Chapters in this book

1. Frontmatter i
2. Contents vii
3. Prologue xvii
4. Acknowledgements xxv
5. ACT I The Nature of Space
6. 1 Euclidean and Non-Euclidean Geometry 1
7. 2 Gaussian Curvature 17
8. 3 Exercises for Prologue and Act I 24
9. ACT II The Metric
10. 4 Mapping Surfaces: The Metric 31
11. 5 The Pseudosphere and the Hyperbolic Plane 51
12. 6 Isometries and Complex Numbers 65
13. 7 Exercises for Act II 83
14. ACT III Curvature 95
15. 8 Curvature of Plane Curves 97
16. 9 Curves in 3-Space 106
17. 10 The Principal Curvatures of a Surface 109
18. 11 Geodesics and Geodesic Curvature 115
19. 12 The Extrinsic Curvature of a Surface 130
20. 13 Gauss’s Theorema Egregium 138
21. 14 The Curvature of a Spike 143
22. 15 The Shape Operator 149
23. 16 Introduction to the Global Gauss–Bonnet Theorem 165
24. 17 First (Heuristic) Proof of the Global Gauss–Bonnet Theorem 175
25. 18 Second (Angular Excess) Proof of the Global Gauss–Bonnet Theorem 183
26. 19 Third (Vector Field) Proof of the Global Gauss–Bonnet Theorem 195
27. 20 Exercises for Act III 219
28. ACT IV Parallel Transport
29. 21 An Historical Puzzle 231
30. 22 Extrinsic Constructions 233
31. 23 Intrinsic Constructions 240
32. 24 Holonomy 245
33. 25 An Intuitive Geometric Proof of the Theorema Egregium 252
34. 26 Fourth (Holonomy) Proof of the Global Gauss–Bonnet Theorem 257
35. 27 Geometric Proof of the Metric Curvature Formula 261
36. 28 Curvature as a Force between Neighbouring Geodesics 269
37. 29 Riemann’s Curvature 280
38. 30 Einstein’s Curved Spacetime 307
39. 31 Exercises for Act IV 334
40. ACT V Forms
41. 32 1-Forms 345
42. 33 Tensors 360
43. 34 2-Forms 370
44. 35 3-Forms 386
45. 36 Differentiation 392
46. 37 Integration 404
47. 38 Differential Geometry via Forms 430
48. 39 Exercises for Act V 465
49. Further Reading 475
50. Bibliography 485
51. Index 491