The Geometry of Geodesics (Pure and Applied Mathematics Volume 6)

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