- 作 者：John M. Lee著
- 出 版 社：北京/西安：世界图书出版公司
- 出版年份：2003
- ISBN：7506265516
- 标注页数：224 页
- PDF页数：242 页
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1 What Is Curvature? 1
The Euclidean Plane 2
Surfaces in Space 4
Curvature in Higher Dimensions 8
2 Review of Tensors,Manifolds,and Vector Bundles 11
Tensors on a Vector Space 11
Manifolds 14
Vector Bundles 16
Tensor Bundles and Tensor Fields 19
3 Definitions and Examples of Riemannian Metrics 23
Riemannian Metrics 23
Elementary Constructions Associated with Riemannian Metrics 27
Generalizations of Riemannian Metrics 30
The Model Spaces of Riemannian Geometry 33
Problems 43
4 Connections 47
The Problem of Differentiating Vector Fields 48
Connections 49
Vector Fields Along Curves 55
Geodesics 58
Problems 63
5 Riemannian Geodesics 65
The Riemannian Connection 65
The Exponential Map 72
Normal Neighborhoods and Normal Coordinates 76
Geodesics of the Model Spaces 81
Problems 87
6 Geodesics and Distance 91
Lengths and Distances on Riemannian Manifolds 91
Geodesics and Minimizing Curves 96
Completeness 108
Problems 112
7 Curvature 115
Local Invariants 115
Flat Manifolds 119
Symmetries of the Curvature Tensor 121
Ricci and Scalar Curvatures 124
Problems 128
8 Riemannian Submanifolds 131
Riemannian Submanifolds and the Second Fundamental Form 132
Hypersurfaces in Euclidean Space 139
Geometric Interpretation of Curvature in Higher Dimensions 145
Problems 150
9 The Gauss-Bonnet Theorem 155
Some Plane Geometry 156
The Gauss-Bonnet Formula 162
The Gauss-Bonnet Theorem 166
Problems 171
10 Jacobi Fields 173
The Jacobi Equation 174
Computations of Jacobi Fields 178
Conjugate Points 181
The Second Variation Formula 185
Geodesics Do Not Minimize Past Conjugate Points 187
Problems 191
11 Curvature and Topology 193
Some Comparison Theorems 194
Manifolds of Negative Curvature 196
Manifolds of Positive Curvature 199
Manifolds of Constant Curvature 204
Problems 208
References 209
Index 213