some remarks on using this book vii
curves 1
introduction i
parametrized curves 2
regular curves; arc length 5
the vector product in ra3 11
the local theory of curves parametrized by arc length 16
the local canonical form 27
global properties of plane curves 30
regular surfaces 51
introduction 51
regular surfaces; inverse images of regular values 52
change of parameters; differential functions on surfaces 69
the tangent plane; the differential of a map 83
the first fundamental form; area 92
orientation of surfaces 102
a characterization of compact orientable surfaces 109
a geometric definition of area 114
appendix: a brief review on continuity
and differentiability 118
3. the geometry of the gauss map 134
3-1 introduction 134
3-2 the definition of the gauss map and
its fundamental properties 135
3-3 the gauss map in local coordinates 153
3-4 vector fields 175
3-5 ruled surfaces and minimal surfaces 188
appendix: self-adjoint linear maps and quadratic forms 214
4. the intrinsic geometry of surfaces 217
4-1 introduction 217
4-2 isometries; conformal maps 218
4-3 the gauss theorem and the equations of compatibility 231
4-4 parallel transport; geodesics 238
4-5 the gauss-bonnet theorem and its applications 264
4-6 the exponential map. geodesic polar coordinates 283
4-7 further properties of geodesics. convex neighborhoods 298
appendix: proofs of the fundamental theorems of
the local theory of curves and surfaces 309
5. global differential geometry 315
5-1 introduction 315
5-2 the rigidity of the sphere 317
5-3 complete surfaces. theorem of hopf-rinow 325
5-4 first and second variations of the arc length;
bonnet's theorem 339
5-5 jacobi fields and conjugate points 357
5-6 covering spaces; the theorems of hadamard 371
5-7 global theorems for curves; the fary-miinor theorem 380
5-8 surfaces of zero gaussian curvature 408
5-9 jacobi's theorems 415
5-10 abstract surfaces; further generalizations 425
5-11 hilbert's theorem 446
appendix: point-set topology of euclidean spaces 456
bibliography and comments 471
hints and answers to some exercises 475
index 497
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