The Geometry of Geodesics
The Geometry of Geodesics
Published on 21. September 2011
421 pages
978-0-08-087314-5 (ISBN)
Description
The Geometry of Geodesics
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Content
- Front Cover
- The Geometry of Geodesics
- Copyright Page
- Table of Contents
- Preface
- Chapter I. The Basic Concepts
- 1. Introduction
- 2. Compact and Finitely Compact Metric Spaces
- 3. Convergence of Point Sets
- 4. Motion and Isometry
- 5. Curves and Their Lengths
- 6. Segments
- 7. Geodesics
- 8. G-Spaces
- 9. Multiplicity, Geodesics Without Multiple Points
- 10. Two-Dimensional G-Spaces
- 11. Plane Metrics Without Conjugate Points
- Chapter II. Desarguesian Spaces
- 12. Introduction
- 13. Planes with the Desargues Property
- 14. Spaces Which Contain Planes
- 15. Riemann and Finsler Spaces. Beltrami's Theorem
- 16. Convex Sets in Affine Space
- 17. Minkowskian Geometry
- 18. Hilbert's Geometry
- Chapter III. Perpendiculars and Parallels
- 19. Introduction
- 20. Convexity of Spheres and Perpendicularity
- 21. Characterization of the Higher-Dimensional Elliptic Geometry
- 22. Limit Spheres and Co-Rays in G-Spaces
- 23. Asymptotes and Parallels in Straight Spaces
- 24. Characterizations of the Higher-Dimensional Minkowskian Geometry
- 25. Characterization of the Minkowski Plane
- Chapter IV. Covering Spaces
- 26. Introduction
- 27. Locally Isometric Spaces
- 28. The Universal Covering Space
- 29. Fundamental Sets
- 30. Locally Minkowskian, Hyperbolie or Spherical Spaces
- 31. Spaces in Which Two Points Determine a Geodesic
- 32. Free Homotopy and Closed Geodesics
- 33. Metrics Without Conjugate Points on the Torus
- 34. Transitive Geodesics on Surfaces of Higher Genus
- Chapter V. The Influence of the Sign of the Curvature on the Geodesics
- 36. Introduction
- 36. Local Properties
- 37. Non-Positive Curvature in the Theory of Parallels
- 38. Straightness of the Universal Covering Space
- 39. The Fundamental Groups of Spaces with Convex Capsules
- 40. Geodesics in Spaces with Negative Curvature
- 41. Relation to Non-Positive Curvature in Standard Sense
- 42. Angular Measure
- 43. Excess and Characteristic
- 44. Simple Monogons, Total Excess. Surfaces with Positive Excess
- Chapter VI. Homogeneous Spaces
- 45. Introduction
- 46. Spaces with Flat Bisectors I
- 47. Spaces with Flat Bisectors II
- 48. Applications of the Bisector Theorem. The Helmholtz-Lie Problem
- 49. Involutoric Motions
- 50. New Characterizations of the Minkowskian Spaces
- 51. Translations Along Two Lines
- 52. Surfaces With Transitive Groups of Motions
- 53. The Hermitian Elliptic and Similar
- 54. Compact Spaces with Pairwise Transitive Groups of Motions
- 55. Odd-Dimensional Spaces with Pairwise Transitive Groups of Motions
- Appendix . Problems and Theorems
- Notes of the Text
- Index
