微分几何基础(英文影印版.第2版修订版)

.3.1. isometries of r3　100
3.2. the tangent map of an isometry　107
3.3. orientation　110
3.4. euclidean geometry　116
3.5. congruence of curves　121
3.6. summary　128
4. calculus on a surface
4.1. surfaces in r3　130
4.2. patch computations　139
4.3. differentiable functions and tangent vectors　149
4.4. differential forms on a surface　158
4.5. mappings of surfaces　166
4.6. integration of forms　174
4.7. topological properties of surfaces　184
4.8. manifolds　193
4.9. summary　201
5. shape operators..
5.1. the shape operator of m c r3　202
5.2. normal curvature　209
5.3. gaussian curvature　216
5.4. computational techniques　224
5.5. the implicit case　235
5.6. special curves in a surface　240
5.7. surfaces of revolution　252
5.8. summary　262
6. geometry of surfaces in r3
6.1. the fundamental equations　263
6.2. form computations　269
6.3. some global theorems　273
6.4. isometries and local isometries　281
6.5. intrinsic geometry of surfaces in r3　289
6.6. orthogonal coordinates　294
6.7. integration and orientation　297
6.8. total curvature　304
6.9. congruence of surfaces　314
6.10. summary　319
7. riemannian geometry
7.1. geometric surfaces　321
7.2. gaussian curvature　329
7.3. covariant derivative　337
7.4. geodesics　346
7.5. clairaut parametrizations　353
7.6. the gauss-bonnet theorem　364
7.7. applications of gauss-bonnet　376
7.8. summary　386
8. global structure of surfaces
8.1. length-minimizing properties of geodesics　388
8.2. complete surfaces　400
8.3. curvature and conjugate points　405
8.4. covering surfaces　416
8.5. mappings that preserve inner products　425
8.6. surfaces of constant curvature　433
8.7. theorems of bonnet and hadamard　442
8.8. summary　449
appendix: computer formulas　451
bibliography　467
answers to odd-numbered exercises　468
index...　495