The Geometry of Geodesics,9780486442372

The Geometry of Geodesics

by
ISBN13: 9780486442372
ISBN10: 0486442373
Format: Paperback
Pub. Date: 2005-05-13
Publisher(s): Dover Publications
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Summary

A comprehensive approach to qualitative problems in intrinsic differential geometry, this text opens with an explanation of the basic concepts and proceeds to discussions of Desarguesian spaces, perpendiculars and parallels, and covering spaces. Concluding chapters examine the influence of the sign of the curvature on geodesics and homogenous spaces. 1955 edition. Includes 66 figures.

Table of Contents

Preface v
CHAPTER I. THE BASIC CONCEPTS 1(64)
1. Introduction
1(2)
2. Compact and finitely compact metric spaces
3(7)
3. Convergence of point sets
10(4)
4. Motion and isometry
14(5)
5. Curves and their lengths
19(8)
6. Segments
27(3)
7. Geodesics
30(6)
8. G-spaces
36(8)
9. Multiplicity. Geodesics without multiple points
44(5)
10. Two-dimensional G-spaces
49(7)
11. Plane metrics without conjugate points
56(9)
CHAPTER II. DESARGUESIAN SPACES 65(50)
12. Introduction
65(1)
13. Planes with the Desargues property
66(10)
14. Spaces which contain planes
76(6)
15. Riemann and Finsler spaces. Beltrami's theorem
82(5)
16. Convex sets in affine space
87(7)
17. Minkowskian geometry
94(11)
18. Hilbert's geometry
105(10)
CHAPTER III. PERPENDICULARS AND PARALLELS 115(50)
19. Introduction
115(2)
20. Convexity of spheres and perpendicularity
117(7)
21. Characterization of the higher-dimensional elliptic geometry
124(6)
22. Limit spheres and co-rays in G-spaces
130(7)
23. Asymptotes and parallels in straight spaces
137(7)
24. Characterizations of the higher-dimensional Minkowskian geometry
144(9)
25. Characterization of the Minkowski plane
153(12)
CHAPTER 1V. COVERING SPACES 165(70)
26. Introduction
165(2)
27. Locally isometric spaces
167(7)
28. The universal covering space
174(7)
29. Fundamental sets
181(7)
30. Locally Minkowskian, hyperbolic, or spherical spaces
188(11)
31. Spaces in which two points determine a geodesic
199(5)
32. Free homotopy and closed geodesics
204(11)
33. Metrics without conjugate points on the torus
215(8)
34. Transitive geodesics on surfaces of higher genus
223(12)
CHAPTER V. THE INFLUENCE OF THE SIGN OF THE CURVATURE ON THE GEODESICS 235(72)
35. Introduction
235(2)
36. Local properties
237(11)
37. Non-positive curvature in the theory of parallels
248(6)
38. Straightness of the universal covering space r
254(4)
39. The. fundamental groups of spaces with convex capsules
258(4)
40. Geodesics in spaces with negative curvature
262(5)
41. Relation to non-positive curvature in standard sense
267(6)
42. Angular measure
273(9)
43. Excess and characteristic
282(10)
44. Simple monogons, total excess, surfaces with positive excess
292(15)
CHAPTER VI. HOMOGENEOUS SPACES 307(96)
45. Introduction
307(2)
46. Spaces with flat bisectors I
309(11)
47. Spaces with flat bisectors II
320(13)
48. Applications of the bisector theorem. The Helmholtz-Lie Problem
333(10)
49. Involutoric motions
343(7)
50. New characterizations of the Minkowskian spaces
350(9)
51. Translations along two lines
359(7)
52. Surfaces with transitive groups of motions
366(8)
53. The hermitian elliptic and similar spaces
374(11)
54. Compact spaces with pairwise transitive groups of motions
385(9)
55. Odd-dimensional spaces with pairwise transitive groups of motions
394(9)
APPENDIX. Problems and Theorems 403(10)
NOTES TO THE TEXT 413(4)
INDEX 417

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