Recent Synthetic Differential Geometry

Herbert Busemann (Autor)

Buch | Softcover

VIII, 112 Seiten

2012 | 1. Softcover reprint of the original 1st ed. 1970
Springer Berlin (Verlag)
978-3-642-88059-9 (ISBN)

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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.

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.

Erscheint lt. Verlag	11.4.2012
Reihe/Serie	Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Zusatzinfo	VIII, 112 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	200 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Differentialgeometrie • Differential Geometry • Geometry • Riemannian Geometry • Synthetic Differential Geometry
ISBN-10	3-642-88059-2 / 3642880592
ISBN-13	978-3-642-88059-9 / 9783642880599
Zustand	Neuware