Table Of ContentProgress in Mathematics 61
Series Editors: Hyman Bass · Joseph Oesterlé
Yuri Tschinkel · Alan Weinstein
Werner Ballmann · Mikhael Gromov
Viktor Schroeder
Manifolds of
Nonpositive
Curvature
Progress in Mathematics
Vol.61
Edited by
J. Coates and
S. Helgason
Springer Science+Business Media, LLC
Werner Ballmann
Mikhael Gromov
Viktor Schroeder
Manifolds of
Nonpositive Curvature
Springer Science+Business Media, LLC 1985
Authors:
Werner Ballmann Viktor Schroeder
Dep!. of Mathematics Math. Institut der Universität
University of Maryland Einsteinstrasse 62
College Park, MD 20742/ USA D-4400 Münster
and and
Math. Institut der Universität Math. Institut der Universität
Wegeierstrasse 10 Rheinsprung 21
D-5300 Bonn / West Germany CH-4051 Basel / Switzerland
Mikhael Gromov
Ins!. des Hautes Etudes Scientifiques
35, Route de Chartres
F-91440 Bures-sur-Yvette / France
Library of Congress Cataloging in Publication Data
Ba/lmann, Werner.
Manifolds of nonpositive curvature.
(Progress in mathematics ; vol. 61)
Based on four lectures delivered by M. Gromov in Feb.
1981 at College de France, Paris, with appendices by
W. BalImann and V. Schroeder.
Bibliography: p.
IncJudes index.
1. Geometry, Differential - - Addresses, essays,
lectures. 2 Manifolds (Mathematics) - - Addresses,
essays, lectures. I. Gromov, Mikhael. 11. Schroeder,
Viktor. IH. Title. IV. Series: Progress in mathe-
matics (Boston, Mass.) ; vol. 6l.
QA649.B35 1985 516.3'6 85-20149
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Ba/lmann, Werner:
Manifolds of nonpositive curvature / Werner
BalImann ; Mikhael Gromov ; Viktor Schroeder.
Boston ; Basel ; Stuttgart : Birkhäuser,
1985.
(Progress in mathematics ; Vol. 61)
ISBN 978-1-4684-9161-6 ISBN 978-1-4684-9159-3 (eBook)
DOI 10.1007/978-1-4684-9159-3
NE: Gromov, Mikhael ; Schroeder, Viktor ; GT
All rights reserved.
No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording or otherwise, without prior permission of the
copyright owner.
© 1985 Springer Science+Business Media New York
Originally published by Birkhäuser Boston, Inc. in 1985.
9 8 7 6 5 4 3 2 1
MANIFOLDS OF NONPOSITIVE CURVATURE
Table of Contents
Prefaee
Introduetion
Lectures on Manifolds of Nonpositive Curvature
by Mikhael Gromov (writlen by Viktor Sehroeder)
Lecture I: Simply Conneeted Manifolds of 1
Nonpositive Curvature
u. Loeal Geometry and Convexity 1
§2. The Theorem of Hadamard-Cartan and 15
Complete Simply Conneeted Manifolds of Nonpositive
Curvature
~3. Ideal Boundary 21
H. The Tits Metrie on X(<») 33
~5. Rigidity and Extensions of Isometrie Maps 60
Lecture II: Groups of Isometries 77
~6. IndividualIsometries 77
~7. Special Groups of Isometries 86
MANIFOLDS OF NONPOSITIVE CURVATURE
Table of Contents (Cont'd)
'58. Discrete Groups of Isometries and the 99
Margulis Lemma
~9. Bieberbach Groups and a Proof of the 103
Margulis Lemma
Lecture III: Finiteness Theorems 110
'510. Manifolds of Bounded Negative Curvature 110
'511. Analytie Manifolds of Nonpositive Curvature 120
'512. Stable Isometries, Essential Volume and 125
Essential Volume of Stable Submanifolds
"i 13. Topology of Analytie Manifolds of Nonpositive 138
Curvature
Lecture IV: Strong Rigidity of Loeally Symmetrie 153
Spaees
'514. Mostow's Rigidity Theorem and its 153
Generalization. An Outline of the Proof.
'515. Proof of the Rigidity Theorem 157
Appendix 1: Manifolds of Higher Rank 177
by Werner Ballmann
MANIFOLDS OF NONPOSITIVE CURVATURE
Table of Contents (Cont'd)
Appendix 2: Finiteness Results for Nonanalytie Manifolds 193
by Viktor Sehroeder
Appendix 3: Tits Metrie and the Aetion of 212
Isometries at Infinity
by Viktor Sehroeder
Appendix 4: Tits Metrie and Asymptotie Rigidity 221
by Viktor Sehroeder
Appendix 5: Symmetrie Spaees of Noneompaet Type 238
by Viktor Sehroeder
Referenees 256
Subjeet Index 260
PREFACE
The final version of this book was completed, while the last
author was a member of the Mathematical Sciences Research Institute
in Berkeley. We would like to thank the institute for its hospitality,
Mike Anderson and Chris Croke for reading parts of the original
manuscript and Faye Yeager for her excellent typing job.
Viktor Schroeder
Berkeley, July 1985
Introduction
This book is based on four lectures deJivered by the see<md
author in February 1981 at College de France in Paris. The goal of
the leetures was to prove the following Theorems 1 dnd 2 which
generaJize the eorresponding facts for locally symmetrie spaees of
non-compaet type.
Theorem 1. Let V· be a eompact irredueible loeally symmetrie spaee
of IR-rank ~ 2 and let V be a compact Riemannian cO<> -manifold with
seetional eurvature K:!; O. If 1\" 1( V) '" 1\" 1 (V\ then V is isometrie
tu V*, provided vol V = vol V*.
Remarks.
(a) This result for V loeally symmetrie is the famous Mostow
rigidity theorem, wh ich remains valid (unlike Theorem 1) for manifolds
=
with IR-rank 1 and with dirn V ~ 3.
(b) An obvious modification of Theorem 1 holds true for
reduci öl e manifolds V*. (One says V is redueible if there is a
loeally isometrie finite eovering V 1 X V 2 - V for some Riemannian
Inilnifolds V 1 and V 2 of positive dimension.) In fact the rigidity of
t.he produet strueture for K:!; 0 is a well known result. of
[Gromoll-Wolf. 1971] and [Lawson-Yau, 19'72]. Furthermore, by the
work of [Eberlein, 1983] the product rigidity yields Theorem 1 when
the universal covering l.j* of V* is reducible.
(c) We prove Theorem 1 in Lecture IV by ildopting the original
argument by Mostow. This argument relies on fundamental results of
Tits relilting the geometry of V" to the (T it s) (J eome t r 1/ of the
i d e al/) 0 und a r 1J of V·.
This ideal boundary ean be dcfined as the base of thc
Euelidean eone obtained as an appropriate limit of metrie spaees
(V*, e:g) for e: - 0, where g is the original metrie in V* The limit
definition of Tits' geometry generalize tn a wide dass of metrie
spaees (X,g) and is especially useful for Riemannian manifolds whose
i
