黎曼幾何

內(nèi)容概要

The object of this book is to familiarize the reader with the basic language of and some fundamental theorems in Riemannian Geometry。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?！　he first four chapters of the book present the basic concepts of Riemannian Geometry （Riemannian metrics， Riemannian connections， geodesics and curvature）。A good part of the study of Riemannian Geometry consists of understanding the relationship between 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 isometric immersion， and prove a generalization of the Theorem Egregium of Gauss。This allows us to relate the notion of curvature in Riemannian manifolds to the classical concept of Gaussian curvature for surfaces。

作者簡介

作者：(葡萄牙)卡莫(Carmo M.p.)

書籍目錄

Preface to the first editionPreface to the second edition Preface to the English edition How to use this book CHAPTER 0-DIFFERENTIABLE MANIFOLDS 　1. Introduction 　2. Differentiable manifolds；tangent space 　3. Immersions and embeddings；examples 　4. Other examples of manifolds，Orientation 　5. Vector fields； brackets，Topology of manifolds CHAPTER 1-RIEMANNIAN METRICS 　1. Introduction 　2. Riemannian Metrics CHAPTER 2-AFFINE CONNECTIONS；RIEMANNIAN CONNECTIONS 　1. Introduction 　2. Affine connections 　3. Riemannian connections CHAPTER 3-GEODESICS；CONVEX NEIGHBORHOODS 　1．Introduction  2．The geodesic flow  3．Minimizing properties ofgeodesics  4．Convex neighborhoodsCHAPTER 4-CURVATURE    1．Introduction  2．Curvature  3．Sectional curvature  4．Ricci curvature and 8calar curvature  5．Tensors 0n Riemannian manifoidsCHAPTER 5-JACOBI FIELDS  1．Introduction  2．The Jacobi equation  3．Conjugate pointsCHAPTER 6-ISOMETRIC IMMERSl0NS    1．Introduction．  2．The second fundamental form  3．The fundarnental equationsCHAPTER 7-COMPLETE MANIFoLDS；HOPF-RINOW AND HADAMARD THEOREMS  1．Introduction．  2．Complete manifolds；Hopf-Rinow Theorem．  3．The Theorem of Hadamazd．CHAPTER 8-SPACES 0F CONSTANT CURVATURE  1．Introduction  2．Theorem of Cartan on the determination ofthe metric by mebns of the　curvature．  3．Hyperbolic space  4．Space forms  5．Isometries ofthe hyperbolic space；Theorem ofLiouvilleCHAPTER 9一VARIATl0NS 0F ENERGY    1．Introduction．  2．Formulas for the first and second variations of enezgy  3．The theorems of Bonnet—Myers and of Synge-WeipJteinCHAPTER 10-THE RAUCH COMPARISON THEOREM    1．Introduction  2．Ttle Theorem of Rauch．  3．Applications of the Index Lemma to immersions  4．Focal points and an extension of Rauch’s TheoremCHAPTER 11—THE MORSE lNDEX THEOREM    1．Introduction  2.The Index TheoremCHAPTER 12-THE FUNDAMENTAL GROUP OF MANIFOLDS 0F NEGATIVE CURVATURE    1．Introduction  2．Existence of closed geodesicsCHAPTER 13-THE SPHERE THEOREMReferencesIndex

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