Elementary Differential Geometry (eBook)

Front Cover 1
Elementary Differential Geometry 4
Copyright Page 5
Table of Contents 8
Preface 6
Introduction 12
Chapter I. Calculus on Euclidean Space 14
1 Euclidean Space 14
2 Tangent Vectors 17
3 Directional Derivatives 22
4 Curves in E 26
5 1-Forms 33
6 Differential Forms 37
7 Mappings 43
8 Summary 52
Chapter II. Frame Fields 53
1 Dot Product 53
2 Curves 62
3 The Frenet Formulas 67
4 Arbitrary-Speed Curves 77
5 Covariant Derivatives 88
6 Frame Fields 92
7 Connection Forms 96
8 The Structural Equations 102
9 Summary 107
Chapter III. Euclidean Geometry 109
1 Isometries of E 109
2 The Derivative Map of an Isometry 115
3 Orientation 118
4 Euclidean Geometry 123
5 Congruence of Curves 127
6 Summary 134
Chapter IV. Calculus on a Surface 135
1 Surfaces in E 135
2 Patch Computations 144
3 Differentiable Functions and Tangent Vectors 154
4 Differential Forms on a Surface 163
5 Mappings of Surfaces 169
6 Integration of Forms 178
7 Topological Properties of Surfaces 187
8 Manifolds 193
9 Summary 198
Chapter V. Shape Operators 200
1 The Shape Operator of M c E 200
2 Normal Curvature 206
3 Gaussian Curvature 214
4 Computational Techniques 221
5 Special Curves in a Surface 234
6 Surfaces of Revolution 245
7 Summary 255
Chapter VI. Geometry of Surfaces in E 256
1 The Fundamental Equations 256
2 Form Computations 262
3 Some Global Theorems 267
4 Isometries and Local Isometries 274
5 Intrinsic Geometry of Surfaces in E 282
6 Orthogonal Coordinates 287
7 Integration and Orientation 291
8 Congruence of Surfaces 308
9 Summary 314
Chapter VII. Riemannian Geometry 315
1 Geometric Surfaces 315
2 Gaussian Curvature 321
3 Covariant Derivative 329
4 Geodesies 337
5 Length-Minimizing Properties of Geodesies 350
6 Curvature and Conjugate Points 363
7 Mappings that Preserve Inner Products 373
8 The Gauss-Bonnet Theorem 383
9 Summary 400
Bibliography 402
Answers to Odd-Numbered Exercises 404
INDEX 416