Differential geometry of curves and surfaces

Preface......Page 6
Preface to the First Edition......Page 7
Translators’ Note......Page 8
Contents......Page 10
 1.1 Concept of Curves......Page 12
 1.2 Plane Curves......Page 13
 1.3 Global Theorems on Plane Curves......Page 23
 1.4 Space Curves......Page 31
 1.5 Global Results on Space Curves......Page 38
 2.1 Concept of Surfaces in the Space......Page 46
 2.2 Fundamental Forms and Curvatures......Page 52
 2.3 Examples and Calculations of Fundamental Forms and Curvatures......Page 65
 2.4 Method of Orthonormal Frames......Page 73
 2.5 Exterior Differential Forms in Two Variables......Page 77
 2.6 Use of Exterior Differential Forms......Page 81
 3.1 Riemannian Metrics on a Surface......Page 87
 3.2 Structure Equations of a Surface......Page 90
 3.3 Vector Fields......Page 96
 3.4 Covariant Derivatives and Parallel Translations......Page 100
 3.5 Geodesics......Page 104
 3.6 Geodesics as Shortest Curves......Page 112
 4.1 Integration of Exterior Differential Forms......Page 119
 4.2 The Gauss-Bonnet Theorem (Domains)......Page 124
 4.3 The Gauss-Bonnet Theorem (Closed Surfaces)......Page 131
 5.1 Mean Curvatures and Minimal Surfaces......Page 142
 5.2 Examples of Minimal Surfaces......Page 144
 5.3 Isothermal Coordinate Systems......Page 150
 5.4 The Weierstrass-Enneper Representation......Page 153
 5.5 Associated Minimal Surfaces......Page 159
 5.6 Curvatures of Minimal Surfaces......Page 161
 5.7 Gauss’ Spherical Maps......Page 163
Appendix......Page 166
Solutions to Problems......Page 168
Postscript......Page 195
Index......Page 199