Geometry of Geodesics (eBook)

Front Cover 1
The Geometry of Geodesics 4
Copyright Page 5
Table of Contents 10
Preface 6
Chapter I. The Basic Concepts 12
1. Introduction 12
2. Compact and Finitely Compact Metric Spaces 14
3. Convergence of Point Sets 21
4. Motion and Isometry 25
5. Curves and Their Lengths 30
6. Segments 38
7. Geodesics 41
8. G-Spaces 47
9. Multiplicity, Geodesics Without Multiple Points 55
10. Two-Dimensional G-Spaces 60
11. Plane Metrics Without Conjugate Points 67
Chapter II. Desarguesian Spaces 76
12. Introduction 76
13. Planes with the Desargues Property 77
14. Spaces Which Contain Planes 87
15. Riemann and Finsler Spaces. Beltrami's Theorem 93
16. Convex Sets in Affine Space 98
17. Minkowskian Geometry 105
18. Hilbert's Geometry 116
Chapter III. Perpendiculars and Parallels 126
19. Introduction 126
20. Convexity of Spheres and Perpendicularity 128
21. Characterization of the Higher-Dimensional Elliptic Geometry 135
22. Limit Spheres and Co-Rays in G-Spaces 141
23. Asymptotes and Parallels in Straight Spaces 148
24. Characterizations of the Higher-Dimensional Minkowskian Geometry 155
25. Characterization of the Minkowski Plane 164
Chapter IV. Covering Spaces 176
26. Introduction 176
27. Locally Isometric Spaces 178
28. The Universal Covering Space 185
29. Fundamental Sets 192
30. Locally Minkowskian, Hyperbolie or Spherical Spaces 199
31. Spaces in Which Two Points Determine a Geodesic 210
32. Free Homotopy and Closed Geodesics 215
33. Metrics Without Conjugate Points on the Torus 226
34. Transitive Geodesics on Surfaces of Higher Genus 234
Chapter V. The Influence of the Sign of the Curvature on the Geodesics 246
36. Introduction 246
36. Local Properties 248
37. Non-Positive Curvature in the Theory of Parallels 259
38. Straightness of the Universal Covering Space 265
39. The Fundamental Groups of Spaces with Convex Capsules 269
40. Geodesics in Spaces with Negative Curvature 273
41. Relation to Non-Positive Curvature in Standard Sense 278
42. Angular Measure 284
43. Excess and Characteristic 293
44. Simple Monogons, Total Excess. Surfaces with Positive Excess 303
Chapter VI. Homogeneous Spaces 318
45. Introduction 318
46. Spaces with Flat Bisectors I 320
47. Spaces with Flat Bisectors II 331
48. Applications of the Bisector Theorem. The Helmholtz-Lie Problem 344
49. Involutoric Motions 354
50. New Characterizations of the Minkowskian Spaces 361
51. Translations Along Two Lines 370
52. Surfaces With Transitive Groups of Motions 377
53. The Hermitian Elliptic and Similar 385
54. Compact Spaces with Pairwise Transitive Groups of Motions 396
55. Odd-Dimensional Spaces with Pairwise Transitive Groups of Motions 405
Appendix . Problems and Theorems 414
Notes of the Text 424
Index 428