杜卡莫—黎曼几何

所需积分/C币:45 2018-09-30 15:53:13 3.54MB PDF

很好的书,是黎曼几何的经典教材,适合本科生,或者非数学系研究生研读
Manfredo perdigao do Carmo TRanslated by Instituto de Matematica Pura francis Flaherty e aplicada Department of Mathematics Edificio Lelio gama Oregon State University Rio de janeiro Corvallis. OR 97331 Brazil U.S.A Library of Congress Cataloging-in-Publication Data Carmo, Manfredo perdigao do [Geometria riemannian. English] Riemannian geometry Manfredo do Carmo translated by Francis flake p. cm.--(Mathematics. Theory and applications Translation of: Geometria riemannian Includes bibliographical references and index. isBN0-8176-3490-8(acid-free) Geometry, Riemannian. I. Title. II. Series: Mathematics (Boston, Mass.) QA649c29131992 91-37377 516373-dc20 CIP Printed on acid-free paper. c Birkhauser Boston 1992 Second Printing, 1993. Copyright is not claimed for works of U.S. Government employees. 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, phe copying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center(CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139US.A ISBN0817634908 SBN37643-3490-8 Typeset in TEX using AMS TEX 2.0 Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the U.S.A 98765432 Manfredo Perdigao do Carmo Riemannian geometry Translated by Francis Flaherty Birkhauser Boston· Basel· Berlin CONTENTS Preface to the first edition ....... Preface to the second edition..........,,.,.. Preface to the English edition How to use this book ............... CHAPTER O-DIFFERENTIABLE MANIFOLDS S1. Introduction 82. Differentiable manifolds; tangent space 83. Immersions and embeddings; examples 11 84. Other examples of manifolds. Orientation 15 85. Vector fields; brackets. Topology of manifolds 25 CHAPTER 1-RIEMANNIAN METRICS 35 1. Introduction 85 82.Riemannian Metrics 。38 CHAPTER 2-AFFINE CONNECTIONS RIEMANNIAN CONNECTIONS,。,,,·,,,48 8 1. Introduction,,,。。、,,.,.,,, 48 2. Affine connections 3. Riemannian connections................ 53 CHAPTER 3-GEODESICS: CONVEX NEIGHBORHOODS . 60 1. Introduction 60 2. The geodesic flow 61 83. Minimizing properties of geodesics ......... 67 $4.Convex neighborhoods 75 CHAPTER A-CURVATURE 。,。。,。88 §1. Introduction 。。88 S2.Curvature 89 ction curvature.,,,。.。, 93 4. Ricci curvature and scalar curvature 97 S5. Tensors on Riemannian manifolds 100 CHAPTER 5-JACOBI FIELDS 110 1. Introduction..。。。 110 2. The Jacobi equation 110 3. Conjugate points 116 CHAPTER 6-ISOMETRIC IMMERSIONS 124 1. Introduction 124 §2. The second fundamental form∴∴∴··125 §3. The fundamental equations∴... 134 CHAPTER 7-COMPLETE MANIFOLDS: HOPF-RINOW AND HADAMARD THEOREMS 1. Introduction 144 §2. Complete manifolds; Hopf-Rinow Theorem∴ 145 $3. The Theorem of Hadamard 149 CHAPTER 8-SPACES OF CONSTANT CURVATURE ... 155 1. Introduction 155 82. Theorem of Cartan on the determination of the metric by means of the curvature 156 §3. Hyperbolic space. 160 §4. Space forms..… 162 85. Isometries of the hyperbolic space; Theorem of Liouville .. 168 CHAPTER 9-VARIATIONS OF ENERGY.......... 191 1. Introduction 191 2. Formulas for the first and second variations of energy ..,. 191 3. The theorems of bonnet-Myers and of Synge-Weinstein....200 CHAPTER 10-THE RAUCH COMPARISON THEOREM., 210 1. Introduction。,,,...。 210 82. The Theorem of Rauch 212 §3. Applications of the Index Lemma to immersions∵∵….21 84. Focal points and an extension of Rauch's Theorem 227 CHAPTER 11-THE MORSE INDEX THEOREM 242 1. Introduction 。。.,.242 2. The Index Theorem 。。.....242 CHAPTER 12-THE FUNDAMENTAL GROUP OF MANI FOLDS OF NEGATIVE CURVATURE 253 1. Introduction 253 2. Existence of closed geodesics 。。,,,,,。..。...254 83. Preissman's Theorem CHAPTER 13-THE SPHERE THEOREM 265 1. Introduction 265 2. The cut locus 267 83. The estimate of the injectivity re 276 §4. The Sphere Theorem 283 85. Some further developments .。,.288 References ..292 Index 297 PREFACE TO THE 1st EDITION The object of this book is to familiarize the reader with the ba sic language of and some fundamental theorems in Riemannian Ge- ometry. To avoid referring to previous knowledge of differentiable manifolds, we include Chapter 0, which contains those concepts and results on differentiable manifolds which are used in an essential way in the rest of the book The first four chapters of the book present the basic concepts of Riemannian Geometry(riemannian metrics, Riemannian connec tions, geodesics and curvature). a good part of the study of rie- mannan Geometry consists of understanding the relationship be- tween geodesics and curvature. Jacobi fields an essential tool for this understanding, are introduced in Chapter 5. In Chapter 6 we introduce the second fundamental form associated with an isomet- ric immersion, and prove a generalization of the Theorem egregium of Gauss. This allows us to relate the notion of curvature in Rie- mannan manifolds to the classical concept of Gaussian curvature for surfaces Starting in Chapter 7, we begin the study of global questions. We emphasize techniques of the Calculus of Variations which we present without assuming a previous knowledge of the subject. among other things, we prove the Theorems of Hadamard(Chap. 7 (Chap. 9)and Synge(Chap. 9), the Rauch Comparison Theorem (Chap. 10), and the Morse Index Theorem( Chap 11). One of the most remarkable applications of these techniques of the alculus of Variations the Sphere Theorem, is presented in Chapter 13. In addition, we include a uniformization" theorem for manifolds of constant curvature(Chap. 8 and a study of the fundamental group of compact manifolds of negative curvature (CI (Chap. 12) Many important topics are absent. Because of limitations of time and space, a choice was necessary; we hope that the references men tioned in each chapter stimulate the reader to complete his knowl- edge in the direction of his own taste Our debt to existing sources(written and oral)is enormous and impossible to catalog. We mention only Chern [Ch 1], Klingenberg- Gromoll-Meyer [KGM and Milnor [Mi] as main infuences This book had its origin in notes of a course given in Berke ley in 1968. Later, with the help of students at IMPA (Instituto de matematica Pura e Aplicada), the notes were translated into Portuguese and published in the Monograph collection of IMPa in 1971. Finally, in a form very close to the present, it was given as a course in the School of Differential Geometry at Fortaleza in July 1978. Throughout all these years, various colleagues and students contributed criticisms and suggestions to improve the text. I want to express, in a most special way, my gratitude to Professor Lucio rodriguez who, in my absence assumed the unpleasant task of cor- recting the proofs and organizing the alphabetical index. To all,my sincerest thanks Manfredo perdigao do carmo Rio de janeiro, June 1979

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