黎曼幾何（簡體書）

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。The 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本書的內容源自于作者1986年在Berkeley講授的課堂筆記，隨後在諸多IMPA（巴西國家數學與應用數學研究所）學子的幫助下，整理並翻譯成了葡萄牙文，並於1971年以IMPA專題論文集出版。本書曾在1978年被Fortaleza大學用作教材講授微分幾何課程，後被越來越多的大學所使用。為適應需要，本書的葡萄牙文版先後兩次再版，並在諸多學者的努力下翻譯成英文出版。本書就是葡萄牙文第二版的英文版。
作者意在使讀者熟悉黎曼幾何當中的一些重要定理及其表述的基本語言。閱讀本書，要求讀者有扎實的微積分學功底，並且熟知微分幾何中有關曲面的各項概念。
讀者對象：本書適用於數學專業的研究生。
作者簡介：
Manfredo P.do Carmo 1963年于UC Berkeley獲得博士學位，目前就職于巴西國家數學與應用數學研究所（IMPA）。
目次：微分流形；黎曼度量；仿射聯絡：黎曼聯絡；測地線：凸鄰域；曲率；Jacobi場；等距侵入；完全流形：Hoft-Rinow及Hadamard定理；常曲率空間；能量的變分；Rauch比較定理；Morse指數定理；負曲率的流形的基本群；球面定理。

Preface to the first edition
Preface 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 neighborhoods
CHAPTER 4-CURVATURE
1．Introduction
2．Curvature
3．Sectional curvature
4．Ricci curvature and 8calar curvature
5．Tensors 0n Riemannian manifoids
CHAPTER 5-JACOBI FIELDS
1．Introduction
2．The Jacobi equation
3．Conjugate points
CHAPTER 6-ISOMETRIC IMMERSl0NS
1．Introduction．
2．The second fundamental form
3．The fundarnental equations
CHAPTER 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 ofLiouville
CHAPTER 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-WeipJtein
CHAPTER 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 Theorem
CHAPTER 11—THE MORSE lNDEX THEOREM
1．Introduction
2.The Index Theorem
CHAPTER 12-THE FUNDAMENTAL GROUP OF MANIFOLDS 0F NEGATIVE CURVATURE
1．Introduction
2．Existence of closed geodesics
CHAPTER 13-THE SPHERE THEOREM
References
Index