A First Course in Geometric Topology and Differential Geometry

Contents......Page 4
Introduction......Page 7
To the Student......Page 9
1.1 Introduction......Page 12
1.2 Open and Closed Subsets of Sets in R^n......Page 13
1.3 Continuous Maps......Page 24
1.4 Homeomorphisms and Quotient Maps......Page 32
1.5 Connectedness......Page 38
1.6 Compactness......Page 45
2.1 Introduction......Page 58
2.2 Arcs, Disks and 1-spheres......Page 60
2.3 Surfaces in R^n......Page 66
2.4 Surfaces Via Gluing......Page 70
2.5 Properties of Surfaces......Page 81
2.6 Connected Sum and the Classification of Compact Connected Surfaces......Page 84
Appendix A2.1 Proof of Theorem 2.4.3 (i)......Page 93
Appendix A2.2 Proof of Theorem 2.6.1......Page 102
3.1 Introduction......Page 121
3.2 Simplices......Page 122
3.3 Simplicial Complexes......Page 130
3.4 Simplicial Surfaces......Page 142
3.5 The Euler Characteristic......Page 148
3.6 Proof of the Classification of Compact Connected Surfaces......Page 152
3.7 Simplicial Curvature and the Simplicial Gauss-Bonnet Theorem......Page 163
3.8 Simplicial Disks and the Brouwer Fixed Point Theorem......Page 168
4.2 Smooth Functions......Page 178
4.3 Curves in R^3......Page 184
4.4 Tangent, Normal and Binormal Vectors......Page 191
4.5 Curvature and Torsion......Page 195
4.6 Fundamental Theorem of Curves......Page 203
4.7 Plane Curves......Page 207
5.2 Smooth Surfaces......Page 213
5.3 Examples of Smooth Surfaces......Page 225
5.4 Tangent and Normal Vectors......Page 234
5.5 First Fundamental Form......Page 239
5.6 Directional Derivatives - Coordinate Free......Page 246
5.7 Directional Derivatives - Coordinates......Page 253
5.8 Length and Area......Page 263
5.9 Isometries......Page 268
Appendix A5.1 Proof of Proposition 5.3.1......Page 275
6.1 Introduction and First Attempt......Page 281
6.2 The Weingarten Map and the Second Fundamental Form......Page 285
6.3 Curvature - Second Attempt......Page 292
6.4 Computations of Curvature Using Coordinates......Page 302
6.5 Theorema Egregium and the Fundamental Theorem of Surfaces......Page 307
7.1 Introduction - \"Straight Lines\" on Surfaces......Page 320
7.2 Geodesies......Page 321
7.3 Shortest Paths......Page 333
8.1 Introduction......Page 339
8.2 The Exponential Map......Page 340
8.3 Geodesic Polar Coordinates......Page 346
8.4 Proof of the Gauss-Bonnet Theorem......Page 356
8.5 Non-Euclidean Geometry......Page 364
Appendix A8.1 Geodesic Convexity......Page 373
Appendix A8.2 Geodesic Triangulations......Page 382
Appendix......Page 392
Further Study......Page 397
References......Page 402
Hints for Selected Exercises......Page 407
Index of Notation......Page 424
Index......Page 427