Nonpositive Curvature: Geometric and Analytic Aspects
The present book contains the lecture notes from a "Nachdiplomvorlesung", a topics course adressed to Ph. D. students, at the ETH ZUrich during the winter term 95/96. Consequently, these notes are arranged according to the requirements of organizing the m
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Jtirgen Jost
Nonpositive Curvature: Geometric and Analytic Aspects
Springer Basel AG
Author's address: Max-PIanck-Institute for Mathematics in the Sciences Inseistr. 22-26 0-04103 Leipzig
A CIP catalogue record for this book is available from the Library of Congress, Washington D.e., USA
Deutsche Bibliothek Cataloging-in-Publication Data Jost, Jiirgen: Nonpositive curvature : geometric and analytic aspects / Jiirgen Jost. Basei ; Boston; Berlin: Birkhiiuser, 1997 (Lectures in mathematics : ETH Ziirich) ISBN 978-3-7643-5736-8 ISBN 978-3-0348-8918-6 (eBook) DOI 10.1007/978-3-0348-8918-6
This work is subject to copyright. AlI rights are reserved, whether the whole or part of the material is concemed, specifically the rights of transiation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained.
© 1997 Springer Basei AG Originally published by Birkhauser Verlag in 1997 Printed on acid-free paper produced from chiorine-free puIp. TCF 00 ISBN 978-3-7643-5736-8 987654321
Contents Preface . . . . .
vii
1 IntrQduction 1.1 Examples of Riemannian manifolds of negative or nonpositive sectional curvature . . . . . . . . . . . Appendix to § 1.1: Symmetric spaces of noncompact type 1.2 Mordell and Shafarevitch type problems 1.3 Geometric superrigidity . . . . . . . . . 2 Spaces of nonpositive curvature 2.1 Local properties of Riemannian manifolds of nonpositive sectional curvature . . . . . . . . 2.2 Nonpositive curvature in the sense of Busemann 2.3 Nonpositive curvature in the sense of Alexandrov 3
4
Convex functions and centers of mass 3.1 Minimizers of convex functions 3.2 Centers of mass . 3.3 Convex hulls . . . . . Generalized harmonic maps 4.1 The definition of generalized harmonic maps . 4.2 Minimizers of generalized energy functionals .
5 Bochner-Matsushima type identities for harmonic maps and rigidity theorems 5.1 The Bochner formula for harmonic one-forms and harmonic maps. . . . . . . . . . . . . . . 5.2 A Matsushima type formula for harmonic maps 5.3 Geometric superrigidity . . . . . . . . . . . . .
1 11
19 23
33 42 54 61 64 67
69 76
85
90 95 99
Bibliography Index . . . .
105 v
Preface The present book contains the lecture notes from a "Nachdiplomvorlesung", a topics course adressed to Ph.D. students, at the ETH ZUrich during the winter term 95/96. Consequently, these notes are arranged according to the requirements of organizing the material for oral exposition, and the level of difficulty and the exposition were adjusted to the audience in Zurich. The aim of the course was to introduce some geometric and analytic concepts that have been found useful in advancing our understanding of spaces of nonpositive curvature. In particular in recent years, it has been realized that often it is useful for a systematic understanding not to restrict the attention to Riemannian manifolds only, but to consider more general classes of metric spaces of gene
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