Preface ......................................................... v
I Curves ...................................................... 1
1 What exactly is a "curve"? ............................... 1
2 Curvature and the Frenet formula ........................ 10
3 Closed curves ........................................... 28
4 Geometry of spirals ..................................... 40
5 Space curves ............................................ 50
II Surfaces ................................................... 57
6 What exactly is a "surface"? ............................ 57
7 The first fundamental form .............................. 68
8 The second fundamental form ............................. 77
9 Principal and asymptotic directions ..................... 89
10 Geodesies and the Gauss-Bonnet theorem .................. 99
11 Proof of the Gauss-Bonnet theorem ...................... 117
III Surfaces from the Viewpoint of Manifolds .................. 131
12 Differential forms .................................... 131
13 Levi-Civita connections ............................... 138
14 The Gauss-Bonnet formula for 2-manifolds .............. 148
15 Poincaré-Hopf index theorem ........................... 152
16 The Laplacian and isothermal coordinates .............. 165
17 The Gauss and Codazzi equations ....................... 172
18 Cycloids as brachistochrones .......................... 181
19 Geodesic triangulations of compact Riemannian
2-manifolds ........................................... 184
Appendix A Supplements ....................................... 193
A.l A review of calculus ................................. 193
A.2 The fundamental theorems for ordinary differential
equations ............................................ 198
A.3 Euclidean spaces ..................................... 200
Appendix В Advanced Topics on Curves and Surfaces ............ 213
B.l Evolutes and the cycloid pendulum .................... 213
B.2 Convex curves and curves of constant width ........... 219
B.3 Line integrals and the isoperimetric inequality ...... 224
B.4 First fundamental forms and maps ..................... 231
B.5 Curvature line coordinates and asymptotic line
coordinates .......................................... 236
B.6 Surfaces with К = 0 .................................. 244
В.7 A relationship between surfaces with constant
Gaussian curvature and with constant mean curvature .. 251
B.8 Surfaces of revolution of negative constant
Gaussian curvature ................................... 259
B.9 Criteria of typical singularities ................... 261
B.10 Proof of the fundamental theorem for surfaces ........ 268
Answers to Exercises .......................................... 279
Bibliography .................................................. 301
List of Symbols ............................................... 305
Index ......................................................... 307
| 
