII(t) for all t with J(t) =/=- O.

Proof. The straightforward computation does the job:

II J 1

1/

= =

o

J',J »'

«

IIJ

IIJII

I1

3 (-

=

1

(

IIJI1 2 < J

< R(J, 'Yh, J

>

1/

II I

,J > J

+<J

" I I I < J',J >2) , J > ,J IIJII

11111 2 + IIJ'11 2 11J11 2 -
== O.

Proof. Clearly (i) =} (ii) =} (iii). Now (iii) Lemma 2.3. If IIJ(t)11 is constant, then

and hence (i). 0

=}

(ii) since IIJ(t)11 is convex in t, see

o = < J,J >" = 2 « J',J' > - < R(J,i'h,J ». J'(t) == 0 since the sectional curvature is nonpositive.

Hence (ii)

=}

Jacobi fields and curvature

69

2.7 Definition. Let M be a manifold of nonpositive curvature and 1 : JR -. M a unit speed geodesic. We say that a Jacobi field J along 1 is stable if IIJ(t)11 :::; 0 for all t 2': 0 and some constant 0 2': O. 2.8 Proposition. Let M be a manifold of nonpositive curvature and 1 : JR -. M a unit speed geodesic. Set p = 1(0). (i) For any X E TpM, there exists a unique stable Jacobi field J x along 1 with Jx(O) = X. (ii) Let (1n) be a sequence of unit speed geodesics in M with 1n -. 1. For each n, let I n be a Jacobi field along 1n. Assume that In(O) -. X E TpM and that IIJn(tn)11 :::; 0, where t n -. 00 and 0 is independent of n. Then I n -. J x and J~ -. J'x. Proof. Uniqueness in (i) follows from the convexity of IIJ(t)ll. Existence follows from (ii) since we may take 1n = 1 and I n the Jacobi field with In(O) = X, In(n) = 0 (recall that 1 has no conjugate points). Now let 1n, I n , t n and 0 be as in (ii). Fix m 2': o. By convexity we have IIJn(t) II :::; 0,

0:::; t :::; m,

for all n with t n 2': m. By a diagonal argument we conclude that any subsequence (nk) has a further subsequence such that the corresponding subsequence of Un) converges to a Jacobi field J along 1 with IIJ(t)11 :::; 0 for all t 2': o. Now the uniqueness in (i) implies J = J x and (ii) follows. 0 2.9 Proposition. Let M be a manifold of nonpositive curvature, 1 : JR -. M a unit speed geodesic and J a stable Jacobi field along 1 perpendicular to "y. (i) If the curvature of M along 1 is bounded from above by /'i, = _a 2 :::; 0, then

IIJ(t)11 :::; IIJ(O)lle- at

and

IIJ'(t)11 2': aIIJ(t)11

for all t 2':

(ii) If the curvature of M along 1 is bounded from below by ,\ = _b

IIJ(t)11 2': IIJ(O)lle-

bt

and

IIJ'(t)11 :::; bIIJ(t)11

2

:::;

for all t 2':

o. 0, then

o.

Proof. Let I n be the Jacobi field along 1 with In(O) = J(O) and In(n) = 0, n 2': l. Then I n - . J and J~ -. J' by the previous lemma. If /'i, = -a 2 :::; 0 is an upper bound for the curvature along 1, then by Proposition 2.4

IIJn(t)11 < sinh(a(n - t)) IIJn(O)11 sinh(an) and IlJnll'(t) :::; -acoth(a(n - t))lIJn(t)ll, where we have sn,,(s) =

~ sinh(as) a

0

< t < n,

(:= s if a = 0) and ct,,(s) = acoth(as) (:=

1 if a = 0). Hence (i). In the same way we conclude (ii) from Proposition 2.5.

0

The following estimate will be needed in the proof of the absolute continuity of the stable and unstable foliations in the Appendix on the ergodicity of geodesic flows.

Chapter IV: Rank rigidity

70

Proposition 2.10. Let M be a Hadamard manifold and assume that the sectional KM of M is bounded by -b 2 ::; K M ::; -a 2 < O. Denote by d s the distance in the unit tangent bundle SM of M. Then for every constant D > 0 there exist constants C = C(a, b) ~ 1 and T = T(a, b) ~ 1 such that

where v, ware inward unit vectors to a geodesic sphere of radius R with foot points x, y of distance d( x, y) ::; D

~

T in M

Proof. Since the curvature of M is uniformly bounded, the metric

is equivalent to ds and therefore it suffices to consider d1 . For the same reason, there is a constant C 1 > 0 such that the interior distance of any pair of points x, y on any geodesic sphere in M is bounded by C 1 d(x, y) as long as d(x, y) ::; D. Now the interior distance of IV(t) and IW(t) in the sphere of radius R - t about p = Iv(R) = Iw(R) can be estimated by a geodesic variation Is, 0::; s ::; 1, such that 10 = Iv, 11 = IW' Is(R) = P and IS(O) E Sp(R), 0 ::; s ::; 1. Arguing as in the proof of Proposition 2.9, we conclude that there is a constant C 2 such that the asserted inequality holds for 0 ::; t ::; R - 1 (recall that we are using the distance dd. Now the differential of the geodesic flow gT, 0 ::; T ::; 1, is uniformly bounded by a constant C 3 = C 3 (b), see Proposition 1.19. Hence C = C 2 C 3 ea is a constant as desired. 0

3. Busemann functions and horospheres We come back to the description of Busemann functions in Exercise II.2.6.

3.1 Proposition. Let M be a Hadamard manifold and let p E M. Then a function f : M ----+ lR is a Busemann function based at p if and only if (i) f(p) = 0; (ii) f is convex; (iii) f has Lipschitz constant 1; (iv) for any q E M there is a point ql EM with f(q) - f(qd = 1. Proof. Let f: M ----+ lR be a function satisfying the conditions (i)-(iv). Let q E M be arbitrary. Set qo := q and define qn, n ~ 1, recursively to be a point in M with d(qn, qn-l) = 1 and f(qn) - f(qn-d = -1. Let IJ n : [n - 1, n] ----+ M be the unit speed geodesic segment from qn-l to qn, n ~ 1. Since f has Lipschitz constant 1, the concatenation IJ q := IJI * IJ2 * ... : [0, (0) ----+ !vI is a unit speed ray with

f(q)-t,

t~O.

Busemann functions and horospheres

71

We show now that all the rays a q , q E M, are asymptotic to a p ' We argue by contradiction and assume that a q(00) -I- a p (00). Let a be the unit speed ray from p with a(oo) = aq(oo). Then d(a(t),ap(t)) ---> 00 and there is a first tl > 0 such that d(a(td,ap(td) = 1. For t;::: tl we have 1 2t 1 '

a=-

by comparison with Euclidean geometry. Hence the midpoint Tnt between a(t) and ap(t) satisfies

Now

f

is convex and Lipschitz, hence

f(Tnt) ::;

1

"2 (f(a(t) + f(ap(t))) 1

::; "2 (J(aq(t)) +f(ap(t)) +d(p,q)) < -t+C, where C = (d(p,q) + f(q))/2 This contradicts (*). It follows easily that f is a Busemann function. The other direction is clear.

o

The characterization in Proposition 3.1 implies that Busemann functions are cf. [BGS, p.24]. Using that Busemann functions are limits of normalized distance functions, we actually get that they are C 2 . For ~ E M ((0) and p E M let ap,E, be the unit speed ray from p to ~ and set vE,(p) = o-p,E,(O).

Cl,

3.2 Proposition (Eberlein, see [HeIH]). Let M be a Hadamard manifold, p E M and ~ E M(oo). If f = b(~,p,.) is the Busemann function at ~ based at p, then f is C 2 and we have gradf = -vE,

and

D x gradf(q) = -J~(O), q E M ,X E TqM,

where J x is the stable Jacobi field along aq,E, with Jx(O)

Proof. Let (Pn) be a sequence in M converging to

~

= X.

and let

Then fn ---> f uniformly on compact subsets of M. The gradient of fn on }vf \ {Pn} is the negative of the smooth field of unit vectors V n pointing at Pn. Now

By comparison with Euclidean geometry, the right hand side tends to 0 uniformly on compact subsets of M as n ---> 00. Hence f is C 1 and grad f = -vE,'

Chapter IV: Rank rigidity

72

Now let X be a smooth vector field on M. Then

DX(q)gradfn = -J~(q,O), where In(q,.) is the Jacobi field along (Jq.Pn with In(O) = X(q) and In(q, t n ) = 0, where t n = d(q,Pn). By Proposition 2.8, J~(q,O) tends to J~(q)(O) uniformly on compact subsets of M. Hence f is C 2 and the covariant derivative of grad f is as asserted. 0 3.3 Remark. In general, Busemann functions are not three times differentiable, even if the metric of M is analytic, see [BBB2]. However, if the first k derivatives of the curvature tensor R are uniformly bounded on M and the sectional curvature of M is pinched between two negative constants, then Busemann functions are C k +2 . For v E SM set v(oo) := T'v(oo). We define

WSO(v) For v fixed and

~ =

= {w

E SM I w(oo)

= v(oo)}.

v( 00), WSO(v)

=

{-gradf(q) I q E M}.

where f is a Busemann function at ~. Proposition 3.2 implies that WSO(v) is a C 1_ submanifold of SM. The submanifolds W so ( v), v E S M, are a continuous foliation of SM. By Proposition 3.2, the tangent distribution of W So is

ESO(v)

= {(X, Y) I Y =

J~(O)},

where J x is the stable Jacobi field along T'v with Jx(O) the foliation WSo is a very intricate question.

=

X. The regularity of

Remark 3.4. The following is known in the case that M is the universal cover (in the Riemannian sense) of a compact manifold of strictly negative curvature: (a) For each v E SM, WSO(v) is the weak stable manifold of v with respect to the geodesic flow and hence smooth. (Compare also Remark 3.3.) (b) The foliation W so is a smooth foliation of S M if and only if M is a symmetric space (of rank one), see [Gh, BFL, BCG]. (c) If the dimension of M is 2 or if the sectional curvature of M is strictly 1 negatively :i-pinched, then the distribution ESO is C 1, see [HoI, HiPu]. If the

dimension of Mis 2 and ESO is C 2, then it is Coo, see [HurK]. (d) The foliation WSo is Holder and absolutely continuous, see [Anol], [AnSi] and the Appendix below.

Rank, regular vectors and flats

73

4. Rank, regular vectors and flats

As before, we let M be a Hadamard manifold and denote by 8M the unit tangent bundle of M. For a vector v E 8M, the rank of v is the dimension of the vector space .:JP(v) of parallel Jacobi fields along "(V and the rank of M is the minimum of rank(v) over v E 8M. Clearly we have rank(w) ~ rank(v) for all vectors in a sufficiently small neighborhood of a given vector v E 8M. We define R, the set of regular vectors in 8M, to be those vectors v in 8M such that rank(w) = rank(v) for all w sufficiently close to v. The set of regular vectors of rank m is denoted by R m . The sets Rand R m are open. Moreover, R is dense in 8M since the rank function is semicontinuous and integer valued. If k = rank(M), then the set R k is nonempty and consists of all vectors in 8M of rank k. For v E 8M we can identify TvTM with the space .:J(v) of Jacobi fields along "(V by associating to (X, Y) E TvTM the Jacobi field J along "( with J(O) = X, J'(O) = Y. We let F(v) be the subspace of Tv8M corresponding to the space .:JP(v) of parallel Jacobi fields along "(v' That is, F(v) consists of all (X,O) such that the stable Jacobi field J x along "(v determined by Jx(O) = X is parallel. The distribution F is tangent to 8M and invariant under isometries and the geodesic flow of M. Note that F has constant rank locally precisely at vectors in R. On each nonempty set R m , m ::::: rank(M), F has constant rank m, and we shall see that F is smooth and integrable on R m . Moreover, for each vector v E R m the integral manifold of F through v contains an open neighborhood of v in the set P (v) of all vectors in 8M that are parallel to v. 4.1 Lemma. For every integer m ::::: rank( M), the distribution F is smooth on R m

.

Proof. For each vector v E R m , we consider the symmetric bilinear form

where X, Yare arbitrary parallel vector fields (not necessarily Jacobi) along "(v' Here R denotes the curvature tensor of M, and T is a positive number. Since the linear transformation w ----+ R( w, "'tv)"'tv is symmetric and negative semidefinite, a parallel vector field X on "(v is in the nullspace of Q'T if and only if for all t E [-T,T] Hence such a vector field X is a Jacobi field on "(v [- T, T]. It follows that for a small neighborhood U of v in R m and a sufficiently large number T the nullspace of Q'T is precisely .:JP(w) for all w E U. For a fixed T the form Q'T depends smoothly on w. Since the dimension of the nullspace is constant in U, F(w) also depends smoothly on w. 0 We now integrate parallel Jacobi fields to produce flat strips in M.

Chapter IV: Rank rigidity

74

4.2 Lemma. The distribution F is integrable on each nonempty set R m C SM, m 2': rank(M). The maximal arc-connected integral manifold through v E R m is an open subset ofP(v). In particular, ifw E P(v) nRm ) then P(v) and P(v) are smooth m-dimensional manifolds near wand Jr( w) respectively. Proof. To prove the first assertion we begin with the following obeservation: let cr : (-E, E) ---+ R m be a smooth curve tangent to F. Then cr( s) is parallel to cr(0) = v for all s. To verify this we consider the geodesic variation 'Ys (t) = (Jr 0 gt) (cr( S )) of 'Yo = 'Yv· The variation vector fields Y. of ('Ys) are given by Ys(t)

=

(dJr

0

d dgt)(ds cr(s)).

The vector fields Ys are parallel Jacobi fields along 'Ys since (djds)cr(s) belongs to F. For each t the curve at : u ---+ 'Yu(t) has velocity (djdu)at(u) = Yu(t), and hence, since each Yu is parallel, the lengths of the curves at (1) are constant in t for any interval I C (-E, E). Therefore, for every s the convex function t ---+ d("(o (t), 'Ys (t)) is bounded on JR and hence constant. This implies that the geodesics 'Ys are parallel to 'Yo and cr(s) = 18(0) is parallel to cr(O) = v. Now let v E R m be given, and let X, Y be smooth vector fields defined in a neighborhood of v in R m that are tangent to F. Denote by ¢~ and ¢~ the flows of X and Y respectively. The discussion above shows that if w E R m is parallel to v then so are ¢~ (w) and ¢~ (w) for every small s. Hence cr( s2) = ¢;;s 0 ¢Zs 0 ¢~ 0 ¢~ (v) is parallel to v for every small s. Therefore, [X,Y](v) = (djds)cr(s)ls=o belongs to F, and it follows that F is integrable on R m . The third assertion of the lemma follows from the second, which we now prove. If Q denotes the maximal arc-connected integral manifold of F through v, then Q C P(v) by the discussion above. Now let w E Q be given, and let o C R m be a normal coordinate neighborhood of w relative to the metric in SM. We complete the proof by showing that

Onp(w)

=

Onp(v) C Q.

If Wi EOn P (v) is distinct from w, let cr( s) be the unique geodesic in M from cr(O) = Jr(w) to cr(l) = Jr(w ' ). The parallel field w(s) along cr(s) with w(O) = w has values in P(v) since either 'Yw = 'Yw ' or 'Yw and 'Yw ' bound a flat strip in M. Hence w[O, 1] is the shortest geodesic in SM from w(O) = w to w(l) = Wi and must therefore lie in 0 C R m . Since the curve w(s), 0 ~ s ~ 1, lies in P(v) n R m it is tangent to F and hence Wi E w[O, 1] C Q. D

4.3 Lemma. Let N* be a totally geodesic submanifold of a Riemannian manifold N. Let 'Y be a geodesic of N that lies in N*) and let J be a Jacobi field in N along 'Y. Let J 1 and h denote the components of J tangent and orthogonal to N* respectively. Then J 1 and J 2 are Jacobi fields in N along T Proof. Let R denote the curvature tensor of N, and let v be a vector tangent to N* at a point p. Since N* is totally geodesic it follows that

R(X, v)v E TpN*

if X E TpN*

Rank, regular vectors and fiats and hence, since Y

Since

J~'

--+

75

R(Y, v)v is a symmetric linear operator, that

is tangent and

Jf

perpendicular to N* we see that

is the component of

0= J"

+ R(J,-rh

tangent to N*, and hence must be zero.

0

4.4 Proposition. Suppose the isometry group of M satisfies the duality condition. Then for every vector v E R m ! the set P(v) is an m-fiat, that is, an isometrically

and totally geodesically embedded m-dimensional Euclidean space. Proof. Let v E R m be given. By Lemma 4.2 for any v' E R m the sets P( v') n R m and P(v') are smooth m-manifolds near v' and 7f( v') respectively, and 7f : P( v') n R m --+ P(v') is an isometry onto an open neighborhood of 7f(v') in P(v'). We can choose a neighborhood U C R m of v and a number E > 0 such that for every v' E U the exponential map at 7f( v') is a diffeomorphism of the open E-ball in T7f(v')P(v') onto its image DE(v') C P(v'). Let q E P(v) be given. We assert that q E DE C P(v), where DE is the diffeomorphic image under the exponential map at q of the open E-ball in some m-dimensional subspace of TqM. Let w = v(q) be the vector at q parallel to v. By Corollary 111.1.3 we can choose sequences Sn --+ 00, V n --+ v and (cPn) in r such that (dcPn 0 gSn )(vn ) --+ was n --+ +00. For sufficiently large n, V n E U and

If DE(v') c P(v') for some v' C R, then clearly DE(gSv') C P(v') = P(gSv') for any real number s since P( v') is the union of flat strips containing IV" Hence

for sufficiently large n. Therefore, the open m-dimensional E-balls DE; (dcPn ogSn (v n )) converge to an open m-dimensional E-ball DE; C P(w) = P(v) as asserted. We show first that DE = BE(q) n P(v), where BE(q) is the open E-ball in M with center q. Since q E P(v) was chosen arbitrary this will show that P(v) is an m-dimensional submanifold without boundary of M, and it will follow that P(v) is complete and totally geodesic since P( v) is already closed and convex. Suppose that DE: is a proper subset of BE(q) n P(v). Choose a point q' E BE(q) n P(v) such that the geodesic from q to q' is not tangent to DE at q. The set P(v) contains the convex hull D' of DE and q' and hence contains a subset D" C D' that is diffeomorphic to an open (m + I)-dimensional ball. Similarly P(v) contains the

Chapter IV: Rank rigidity

76

convex hull of D" and p = 1T(V), but this contradicts the fact that P(v) has dimension m at p. We show next that P( v) has zero sectional curvature in the induced metric. By a limit argument we can choose a neighborhood 0 of v in Trr(v)P(v) such that o C R m and for any w E 0 the geodesic Iw admits no nonzero parallel Jacobi field that is orthogonal to P( v). Let w E 0 be given, and let Y be any parallel Jacobi field along Iw' If Z is the component of Y orthogonal to P( v), then Z is a Jacobi field on Iw by Lemma 4.3, and IIZ(t)11 :::; IIY(t)11 is a bounded convex function on R By Lemma 2.6, Z is parallel on IW and hence identically zero since wE O. Hence

Trr(w)P(w)

= span{Y(O)

: Y E J"P(w)} C Trr(v)P(v).

Since P( v) and P( w) are totally geodesic submanifolds of M of the same dimension m it follows that P(v) = P(w) for all wE O. The vector field p -> w(p) is globally parallel on P(w) = P(v). Since this is true for any w E 0 we obtain m linearly independent globally parallel vector fields in P(v). Hence P(v) is flat. 0 5. An invariant set at infinity In this section we discuss Eberlein's modification of the Angle Lemma in [BBE], see [EbHe]. We will also use the notation Pv for P(v). Since the distribution :F is smooth on R, we have the following conclusion of Proposition 4.4. 5.1 Lemma. The function (v, p)

f---+

d(p, Pv ) is continuous on R x M.

0

5.2 Lemma. Let k = rank(M), v E R k and ~ = IV(-oo), 1] = Iv(oo). Then there exists an c > 0 such that if F is a k-fiat in M with d(p, F) = 1, where p = 1T( v) is

the foot point of v, then L(~,F(oo))+L(1],F(oo)) ~

c.

Proof. Suppose there is no c > 0 with the asserted properties. Then, by the semicontinuity of L, there is a k-flat F in M with d(p, F) = 1 and ~,1] E F( 00). Since F is a flat and L(~, 1]) = 1T this implies Lq(~, 1]) = 1T for all q E F. Hence F consists of geodesics parallel to Iv and therefore F C Phv) = Pv . But Pv is a k-flat, hence F = P v . This is a contradiction since d(p, F) = 1. 0 5.3 Lemma. Let k = rank M. Let v E Rk be r -recurrent and let 1] = Iv (00). Then there exists an a > 0 such that L(1], () ~ a for any ( E M( 00) \ Pv (00).

Proof. Let c > 0 be as in Lemma 5.2. Choose 15 > 0 such that any unit vector w at the foot point p of v with L(v, w) < 15 belongs to R k and set a = min{c, 15}. Let ( be any point in M (00 ) \ P v ( 00) and suppose L (1], () < a. In particular, L(1],() < 1T and hence there is a unique L-geodesic a: [0,1] -+ M(oo) from 1] to (. Let v(s) be the vector at p pointing at a(s),O:::; s:::; 1. Note that L(v(s),v(t)) :::; L(a(s),a(t))

=

It -

sIL(1],()

77

An invariant set at infinity

by the definition of the L-metric on M(oo). Hence

L(v,v(s))

~

L('T],a-(s)) < 8

and hence v(s) E R k , 0 ~ S ~ 1. Let w = v(l) be the vector at p pointing at (. Note that 'T] Pw(oo). Otherwise we would have IV C Pw since Pw is convex and p E Pw. But then Pv = Pw , a contradiction since ( Pv(oo). We conclude that (the convex function) d(rv(t),Pw ) ----; 00 as t ----; 00. Since v is f-recurrent, there exist sequences t n ----; 00 and (IPn) in f such that dIPn (gt nv) ----; v as n ----; 00. By what we just said, there is an N E N such that d(rv(tn), Pw ) 2:: 1 for all n 2:: N. By Lemma 5.1 there exists an Sn E [0,1] with d(rv(tn),PV(Sn») = 1, n 2:: N. We let Fn = IPn(P(V Sn ))' Then d(p,Fn ) ----; 1. Passing to a subsequence if necessary, the sequence (Fn ) converges to a k-flat F with d(p, F) = 1. Let In be the geodesic defined by In(t) = IPn(rv(t + t n )), t E R Then d(rn(t), Fn ) is convex in t and increases from 0 to 1 on the interval [-tn,O]. By the choice of t n and IPn we have In ----; Iv and hence d(rv(t), F) ~ 1 on (-00,0]. We conclude that IV(-oo) E F(oo). Therefore L(rv(oo)),F(oo)) 2:: c, where c > 0 is the constant from Lemma 5.2. Let (n = a-(sn). By passing to a further subsequence if necessary, we can assume that IPn((n) E Fn(oo) converges. Then the limit ~ is in F(oo). Also note that IPn ('T]) = In (00) converges to 'T] = IV (00) since In ----; Iv' Hence

rt

rt

c ~ L(rv(oo), F(oo)) ~ L(rv(oo),~) ~ liminf =

This is a contradiction.

L(IPn('T]), IPn((n))

liminf L('T], (n) ~ L('T], ()

< a.

0

For any p EM, let IPp : M (00) ----; SpM be the homeomorphism defined by where v~(p) is the unit vector at p pointing at ~.

IPp(~) = v~(p),

5.4 Lemma. Let v, 'T] and a be as in Lemma 5.3, and let F be a k-fiat with 'T] IV(oo) E F(oo). Then

Bo:('T])

= {( E

M(oo)

I L('T],() < a} C

=

F(oo)

and for any q E F, IPq maps B o ('T]) isometrically onto the ball B o (v'l) (q)) of radius a in SqF.

Proof. Lemma 5.3 implies Bo('T]) C Pv(oo). For q E F, IP;;l maps SqF isometrically into M(oo). In particular, IP;;l maps Bo(v'l)(q)) isometrically into Bo('T]). Hence IPpOIP;;l : Bo(v'l)(q)) ----; Bo(v) is an isometric embedding mapping v'l)(q) to v. Since both, B o (v'l)(q)) and B o (v), are balls of radius a in a unit sphere of dimension k-l, IPp 0 IP;;l is surjective and hence an isometry. The lemma follows. 0

Chapter IV: Rank rigidity

78

5.5 Corollary. Let V,7) and ex be as in Lemma 5.3. Then L q((1,(2) all q E M and (1, (2 E B a (7)).

= L((1,(2) for

Proof. Let q E M and let '"'( be the unit speed geodesic through q with '"'((00) = 7). By Proposition 4.4, '"'( is contained in a k-flat F. By Lemma 5.4, we have B a (7)) C F(oo). Since F is a flat we have Lq((l, (2) = L((l, (2) for (1, (2 E B a (7)). 0 5.6 Proposition. If rank(M) = k ?: 2, if the isometry group f of M satisfies the duality condition and if M is not fiat, then M (00) contains a proper, nonempty, closed and f -invariant subset.

Proof. Suppose first that M has a non-trivial Euclidean de Rahm factor Mo. Then M = M o X M l with M l non-trivial since M is not flat. But then every isometry of M leaves this splitting invariant, and hence M o(00) and M l (00) are subsets of M(oo) as asserted. Hence we can assume from now on that M has no Euclidean de Rahm factor. Let X 8 be the set of ~ E M (00) such that there is an 7) E M (00) with Lq(~, 7)) = 8 for all q E M. Corollary 5.5 shows that X8 -I- 0 for some 8 > 0. Clearly, X 8 is closed and f-invariant for any 8. Let {3 = sup{81 X 8 -I- 0}. Then X{3 -I- 0. We want to show X{3 -I- M(oo). We argue by contradiction and assume X{3 = M(oo). Suppose furthermore (3 < 1f. Let v E R k be f-recurrent, 7) = '"'(v(oo) and let ex be as in Lemma 5.3. Choose 8 > with (3 + 8 < Jr and 8 < ex. Since X{3 = M(oo), there is a point ( E M(oo) with L q (7), () = {3 < Jr for all q E M. Let (Y : [0, (3] -+ M(oo) be the unique unit speed L-geodesic from (to 7). Now the last piece of (Y is in B a (7)) and B a (7)) is isometric to a ball of radius ex in a unit sphere. Hence (Y can be prolongated to a L-geodesic (Y* : [0,{3 + 8] -+ M(oo). Let q E M. Since L('Pq(6),'Pq(6)) :s; L(6,6) for all 6,6 E M(oo), where 'Pq(~) E SqM denotes the unit vector at q pointing at ~ E M(oo), and since L q (7), () = (3 = L(7), (), we conclude

°

L('Pq((Y(s)), 'Pq((Y(t)))

= L(rr(S), (Y(t)) = Is - tl

for all s, t E [0, {3]. Hence 'Pq 0 (Y is a unit speed geodesic in SqM. The geodesic '"'( through q with '"'((00) = 7) is contained in a k-flat F, and hence 'Pq 0 (Y* : ({3 - ex, {3 + 8] is also a unit speed geodesic in SqM. Therefore 'Pq 0 (Y* is a unit speed geodesic in SqM of length {3 + 8 and hence

Since q was chosen arbitrary we get X{3+8 -I- 0. This contradicts the choice of {3, hence {3 = Jr. But X7l" -I- 0 implies that M is isometric to M' x R This is a contradiction to the assumption that M does not have a Euclidean factor. Hence X{3 is a proper, nonempty, closed and f-invariant set as claimed. 0 5.7 Remark. The set X{3 as in the proof catches endpoints of (certain) singular geodesics in the case that X is a symmetric space of non-compact type.

Rank rigidity

79

6. Proof of the rank rigidity

From now on we assume that M is a Hadamard manifold of rank ~ 2 and that the isometry group of M satisfies the duality condition. In the last section we have seen that there is a f-invariant proper, nonempty compact subset Z C M(oo). By Theorem III.2.4 and Formula III.2.5.

f(v) = L(V,Z) := inf{Lrr(v)bv(oo),~) I ~ E Z}

(6.1)

defines a f -invariant continuous first integral for the geodesic flow on S M and

(6.2)

f(v) = f(w)

6.3 Lemma. f : SM

-+

if IV(OO) = IW(OO)

or

IV(-OO) = IW(-OO).

IR is locally Lipschitz, hence differentiable almost every-

where. Proof. Instead of the canonical Riemannian distance ds on SM, we consider the metric d1(v,w) = dbv(O)"w(O)) +dbv(1)"w(l)) which is locally equivalent to ds. Let v,w E SM and p = 1f(v),q = 1f(w). Let Vi be the vector at q asymptotic to v. Then f(v ' ) = f(v) by (6.2). Now

/ d1(w,v )

= dbw(l),'v,(l)):::; dbw(l),'v(1)) +dbv(l)"v,(l)) :::; dbw(1),'v(l))

+ dbv(O)"v'(O)) =

d1(v,w)

since v and Vi are asymptotic and IV/(O) = IW(O). By comparison we obtain

If(v) - f(w)1

If(v / ) - f(w)1

< LV, w ( ')

. (d1(V'W)) :::; 2 arcsm 2

o

Recall the definition of the sets W 80 (v), v E S M, at the end of Section 3,

WSO(v) = {w E SM I w(oo) = v(oo)}. Correspondingly we define

WUD(v) = {w E SM I w(-oo) =v(-oo)}. Note that WUO(v) = -WSD(-V). One of the central observations in [Ba2] is as follows. 6.4 Lemma. Let v E SM. Then TvWSO(v)

+ TvWUO(v)

contain the horizontal

subspace of TvSM. Proof. Note that TV W 80 (v) consists of all pairs of the form (X,B+(X)), where B+(X) denotes the covariant derivative of the stable Jacobi field along Iv determined by X. It follows that Tv W Uo ( v) consists of all pairs of the form (X, B - (X) ),

Chapter IV: Rank rigidity

80

where B- (X) denotes the covariant derivative of the unstable Jacobi field along "tv determined by X. Observe that B+ and B- are symmetric linear operators of T 1r (v)M, and Eo = {X E T 1r (v)M I B+(X) = B-(X) = O} consists of all X which determine a parallel Jacobi field along "tv. (In particular, v E Eo). Since B+ and B- are symmetric, they map T 1r (v)M into the orthogonal complement Et of Eo. Given a horizontal vector (X,O), we want to write it in the form

which is equivalent to

Because of the fact that

it suffices to solve B+(Xr) - B-(Xr)

= -B-(X).

Since B- (X) is contained in Et and (B+ - B-)(Et) c Et, the latter equation has a solution if (B+ - B-) I Et is an isomorphism. This is clear, however, since (B+ - B-)(Y) = 0 implies that the stable and unstable Jacobi fields along "tv determined by Y have identical covariant derivative in "tv(O); hence Y E Eo. 0 6.5 Corollary. If f is differentiable at v and X is a horizontal vector in TvSM, then dfv(X) = O. Proof. By (6.2), f is constant on W80(V) and WUO(v). By Lemma 6.4 we have X E T v W80(V) + TvWUO(v). 0 6.6 Lemma. If c is a piecewise smooth horizontal curve in S M then f oc is constant. Proof. Since f is continuous, it suffices to prove this for a dense set (in the 0 0 _ topology) of piecewise smooth horizontal curves. Now f is differentiable in a set D c SM of full measure. It is easy to see that we can approximate c locally by piecewise smooth horizontal curves c such that c(t) E D for almost all t in the domain of c. Corollary 6.5 implies that foe = const. for such a curve C. 0 Let p E M. Since the subset Z C M(oo) is proper, the restriction of f to SpM is not constant. Lemma 6.6 implies that f is invariant under the holonomy group G of M at p. In particular, the action of G on SpM is not transitive. Now the theorem of Berger and Simons [Be, Si] applies and shows that M is a Riemannian product or a symmetric space of higher rank. This concludes the proof of the rank rigidity.

Appendix. Ergodicity of the geodesic flow

M. BRIN

We present here a reasonably self-contained and short proof of the ergodicity of the geodesic flow on a compact n-dimensional manifold of strictly negative sectional curvature. The only complete proof of specifically this fact was published by D.Anosov in [Anol]. The ergodicity of the geodesic flow is buried there among quite a number of general properties of hyperbolic systems, heavily depends on most of the rest of the results and is rather difficult to understand, especially for somebody whose background in smooth ergodic theory is not very strong. Other works on this issue either give only an outline of the argument or deal with a considerably more general situation and are much more difficult. 1. Introductory remarks

KHopf (see [HoI], [Ho2]) proved the ergodicity of the geodesic flow for compact surfaces of variable negative curvature and for compact manifolds of constant negative sectional curvature in any dimension. The general case was established by D.Anosov and Ya.Sinai (see [Anol], [AnSi]) much later. Hopf's argument is relatively short and very geometrical. It is based on the property of the geodesic flow which Morse called "instability" and which is now commonly known as "hyperbolicity". Although the later proofs by Anosov and Sinai follow the main lines of Hopf's argument, they are considerably longer and more technical. The reason for the over 30 year gap between the special and general cases is, in short, that length is volume in dimension I but not in higher dimensions. More specifically, these ergodicity proofs extensively use the horosphere H (v) as a function of the tangent vector v and the absolute continuity of a certain map p between two horospheres (the absolute continuity of the horospheric foliations). Since the horosphere H (v) depends on the infinite future of the geodesic '"Yv determined by v, the function H (.) is continuous but, in the general case, not differentiable even if the Riemannian metric is analytic. As a result, the map p is, in general, only continuous. For n = 2 the horospheres are I-dimensional and the property of bounded volume distortion

82

Appendix: Ergodicity of the geodesic Bow

for p is equivalent to its Lipschitz continuity which holds true in special cases but is false in general. The general scheme of the ergodicity argument below is the same as Hopf's. Most ideas in the proof of the absolute continuity of the stable and unstable foliations are the same as in [Anol] and [AnSi]. However, the quarter century development of hyperbolic dynamical systems and a couple of original elements have made the argument less painful and much easier to understand. The proof goes through with essentially no changes for the case of a manifold of negative sectional curvature with finite volume, bounded and bounded away from 0 curvature and bounded first derivatives of the curvature. For surfaces this is exactly Hopf's result. The assumption of bounded first derivatives cannot be avoided in this argument. The proof also works with no changes for any manifold whose geodesic flow is Anosov. In addition to being ergodic, the geodesic flow on a compact manifold of negative sectional curvature is mixing of all orders, is a K-flow, is a Bernoulli flow and has a countable Lebesgue spectrum, but we do not address these stronger properties here. In Section 2 we list several basic facts from measure and ergodic theory. Continuous foliations with C1-Ieaves and their absolute continuity are discussed in Section 3. In Section 4 we recall the definition of an Anosov flow and some of its basic properties and prove the Holder continuity of the stable and unstable distributions. The absolute continuity of the stable and unstable foliations for the geodesic flow and its ergodicity are proved in Section 5. 2. Measure and ergodic theory preliminaries A measure space is a triple (X, 2(, JL), where X is a set, 2( is a O"-algebra of measurable sets and JL is a O"-additive measure. We will always assume that the measure is finite or O"-finite. Let (X, 2(, JL) and (Y, 23, v) be measure spaces. A map 7j; : X ----+ Y is called measurable if the preimage of any measurable set is measurable; it is called nonsingular if, in addition, the preimage of a set of measure 0 has measure O. A nonsingular map from a measure space into itself is called a nonsingular transformation (or simply a transformation). Denote by A the Lebesgue measure on R A measurable flow ¢ in a measure space (X, 2(, JL) is a map ¢ : X x JE. ----+ X such that (i) ¢ is measurable with respect to the product measure JL x A on X x JE. and measure JL on X and (ii) the maps ¢t(.) = ¢(', t) : X ----+ X form a one-parameter group of transformations of X, that is ¢t 0 ¢s = ¢HS for any t, s E R The general discussion below is applicable to the discrete case t E Z and to a measurable action of any locally compact group. A measurable function f : X ----+ JE. is ¢-invariant if JL({x EX: f(¢t x ) if(x)}) = 0 for every t ERA measurable set B E 2( is ¢-invariant if its characteristic function IE is ¢-invariant. A measurable function f : X ----+ JE. is strictly ¢-invariant if f (¢t x) = f (x) for all x EX, t ERA measurable set B is strictly ¢-invariant if IE is strictly ¢-invariant.

Measure and ergodic theory preliminaries

83

We say that something holds true mod 0 in X or for fL-a.e. x if it holds on a subset of full fL-measure in X. A null set is a set of measure O.

2.1 Proposition. Let ¢ be a measurable flow in a measure space (X, 2l, fL) and let f : X ~ ~ be a ¢-invariant function. Then there is a strictly ¢-invariant measurable function J such that f(x) = J(x) mod o.

Proof. Consider the measurable map cI> : X x ~ ~ X, cI>(x, t) = ¢t x , and the product measure v in X x R Since f is ¢-invariant, v({(x,t) : f(¢t x ) = f(x)}) = 1. By the Fubini theorem, the set Af

= {x EX: f(¢t x ) = f(x) for a.e. t E ~}

has full fL-measure. Set J(x)

= { ~(y)

if ¢t x = y E A f for some t E ~ otherwise.

If ¢t x = Y E A f and ¢sx = Z E A f then clearly f(y) defined and strictly ¢-invariant. D

= f(z). Therefore J is well

2.2 Definition (Ergodicity). A measurable flow ¢ in a measure space (X, 2l, fL) is called ergodic if any ¢-invariant measurable function is constant mod 0, or equivalently, if any measurable oo f:j(x) and f-(x) = limT->oo fi(x) exist and are equal for fL-a.e. x E X, moreover f+, f- are fL-integrable and ¢-invariant. (2) (von Neumann) If f E L 2 (X, 2l, It) then f:j and fi converge in L 2 (X, 2l, fL) to ¢-invariant functions f+ and f-, respectively. D

2.4 Remark. If f E L 2 (X, 2l, fL), that is, if j2 is integrable, then f+ and f- represent the same element J of L 2 (X, 2l, fL) and J is the projection of f onto the subspace of ¢-invariant L 2-functions. To see this let h be any ¢-invariant L 2 -function.

Appendix: Ergodicity of the geodesic flow

84

Since ¢ preserves fJ we have

r(J(x) - f(x))hdfJ(x) = Jrx f(x)h(x)dfJ(x) - JXrTlimooT~ Jro f(¢tx)h(x)dtdfJ(x) Jx = r f(x)h(x)dfJ(x) - r lim -T r f(y)h(¢-ly)dtdfJ(Y) Jx Jx T oo h T

-+

T

1

=

L

f(x)h(x)dfJ(x) -

L

-+

f(y)h(y)dfJ(Y) =0.

In the topological setting invariant functions are mod 0 constant on sets of orbits converging forward or backward in time. Let (X, 2l, fJ) be a finite measure spaee such that X is a compact metric space with distance d, fJ is a measure positive on open sets and 2l is the fJ-completion of the Borel a-algebra. Let ¢ be a continuous flow in X, that is the map (x, t) ---> ¢t x is continuous. For x E X define the stable VS(x) and unstable VU(x) sets by the formulas

(2.5)

VS(x)

=

{y EX: d(¢tx,¢t y ) ---> 0 as t

--->

oo},

VU(x)

=

{y EX: d(¢t x , ¢t y ) ---> 0 as t

--->

-oo}.

2.6 Proposition. Let ¢ be a continuous flow in a compact metric space X preserving a finite measure fJ positive on open sets and let f : X ---> lR. be a ¢-invariant

measurable function. Then f is mod 0 constant on stable sets and unstable sets, that is, there are null sets N s and N u such that f(y) = f(x) for any x, y E X \ N s , y E VS(x) and f(z) = f(x) for any x, z E X \ N u , z E VU(x). Proof. We will only deal with the stable sets. Reversing the time gives the same for the unstable sets. Assume WLOG that f is nonnegative. For C E lR. set fdx) = min(J(x), C). Clearly fe is ¢-invariant and it is sufficient to prove the statement for fe with arbitrary C. For a natural m let h m : X ---> lR. be a continuous function such that Ife - hml dfJ(x) < Tk. By the Birkhoff Ergodic Theorem,

Ix

exists a.e. Since fJ and fe are ¢-invariant, we have that for any t E lR.

~> m

r Ifdx) - hm(x)1 dfJ(x) hr Ifd¢t y ) - hm(¢ty)1 dfJ(Y)

h

=

=

L

Ife(Y) - hm(¢ty)1 dfJ(Y)·

Absolutely continuous foliations

85

Therefore

Note that since h m is uniformly continuous, ht(y) = ht(x) whenever y E VS(x) and ht (x) is defined. Hence there is a set N m of J.1.- measure 0 such that ht exists and is constant on the stable sets in X \ N m . Therefore, f;j(x) := limm--->CXJ ht(x) is constant on the stable sets in X \ (UNm ). Clearly fc(x) = f;j(x) mod o. D For differentiable hyperbolic flows in general and for the geodesic flow on a negatively curved manifold in particular, the stable and unstable sets are differentiable submanifolds of the ambient manifold Mn. The foliations WS and W U into stable and unstable manifolds are called the stable and unstable foliations, respectively. Together with the I-dimensional foliation WO into the orbits of the flow they form three transversal foliations W s, WU and WO of total dimension n. Any mod 0 invariant function f is, by definition, mod 0 constant on the leaves of W O and, by Proposition 2.6, is mod 0 constant on the leaves of W S and WU. To prove ergodicity one must show that f is mod 0 locally constant. This would follow immediately if one could apply a version of the Fubini theorem to the three foliations. Unfortunately the foliations WS and W U are in general not differentiable and not integrable (see the next section for definitions). The applicability of a Fubini-type argument depends on the absolute continuity of W S and WU which is discussed in the next section. We will need the following simple lemma in the proof of Theorem 5.1.

2.7 Lemma. Let (X, 2t, J.1.), (Y, 23, v) be two compact metric spaces with Borel (Jalgebras and (J-additive Borel measures and let Pn : X -> Y, n = 1,2, ... , P : X -> Y be continuous maps such that

(1) each Pn and P are homeomorphisms onto their images, (2) Pn converges to P uniformly as n -> 00, (3) there is a constant C such that v(Pn(A))::; CJ.1.(A) for any A E 2t. Then v(p(A)) ::; CJ.1.(A) for any A E 2t. Proof. It is sufficient to prove the statement for an arbitrary open ball B in X. For 8 > 0 let B o denote the 8-interior of B. Then p(B o) C Pn(B) for n large enough, and hence, v(p(B o)) ::; v(Pn(B)) ::; CJ.1.(B). Observe now that v(p(Bo)) / v(p(B)) as 8'\.0. D

3. Absolutely continuous foliations Let M be a smooth n-dimensional manifold and let Bk denote the closed unit ball centered at 0 in ~k. A partition W of M into connected k-dimensional C 1 _ submanifolds W(x) 3 x is called a k-dimensional CO-foliation of M with C 1 -leaves (or simply a foliation) iffor every x E M there is a neighborhood U = Ux 3 x and a homeomorphism w = W x : B k X B n - k -> U such that w(O,O) = x and

86

Appendix: Ergodicity of the geodesic flow

(i) w(B k , z) is the connected component Wu(w(O, z)) of W(w(O, z)) n U containing w(O, z), (ii) w(·, z) is a Cl-diffeomorphism of B k onto Wu(w(O, z)) which depends continuously on z E Bn-k in the Cl-topology. We say that W is a Cl-foliation if the homeomorphisms W x are diffeomorphisms. 3.1 Exercise. Let W be a k-dimensional foliation of M and let L be an (n - k)dimensional local transversal to W at x E M, that is, TxM = TxW(x)EBTxL. Prove that there is a neighborhood U :1 x and a C l coordinate chart w : B k X B n - k ----+ U such that the connected component of L n U containing x is w(O, Bn-k) and there are Cl-functions fy : B k ----+ Bn~k, y E Bn~k with the following properties: (i) fy depends continuously on y in the Cl-topology; (ii) w(graph (fy)) = Wu(w(O, y)).

We assume that M is a Riemannian manifold with the induced distance d. Denote by m the Riemannian volume in M and by mN denote the induced Riemannian volume in a Cl-submanifold N.

3.2 Definition (Absolute continuity). Let L be a (n - k)-dimensional open (local) transversal for a foliation W, that is, TxL EB T x W(x) = TxM for every x E L. Let U c !vi be an open set which is a union oflocal "leaves", that is U = UxEL Wu(x), where Wu(x) ~ B k is the connected component of W(x) n U containing x. The foliation W is called absolutely continuous if for any Land U as above there is a measurable family of positive measurable functions 6x : Wu(x) ----+ JR (called the conditional densities) such that for any measurable subset A c U

Note that the conditional densities are automatically integrable. Instead of absolute continuity we will deal with "transversal absolute continuity' which is a stronger property, see Proposition 3.4.

3.3 Definition (Transversal absolute continuity). Let W be a foliation of M, Xl E M, x2 E W(xd and let L i :1 Xi be two transversals to W, i = 1,2. There are neighborhoods Ui :1 Xi, Ui eLi, i = 1,2, and a homeomorphism p : U l ----+ U2 (called the Poincare map) such that p(xd = X2 and p(y) E 'W(y), Y E U l . The foliation W is called transversally absolutely continuous if the Poincare map p is absolutely continuous for any transversals L i as above, that is, there is a positive measurable function q : U 1 ----+ JR (called the Jacobian of p) such that for any measurable subset A C U l

Absolutely continuous foliations

87

If the Jacobian q is bounded on compact subsets of U 1 then W is called transversally absolutely continuous with bounded Jacobians.

3.4 Exercise. Is the Poincare map in Definition 3.3 uniquely defined? 3.5 Proposition. If W is transversally absolutely continuous, then it is absolutely continuous.

Proof. Let Land U be as in Definition 3.2, x ELand let F be an (n - k)dimensional C 1 -foliation such that F(x) ~ L, Fu(x) = Land U = UyEwu(x)Fu(y), see Figure 2. Obviously F is absolutely continuous and transversally absolutely continuous. Let lR be a measurable function which is mod 0 constant on the leaves of W. Then for any transversal L to W there is a measurable_subset L C L of full induced Riemannian volume in L such that for every x E L there is a subset W(x) C W(x) of full induced Riemannian volume in W(x) on which f is constant.

Proof. Given a transversal L break it into smaller pieces L i and consider neighborhoods Ui as in the definition of absolute continuity. Let N be the null set such

Absolutely continuous foliations

89

that f is constant on the leaves of W in M \ N an~ let N i = N n Ui . By the absolute continuity of W for each i there is a subset L i of full measure in L i such that for each x E Li the complement W Ui (x) of N i in W Ui (x) has full measure in the local leaf. D Two foliations WI and W 2 of a Riemannian manifold M are transversal if T x WI (x) n T x W 2 (x) = {O} for every x E M. 3.12 Proposition. Let]\;1 be a connected Riemannian manifold and let WI, W 2 be two transversal absolutely continuous foliations on M of complementary dimensions, that is TxM = TxWI(x) EEl T x W 2 (x) for every x E M (the sum is direct but not necessarily orthogonal). Assume that f : M ----+ lR is a measurable function which is mod 0 constant on the leaves of WI and mod 0 constant on the leaves of

W2 ·

f is mod 0 constant in M. Proof. Let N i C M, i = 1,2, be the null sets such that f of Wi in M i = M \ N i . Let x E M and let U 3 x be a Then

is constant on the leaves small neighborhood. By Lemma 3.11, since WI is absolutely continuous, there is y arbitrarily close to x whose local leaf W U, (y) intersects M I by a set MI(y) of full measure. Since W 2 is absolutely continuous, for almost every z E ]\;11 (y) the intersection W 2 n M 2 has full measure. Therefore f is mod 0 constant in a neighborhood of x. Since M is connected, f is mod 0 constant in M. D The previous proposition can be applied directly to the stable and unstable foliations of a volume preserving Anosov diffeomorphism to prove its ergodicity (after establishing the absolute continuity of the foliations). The case of flows is more difficult since there are three foliations involved - stable, unstable and the foliation into the orbits of the flow. In that case one gets a pair of transversal absolutely continuous foliations of complementary dimensions by considering the stable foliation WS and the weak unstable foliation WUo whose leaves are the orbits of the unstable leaves under the flow. Two transversal foliations Wi of dimensions di , i = 1,2, are integrable and their integral hull is a (d I + d 2 )-dimensional foliation W if

W(x) = UyEW,(x) W 2 (y) = UZE W2 (x) WI (z). 3.13 Lemma. Let Wi, i = 1,2, be transversal integrable foliations of a Riemannian manifold M with integral hull W such that WI is a CI-foliation and W 2 is absolutely continuous. Then W is absolutely continuous.

Proof. Let L be a transversal for W. For a properly chosen neighborhood U in M the set L = UxEL W w (x) is a transversal for W 2 . Since W 2 is absolutely continuous, we have for any measurable subset A C U m(A)

=! r L

}w2 u(Y)

lA(y, z)8y (z) dmW2U(Y)(Z) dmL (y).

Appendix: Ergodicity of the geodesic flow

90

Since L is foliated by WI-leaves,

~

h

drnL(x)

=

rr

j(x,y)drnW(x)(y)drnL(x)

}L}Ww(x)

for some positive measurable function j. The absolute continuity of immediately.

~V

follows 0

We will need several auxiliary statements to prove the absolute continuity of the stable and unstable foliations of the geodesic flow.

4. Anosov flows and the Holder continuity of invariant distributions For a differentiable flow denote by WO the foliation into the orbits of the flow and by EO the line field tangent to ~VO.

4.1 Definition (Anosov flow). A differentiable flow ¢/ in a compact Riemannian manifold !vI is called Anosov if it has no fixed points and there are distributions K" E 1I c TM and constants C, A > 0, A < 1, such that for every x E !vI and every t 2': 0

(a) E'9(.7:) E8 E"(x) E8 EO(x) = TxM, (b) Ild¢~vsll ::s; CAtllvsl1 for any (c)

Ild¢;t vll ll ::s; CAtllv,,11

for any

Vs

E E"(:r) ,

V 1l

E

E"(x)

It follows automatically from the definition that the distributions ES and E U (called the stable and unstable distributions, respectively) are continuous, invariant under the derivative dq} and their dimensions are (locally) constant and positive. To see this observe that if a sequence of tangent vectors Vk E T!vI satisfies (b) and Vk ----> vETM as k ----> CXJ, then v satisfies (b), and similarly for (c). The stable and unstable distributions of an Anosov flow are integrable in the sense that there arestable and unstable foliations WS and W" whose tangent distributions are precisely E S and E 1I • The manifolds WS(x) and W"(x) are the stable and unstable sets as described in (2.5). This is proved in any book on differentiable hyperbolic dynamics. The geodesic flow on a manifold NI of negative sectional curvature is Anosov. Its stable and unstable subspaces are the spaces of stable and unstable Jacobi fields perpendicular to the geodesic and the (strong) stable and unstable manifolds are unit normal bundles to the horospheres (sec Chapter IV). If the sectional curvature of M is pinched between _0,2 and -b 2 with 0 < a < b then Inequalities (b) and (c) of Definition 4.1 hold true with A = e- a (see Proposition IV.2.9). For subspaces HI, H 2 C T x N1 define the distance dist(H I ,H2) as the Hausdorff distance between the unit spheres in HI and H 2 . We say that H 2 is ()transversal to HI if min IlvI - v211 2': (), where the minimum is taken over all unit vectors VI E HI, v2 E H 2 · Set E'90(X) = ES(x) E8 EO(x), E"O(x) = gU(x) E8 EO(x). By compactness, the pairs ES(x), EUO(x) and E"(x), ESO(x) are ()-transversal with some () > 0 independent of x.

Holder continuity of invariant distributions

91

4.2 Lemma. Let qi be an Anosov flow. Then for every () > 0 there is C 1 > 0 such that for any subspace H C T x "TVI with the same dimension as Es (x) and ()-transverssal to EnO(x) and any t ;::: 0

Proof. Let v E H, Ilvll = 1, v = v, + V uo , V s E ES(x), V uo E EUO(x). Then IIv 8 1 > const . (), d¢-t(x)v s E ES(¢-t x ), d¢-t(x)1I uo E E710(¢-t X ) and Ild¢-t(x)vuoll :::; const ·llvnoll, Ild¢-t(:r)1I s ll;::: C- 1 A- t I11l s ll. 0 A distribution E C T!vI is called Holder continuous if there are constants A, ex > 0 such that for any :r, y E Al

dist(E(x),E(y)) :::; Ad(x,y)" , where dist(E(x), E(y)) denotes, for example, the Hausdorff distance in TM between the unit spheres in E(x) and E(y). The numbers A and ex are called the Holder constant and exponent, respectively. 4.3 Adjusted metric. Many arguments in hyperbolic dynamical systems become technically easier if one uses the so-called adjusted metric in !vI which is equivalent to the original Riemannian metric. Let ¢t : M ---t !vI be an Anosov flow. Let f3 E (A, 1). For Vo E EO, V 8 E ES, V71. E E71. and T > 0 set

111 0

Note that since f3 For t > 0 we have

+ V + 11 8

2

71

1

> A, the integrals

=

2

111 0 1

fo=

+ Iv 12 + 11171.1 2 . 8

Ild~>, I dT and- fo = Ild¢~>, I dT converge.

For T large enough the second integral in parentheses is less than the first one and we get and similarly for V71 . Note that ES, En and EO are orthogonal in the adjusted metric. The metric 1·1 is clearly smooth and equivalent to 11·11 and in the hyperbolicity inequalities of Definition 4.1 A is replaced by f3 and C by 1.

92

Appendix: Ergodicity of the geodesic flow

In the next proposition we prove the Holder continuity of g' and EU. This was first proved by Anosov (see [An02]). To make the exposition complete we present a similar but shorter argument here. It uses the main idea of a more general argument from [BrK1] (see Theorem 5.2). To eliminate a couple of constants we assume that the flow is C 2 but the argument works equally well for a C1-flow whose derivative is Holder continuous. 4.4 Proposition. Distributions ES, E U, ESo, EUO of a C 2 A nosov flow der continuous.

q}

are H ol-

Proof. Since EO is smooth and since the direct sum of two transversal Holder continuous distribution is clearly Holder continuous, it is sufficient to prove that ES and E U are Holder continuous. We consider only E S , the Holder continuity of EU can be obtained by reversing the time. As we mentioned above, the hyperbolicity conditions (see Definition 4.1) are closed, and hence, all four distributions are continuous. Obviously a distribution which is Holder continuous in some metric is Holder continuous in any equivalent metric. Fix (3 E ('\,1) and use the adjusted metric for (3 with an appropriate T. We will not attempt to get the best possible estimates for the Holder exponent and constant here. The main idea is to consider two very close points x, y E M and their images qrx and ¢m y . If the images are reasonably close, the subspaces ES(¢/,'x) and Es(¢m y ) are (finitely) close by continuity. Now start moving the subspaces "back" to x and y by the derivatives d¢-l. By the invariance of ES under the derivative, d¢-k Es(¢m x ) = Es(¢m-k x ). By Lemma 4.2, the image of Es(¢m y ) under d¢"'x¢-k is exponentially in k close to Es(¢m-k x ). An exponential estimate on the distance between the corresponding images of x and y allows us to replace d¢-l along the orbit of y by d¢-l along the orbit of x. To abbreviate the notation we use ¢ instead of ¢l. Fix, E (0,1), let x, y E M and choose q such that ,q+l < d(x, y) ~ ,q. Let D :::: max Ild¢11 and fix E > 0. Let m be the integer part of (log E - q log,) / log D. Then

Assume that , is small enough so that m is large. Consider a system of small enough coordinate neighborhoods Ui =:> Vi 3 ¢i x identified with small balls in T¢ixM by diffeomorphisms with uniformly bounded derivatives and such that ¢-l Vi CUi-I. Assume that E is small enough, so that ¢i y E Vi for i ~ m. We identify the tangent spaces TzM for z E Vi with T¢i x Jvl and the derivatives (dz¢)-l with matrices which may act on tangent vectors with footpoints anywhere in Vi. In the argument below we estimate the distance between ES (x) and the parallel translate of ES (y) from y to x. Since the distance function between points is Lipschitz continuous, our estimate implies the Holder continuity of E S • Let v y E Es(¢m y ), Ivyl > 0, Vk = d¢-k Vy = vI:: +v ko, Vs E Es(¢m-k x ), Vno E Euo(¢m-k x ), k=O,l, ... ,m,. FixKE (V73,1). We use induction on ktoshowthat if E is small enough then Ilvkoll/llvl::ll ~ 8K k for k = 0, ... m and a small 8 > 0.

Proof of absolute continuity and ergodicity

93

For k = 0 the above inequality is satisfied for a small enough E since ES is continuous. Assume that the inequality holds true for some k. Set (dq,~-kx4»-l = Ak, (dq,~-ky4»-l = B k . By the choice of D we have IIAk-Bkll :::; const·'YqDm-k =: T)k :::; const· E, where the constant depends on the maximum of the second derivatives of 4>. We have

Vk+l

= BkVk = AkVk + (Bk - Ak)Vk = Ak(V k + vkO) + (B k - Ak)Vk.

Therefore

By the choice of m and D above, T)k = const· 'YqDm-k :::; const· ED- k , and hence the inequality for k + 1 holds true provided D is big enough, E is small enough, v7J < r;, and {) is chosen so that the last factor is less than 1. For k = m we get that Iv~I/lv~1 :::; {)r;,m and, by the choice of m,

Iv uOI < const . Iv;"I -

-'2':..-

'Y I

(q+ 1~ ) log D < const . dist (x y) -

Note that by varying v y we obtain all vectors

~ log D

.

,

Vm

E E"(y).

o

5. Proof of absolute continuity and ergodicity We assume now that M is a compact boundaryless m-dimensional Riemannian manifold of strictly negative sectional curvature. For v E 8M set E S (v) = {(X, Y) : Y = J~(O)}, where Jx is the stable field along 'Yv which is perpendicular to -rv and such that Jx(O) = X. Set EU(v) = -ES(-v). By Section IV.2, the distributions E" and EU are the stable and unstable distributions of the geodesic flow gt in the sense of Definition 4.1. By Section IV.3, the distributions E S and EU are integrable and the leaves of the corresponding foliations W S and WU of 8.M are the two components of the unit normal bundles to the horospheres. 5.1 Theorem. Let 1\,£ be a compact Riemannian manifold with a C3- metric of negative sectional curvature. Then the foliations WS and WU of 8M into the normal bundles to the horospheres are transversally absolutely continuous with bounded Jacobians.

Proof. We will only deal with the stable foliation W S • Reverse the time for the geodesic flow gt to get the statement for WU. Let L i be two Cl-transversals to WS and let Ui and p be as in Definition 3.3. Denote by I: n the foliation of 8M

Appendix: Ergodicity of the geodesic flow

94

into "inward" spheres, that is, each leaf of 2: n is the set of unit vectors normal to a sphere of radius n in ]1.;1 and pointing inside the sphere. Let Vi C Ui be closed subsets such that V2 contains a neighborhood of p(Vd. Denote by Pn the Poincare map for 2: n and transversals L i restricted to V1 . We will usc Lemma 16 to prove that the Jacobian q of P is bounded. By the construction of the horospheres, Pn ===l P as n ----> 00. We must show that the Jacobians qn of Pn are uniformly bounded. To show that the Jacobians qn of the Poincare maps Pn : L 1 ----> L 2 arc uniformly bounded we represent Pn as the following composition

(5.2)

Pn =

g~n 0

Po

0

gn ,

where gn is the geodesic flow restricted to the transversal L] and Po : gn(Ld ----> gn(L 2) is the Poincare map along the vertical fibers of the natural projection 7r : SM ----> M. For a large enough n, the spheres 2: n are close enough to the stable horospheres WS, and hence, are uniformly transverse to L 1 and L 2 so that Pn is well defined on V1. Let Vi E Vi, i = 1,2, andpn(vd = V2. Then 7r(gn v d = 7r(gn V2 ) and po(gnvd = gn V2 . Let Jk denote the Jacobian of the time 1 map gl in the direction of Tl; = Tgk vi L i at gk vi ELi, i = 1,2, that is Jic = Idct(dg 1(gk Vi ) ITJ I. k If J o denotes the Jacobian of Po, we get from (5.2)

(5.3)

n-1 n-1 n~j qn(vd= rr(J~)-l·Jo· rr(J~)=Jo(gnvd·rr(JUJ~). k=O k=O k=O

By Lemma 4.2, for a large n the tangent plane at gn v1 to the image gn(Ld is close to Eso(gnvd, and hence, is uniformly (in vd transverse to the unit sphere SxAI at the point x = 7r(gn v d = 7r(gn V2 ). Note also that the unit spheres form a fixed smooth foliation of SNI. Therefore the Jacobian J o is bounded from above uniformly in Vj. We will now estimate from above the last product in (5.3). By Proposition

IV.2.1O, d(gk v1 , gk V2 ) :::; const . exp( -ak) . d(V1, V2) . Hence, by the Holder continuity of EVa,

Together with Lemma 4.2 this implies that dist(T~, T~) :::; const· exp( -jJk)

with some jJ > o. Our assumptions on the metric imply that the derivative dg 1 (v) is Lipschitz continuous in v. Therefore its determinants in two exponentially close

Proof of absolute continuity and ergodicity

95

directions are exponentially close, that is IIJ~ - J~II ~ const· exp(-rk). Observe now that IIJ':.II is uniformly separated away from 0 by the compactness of 1\;1. Therefore the last product in (5.3) is uniformly bounded in n and VI. By Lemma 2.7, the Jacobian q of the Poincare map p is bounded. 0 We will need the following property of absolutely continuous foliations in the proof of Theorem 5.5.

5.4 Lemma. Let ~V be an absolutely continuous foliation of a manifold Q and let N c Q be a null set. Then there is a null set N I such that for any x E Q \ N I the intersection ~Y(:r) n N has conditional measure 0 in W(x). Proof. Consider any local transversal L for W. Since W is absolutely continuous, there is a subset L of full mL measure in L such that mw(x)(N) = 0 for x E L. Clearly the set UxEi lY(x) has full measure. 0

5.5 Theorem. Let 1\;1 be a compact Riemannian manifold with a C 3 metric of negative sectional curvature. Then the geodesic flow gt : SM -+ SM is ergodic. Proof. Let m be the Riemannian volume in 1\;1, Ax be the Lebesgue measure in SxM, x E M and M be the Liouville measure in SM, dM(X, v) = dm(x) x dAy (v). Since M is compact, m(M) <x and ~l(SM) = m(M) . Ar(SxM). The local differential equation determining the geodesic flow l is ;j;

= v,iJ = O.

By the divergence theorem, the Liouville measure is invariant under gt (see also Section IV.l). Hence the Birkhoff Ergodic Theorem and Proposition 2.6 hold true for gt. Since the unstable foliation ~Vll is absolutely continuous and the foliation W O into the orbits of gt is C I , the integral hull 1:V"o is absolutely continuous by Lemma 3.13. Let f : S1\;1 -+ lE. be a measurable g-invariant function. By Proposition 2.1, f can be corrected on a set of Liouville measure 0 to a strictly g- invariant function .f. By Proposition 2.6, there is a null set N" such that J is constant on the leaves of Wll in S1\;1 \ N". Applying Lemma 5.4 twice we obtain a null set N I such that N u is a null set in W"O(v) and in Wll(v) for any v E SM \ N I . It follows that f is mod 0 constant on the leaf Wll (v) and, since it is strictly g- invariant, is mod o constant on the leaf W1l0 (v). Hence f is mod 0 constant on the leaves of W 1lO . Now apply Proposition 3.12 to WS and W1lO. 0

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