Differential Geometry of Curves and Surfaces

Title......Page 1
Contents......Page 3
Preface......Page 5
Some Remarks on Using this Book......Page 7
1-1 Introduction......Page 9
1-2 Parametrized Curves......Page 10
1-3 Regular Curves; Arc Length......Page 13
1-4 The Vector Product in R3......Page 19
1-5 The Local Theory of Curves Parametrized by Arc Length......Page 24
1-6 The Local Canonical Form......Page 35
1-7 Global Properties of Plane Curves......Page 38
2-1 Introduction......Page 59
2-2 Regular Surfaces; Inverse Images of Regular Values......Page 60
2-3 Change of Parameters; Differential Functions on Surfaces......Page 77
2-4 The Tangent Plane; the Differential of a Map......Page 91
2-5 The First Fundamental Form; Area......Page 100
2-6 Orientation of Surfaces......Page 110
2-7 A Characterizat;on of Compact Orientable Surfaces......Page 117
2-8 A Geometric Definition of Area......Page 122
Appendix: A Brief Review on Continuity and Differentiability......Page 126
3-1 Introduction......Page 142
3-2 The Definition of the Gauss Map and Its Fundamental Properties......Page 143
3-3 The Gauss Map in Local Coordinates......Page 161
3-4 Vector Fields......Page 183
3-5 Ruled Surfaces and Minimal Surfaces......Page 196
Appendix: Self-Adjoint Linear Maps and Quadratic Forms......Page 222
4-1 Introduction......Page 225
4-2 Isometries; Conformal Maps......Page 226
4-3 The Gauss Theorem and the Equations of Compatibility......Page 239
4-4 Parallel Transport. Geodesics......Page 246
4-5 The Gauss-Bonnet Theorem and its Applications......Page 272
4-6 The Exponential Map. Geodesic Polar Coordinates......Page 291
4-7 Further Properties of Geodesics; Convex Neighborhoods......Page 306
Appendix: Proofs of the Fundamental Theorems of The Local Theory of Curves and Surfaces......Page 317
5-1 Introduction......Page 323
5-2 The Rigidity of the Sphere......Page 325
5-3 Complete Surfaces. Theorem of Hopf-Rinow......Page 333
5-4 First and Second Variations of the Arc Length; Bonnet\'s Theorem......Page 347
5-5 Jacobi Fields and Conjugate Points......Page 365
5-6 Covering Spaces; the Theorems of Hadamard......Page 379
5-7 Global Theorems for Curves; the Fary-Milnor Theorem......Page 398
5-8 Surfaces of Zero Gaussian Curvature......Page 416
5-9 Jacobi\'s Theorems......Page 423
5-10 Abstract Surfaces; Further Generalizations......Page 433
5-11 Hilbert\'s Theorem......Page 454
Appendix: Point-Set Topology of Euclidean Spaces......Page 464
Bibliography and Comments......Page 479
Hints and Answers to Some Exercises......Page 483
Index......Page 505