A First Course in Geometric Topology and Differential Geometry

✍ Scribed by Ethan D. Bloch

• Publisher: Birkhäuser Boston
• Year: 1997
• Tongue: English
• Leaves: 440
• Edition: 1
• Category: Library

✦ Synopsis

The uniqueness of this text in combining geometric topology and differential geometry lies in its unifying thread: the notion of a surface. With numerous illustrations, exercises and examples, the student comes to understand the relationship between modern axiomatic approach and geometric intuition. The text is kept at a concrete level, 'motivational' in nature, avoiding abstractions. A number of intuitively appealing definitions and theorems concerning surfaces in the topological, polyhedral, and smooth cases are presented from the geometric view, and point set topology is restricted to subsets of Euclidean spaces. The treatment of differential geometry is classical, dealing with surfaces in R3 . The material here is accessible to math majors at the junior/senior level.

✦ Table of Contents

Cover......Page 1
Title Page......Page 5
Contents......Page 7
Introduction......Page 11
Prerequisites......Page 13
Exercises......Page 14
1.1. Introduction......Page 17
1.2. Open and Closed Subsets of Sets in ℝⁿ......Page 18
1.3. Continuous Maps......Page 29
1.4. Homeomorphisms and Quotient Maps......Page 37
1.5. Connectedness......Page 43
1.6. Compactness......Page 50
Endnotes......Page 62
2.1 Introduction......Page 63
2.2 Arcs, Disks and 1-Spheres......Page 65
2.3 Surfaces in ℝⁿ......Page 71
2.4 Surfaces via Gluing......Page 75
2.5 Properties of Surfaces......Page 86
2.6 Connected Sum and the Classification of Compact Connected Surfaces......Page 89
Appendix A2.1 Proof of Theorem 2.4.3 (i)......Page 98
Appendix A2.2 Proof of Proposition 2.6.1......Page 107
Endnotes......Page 124
3.1 Introduction......Page 126
3.2 Simplices......Page 127
3.3 Simplicial Complexes......Page 135
3.4 Simplicial Surfaces......Page 147
3.5 The Euler Characteristic......Page 153
3.6 Proof of the Classification of Compact Connected Surfaces......Page 157
3.7 Simplicial Curvature and the Simplicial Gauss-Bonnet Theorem......Page 168
3.8 Simplicial Disks and the Brouwer Fixed Point Theorem......Page 173
Endnotes......Page 181
4.2. Smooth Functions......Page 183
4.3 Curves in ℝ³......Page 189
4.4 Tangent, Normal and Binormal Vectors......Page 196
4.5 Curvature and Torsion......Page 200
4.6 Fundamental Theorem of Curves......Page 208
4.7 Planar Curves......Page 212
Endnotes......Page 217
5.2 Coordinate Patches and Smooth Surfaces......Page 218
5.3 Examples of Smooth Surfaces......Page 230
5.4 Tangent and Normal Vectors......Page 239
5.5 First Fundamental Form......Page 244
5.6 Directional Derivatives — Coordinate-Free......Page 251
5.7 Directional Derivatives — Coordinates......Page 258
5.8 Length and Area......Page 268
5.9 Isometries......Page 273
Appendix A5.1 Proof of Proposition 5.3.1......Page 280
Endnotes......Page 283
6.1 Introduction......Page 286
6.2 The Weingarten Map and the Second Fundamental Form......Page 290
6.3 Curvature — Second Attempt......Page 297
6.4 Computations of Curvature Using Coordinates......Page 307
6.5 Theorema Egregium and the Fundamental Theorem of Surfaces......Page 312
Endnotes......Page 322
7.1 Introduction......Page 325
7.2 Geodesics......Page 326
7.3 Shortest Paths......Page 338
Endnotes......Page 342
8.1 Introduction......Page 344
8.2 The Exponential Map......Page 345
8.3 Geodesic Polar Coordinates......Page 351
8.4 Proof of the Gauss-Bonnet Theorem......Page 361
8.5 Non-Euclidean Geometry......Page 369
Appendix A8.1 Geodesic Convexity......Page 378
Appendix A8.2 Geodesic Triangulations......Page 387
Endnotes......Page 394
Affine Linear Algebra......Page 397
1. Collateral Reading......Page 402
3. Algebraic Topology......Page 403
4. Geometric Topology......Page 404
5. Differential Geometry......Page 405
6. Differential Topology......Page 406
References......Page 407
Section 1.3......Page 412
Section 1.4......Page 413
Section 1.6......Page 414
Section 2.2......Page 415
Section 2.4......Page 416
Appendix A2.2......Page 417
Section 3.2......Page 418
Section 3.7......Page 419
Section 4.2......Page 420
Section 4.5......Page 421
Section 5.4......Page 422
Section 5.9......Page 423
Section 6.3......Page 424
Section 7.2......Page 425
Section 8.3......Page 426
Appendix A8.1......Page 427
Appendix A8.2......Page 428
Index of Notation......Page 429
Index......Page 432
Back Cover......Page 440