Chapter 0 Calculus in Euclidean Space 1
0.1 Euclidean Space 1
0.2 The Topology of Euclidean Space 2
0.3 Differentiation in Rn 3
0.4 Tangent Space 5
0.5 Local Behavior of Differentiable Functions（Injective and Surjective Functions） 6
Chapter 1 Curves 8
1.1 Definitions 8
1.2 The Frenet Frame 10
1.3 The Frenet Equations 11
1.4 Plane Curves;Local Theory 15
1.5 Space Curves 17
1.6 Exercises 20
Chapter 2 Plane Curves;Global Theory 21
2.1 The Rotation Number 21
2.2 The Umlaufsatz 24
2.3 Convex Curves 27
2.4 Exercises and Some Further Results 29
Chapter 3 Surfaces：Local Theory 33
3.1 Definitions 33
3.2 The First Fundamental Form 35
3.3 The Second Fundamental Form 38
3.4 Curves on Surfaces 43
3.5 Principal Curvature,Gauss Curvature,and Mean Curvature 45
3.6 Normal Form for a Surface,Special Coordinates 49
3.7 Special Surfaces,Developable Surfaces 54
3.8 The Gauss and Codazzi-Mainardi Equations 61
3.9 Exercises and Some Further Results 66
Chapter 4 Intrinsic Geometry of Surfaces：Local Theory 73
4.1 Vector Fields and Covariant Differentiation 74
4.2 Parallel Translation 76
4.3 Geodesics 78
4.4 Surfaces of Constant Curvature 83
4.5 Examples and Exercises 87
Chapter 5 Two-dimensional Riemannian Geometry 89
5.1 Local Riemannian Geometry 90
5.2 The Tangent Bundle and the Exponential Map 95
5.3 Geodesic Polar Coordinates 99
5.4 Jacobi Fields 102
5.5 Manifolds 105
5.6 Differential Forms 111
5.7 Exercises and Some Further Results 119
Chapter 6 The Global Geometry of Surfaces 123
6.1 Surfaces in Euclidean Space 123
6.2 Ovaloids 129
6.3 The Gauss-Bonnet Theorem 138
6.4 Completeness 144
6.5 Conjugate Points and Curvature 148
6.6 Curvature and the Global Geometry of a Surface 152
6.7 Closed Geodesics and the Fundamental Group 156
6.8 Exercises and Some Further Results 161
References 167
Index 171
Index of Symbols 177
