Recent Synthetic Differential Geometry

ISBN-13: 9783642880599 / Angielski / Miękka / 2012 / 112 str.

Herbert Busemann

Opis

A synthetic approach to intrinsic differential geometry in the large and its connections with the foundations of geometry was presented in "The Geometry of Geodesics" (1955, quoted as G). It is the purpose of the present report to bring this theory up to date. Many of the later ip.vestigations were stimulated by problems posed in G, others concern newtopics. Naturally references to G are frequent. However, large parts, in particular Chapters I and III as weIl as several individual seetions, use only the basic definitions. These are repeated here, sometimes in a slightly different form, so as to apply to more general situations. In many cases a quoted result is quite familiar in Riemannian Geometry and consulting G will not be found necessary. There are two exceptions : The theory of paralleIs is used in Sections 13, 15 and 17 without reformulating all definitions and properties (of co-rays and limit spheres). Secondly, many items from the literature in G (pp. 409-412) are used here and it seemed superfluous to include them in the present list of references (pp. 106-110). The quotations are distinguished by [ ] and ( ), so that, for example, FreudenthaI [1] and (I) are found, respectively, in G and here.

Spis treści

I. Completeness, Finite Dimensionality, Differentiability.- 1. The Theorem of Hopf and Rinow.- 2. Geodesic Completeness. Local Homogeneity.- 3. The Topology of r-Spaces.- 4. Finite-Dimensional G-Spaces.- 5. Differentiability.- II. Desarguesian Spaces.- 6. Similarities.- 7. Imbeddings of Desarguesian Spaces.- 8. A Characterization of Hilbert’s and Minkowski’s Geometries.- III. Length Preserving Maps.- 9. Shrinkages, Equilong Maps, Local Isometries.- 10. Spaces without Proper Local Isometries.- 11. Proper Equilong Maps.- IV. Geodesics.- 12. Closed Hyperbolic Space Forms.- 13. Axes of Motions and Closed Geodesics.- 14. Plane Inverse Problems. Higher Dimensional Collineation Groups.- 15. One-Dimensional and Discrete Collineation Groups.- 16. Bonnet Angles. Quasi-Hyperbolic Geometry.- 17. Various Aspects of Conjugacy.- V. Motions.- 18. Finite and One-Parameter Groups of Motions.- 19. Transitivity on Pairs of Points and on Geodesies.- VI. Observations on Method and Content.- Literature.