Riemannian Geometry and Geometric Analysis
This textbook introduces techniques from nonlinear analysis at an early stage. Such techniques have recently become an indispensable tool in research in geometry, and they are treated here for the first time in a textbook. Topics treated include: Differen
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Springer-Verlag Berlin Heidelberg GmbH
Jiirgen Jost
Riemannian Geometry
and
Geometric Analysis
Springer
Jiirgen Jost Institut fiir Mathematik Universitat Bochum UniversitiitsstraBe 150 D-44801 Bochum, Germany
Mathematics Subject Classification (1991): 53821, 53C20, 32Cl7, 35160, 49-XX
ISBN 978-3-540-57113-1
Ubrary of Congress Cataloging-in-Publication Data Jost, Jiirgen, 1956-. Riemannian geometry and geometric analysis I Jiirgen Jost. p. em. - (Universitext) Includes index. ISBN 978-3-662-03118-6 (eBook) ISBN 978-3-540-57113-1 DOI 10.1007/978-3-662-03118-6 I. Geometry, Riemannian. I. Title. QA649.J67 1995 516.3'73-dc20 95-2775 CIP 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. ©Springer-Verlag Berlin Heidelberg 1995 Originally published by Springer-Verlag Berlin Heidelberg New York in 1995 Typesetting: Camera-ready copy from the author using a Springer T EX macro package SPIN: 10062850 4113143-543210- Printed on acid-free paper
Dedicated to
Shing-Tung Yau for so many discussions about mathematics and Chinese culture
Preface
The present textbook is a somewhat expanded version of the material of a three-semester course I gave in Bochum. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. In the first chapter, we introduce the basic geometric concepts, like differentiable manifolds, tangent spaces, vector bundles, vector fields and oneparameter groups of diffeomorphisms, Lie algebras and groups and in particular Riemannian metrics. We also derive some elementary results about geodesics. The second chapter introduces de Rham cohomology groups and the essential tools from elliptic PDE for treating these groups. In later chapters, we shall encounter nonlinear versions of the methods presented here. The third chapter treats the general theory of connections and curvature. In the fourth chapter, we introduce Jacobi fields, prove the Rauch comparison theorems for Jacobi fields and apply these results to geodesics. These first four chapters treat the more elementary and basic aspects of the subject. Their results will be used in the remaining, more advanced chapters that are essentially independent of each other. In the fifth chapter, we develop Morse theory and apply it to the study of geodesics. The sixth chapter treats symmetric spaces as important examples of Riemannian manifolds in detail. While these two chapters emphasize the geometric aspect, the next ones will be of a more analytical character.
