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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