Riemannian Manifolds : An Introduction to Curvature

Cover......Page 1
Series: Graduate Texts in Mathematics 176......Page 2
Riemannian Manifolds: An Introduction to Curvature......Page 4
Copyright......Page 5
Preface......Page 8
Contents......Page 14
1. What Is Curvature?......Page 18
The Euclidean Plane......Page 19
Surfaces in Space......Page 21
Curvature in Higher Dimensions......Page 25
Tensors on a Vector Space......Page 28
Manifolds......Page 31
Vector Bundles......Page 33
Tensor Bundles and Tensor Fields......Page 36
Riemannian Metrics......Page 40
Elementary Constructions Associated with Riemannian Metrics......Page 44
Generalizations of Riemannian Metrics......Page 47
The Model Spaces of Riemannian Geometry......Page 50
Problems......Page 60
4. Connections......Page 64
The Problem of Differentiating Vector Fields......Page 65
Connections......Page 66
Vector Fields Along Curves......Page 72
Geodesics......Page 75
Problems......Page 80
The Riemannian Connection......Page 82
The Exponential Map......Page 89
Normal Neighborhoods and Normal Coordinates......Page 93
Geodesics of the Model Spaces......Page 98
Problems......Page 104
Lengths and Distances on Riemannian Manifolds......Page 108
Geodesics and Minimizing Curves......Page 113
Completeness......Page 125
Problems......Page 129
Local Invariants......Page 132
Flat Manifolds......Page 136
Symmetries of the Curvature Tensor......Page 138
Ricci and Scalar Curvatures......Page 141
Problems......Page 145
8. Riemannian Submanifolds......Page 148
Riemannian Submanifolds and the Second Fundamental Form......Page 149
Hypersurfaces in Euclidean Space......Page 156
Geometric Interpretation of Curvature in Higher Dimensions......Page 162
Problems......Page 167
9. The Gauss–Bonnet Theorem......Page 172
Some Plane Geometry......Page 173
The Gauss–Bonnet Formula......Page 179
The Gauss–Bonnet Theorem......Page 183
Problems......Page 188
10. Jacobi Fields......Page 190
The Jacobi Equation......Page 191
Computations of Jacobi Fields......Page 195
Conjugate Points......Page 198
The Second Variation Formula......Page 202
Geodesics Do Not Minimize Past Conjugate Points......Page 204
Problems......Page 208
11. Curvature and Topology......Page 210
Some Comparison Theorems......Page 211
Manifolds of Negative Curvature......Page 213
Manifolds of Positive Curvature......Page 216
Manifolds of Constant Curvature......Page 221
Problems......Page 225
References......Page 226
Index......Page 230