The link between the physical world and its visualization is geometry. This easy-to-read, generously illustrated textbook presents an elementary introduction to differential geometry with emphasis on geometric results. Avoiding formalism as much as possible, the author harnesses basic mathematical skills in analysis and linear algebra to solve interesting geometric problems, which prepare students for more advanced study in mathematics and other scientific fields such as physics and computer science. The wide range of topics includes curve theory, a detailed study of surfaces, curvature, variation of area and minimal surfaces, geodesics, spherical and hyperbolic geometry, the divergence theorem, triangulations, and the Gauss-Bonnet theorem. The section on cartography demonstrates the concrete importance of elementary differential geometry in applications. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book ideal for courses and self-study. The only prerequisites are one year of undergraduate calculus and linear algebra.
Author(s): Christian Bär
Edition: 1
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 335
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
Notation......Page 13
1.1 The axiomatic approach......Page 15
1.2 The Cartesian model......Page 27
2.1 Curves in Rn......Page 36
2.2 Plane curves......Page 48
2.3 Space curves......Page 71
3.1 Regular surfaces......Page 95
3.2 The tangent plane......Page 107
3.3 The first fundamental form......Page 112
3.4 Normal fields and orientability......Page 117
3.5 The second fundamental form......Page 120
3.6 Curvature......Page 124
3.7 Surface area and integration on surfaces......Page 140
3.8.1 Ruled surfaces......Page 146
3.8.2 Minimal surfaces......Page 150
3.8.3 Surfaces of revolution......Page 157
3.8.4 Tubular surfaces......Page 159
4.1 Isometries......Page 167
4.2 Vector fields and the covariant derivative......Page 170
4.3 Riemann curvature tensor and Theorema Egregium......Page 178
4.4 Riemannian metrics......Page 186
4.5 Geodesics......Page 189
4.6 The exponential map......Page 201
4.7 Parallel transport......Page 210
4.8 Jacobi fields......Page 214
4.9 Spherical and hyperbolic geometry......Page 219
4.10 Cartography......Page 228
4.11 Further models of hyperbolic geometry......Page 235
5.1 The divergence theorem......Page 241
5.2 Variation of the metric......Page 251
6.1 Polyhedra......Page 257
6.2 Triangulations......Page 260
6.3 The Gauss-Bonnet theorem......Page 277
6.4 Outlook......Page 280
Appendix A Hints for solutions to (most) exercises......Page 284
Appendix B Formulary......Page 323
Appendix C List of symbols......Page 327
References......Page 329
Index......Page 331
