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黎曼流形PDF|Epub|txt|kindle电子书版本网盘下载
- John M. Lee著 著
- 出版社： 北京;西安：世界图书出版公司
- ISBN：7506265516
- 出版时间：2003
- 标注页数：224页
- 文件大小：28MB
- 文件页数：242页
- 主题词：
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图书目录
1 What Is Curvature?1
The Euclidean Plane2
Surfaces in Space4
Curvature in Higher Dimensions8
2 Review of Tensors,Manifolds,and Vector Bundles11
Tensors on a Vector Space11
Manifolds14
Vector Bundles16
Tensor Bundles and Tensor Fields19
3 Definitions and Examples of Riemannian Metrics23
Riemannian Metrics23
Elementary Constructions Associated with Riemannian Metrics27
Generalizations of Riemannian Metrics30
The Model Spaces of Riemannian Geometry33
Problems43
4 Connections47
The Problem of Differentiating Vector Fields48
Connections49
Vector Fields Along Curves55
Geodesics58
Problems63
5 Riemannian Geodesics65
The Riemannian Connection65
The Exponential Map72
Normal Neighborhoods and Normal Coordinates76
Geodesics of the Model Spaces81
Problems87
6 Geodesics and Distance91
Lengths and Distances on Riemannian Manifolds91
Geodesics and Minimizing Curves96
Completeness108
Problems112
7 Curvature115
Local Invariants115
Flat Manifolds119
Symmetries of the Curvature Tensor121
Ricci and Scalar Curvatures124
Problems128
8 Riemannian Submanifolds131
Riemannian Submanifolds and the Second Fundamental Form132
Hypersurfaces in Euclidean Space139
Geometric Interpretation of Curvature in Higher Dimensions145
Problems150
9 The Gauss-Bonnet Theorem155
Some Plane Geometry156
The Gauss-Bonnet Formula162
The Gauss-Bonnet Theorem166
Problems171
10 Jacobi Fields173
The Jacobi Equation174
Computations of Jacobi Fields178
Conjugate Points181
The Second Variation Formula185
Geodesics Do Not Minimize Past Conjugate Points187
Problems191
11 Curvature and Topology193
Some Comparison Theorems194
Manifolds of Negative Curvature196
Manifolds of Positive Curvature199
Manifolds of Constant Curvature204
Problems208
References209
Index213