Recent Synthetic Differential Geometry

1. Front Matter

   Pages I-VII

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2. Completeness, Finite Dimensionality, Differentiability

   • Herbert Busemann

   Pages 1-26

3. Desarguesian Spaces

   • Herbert Busemann

   Pages 27-40

4. Length Preserving Maps

   • Herbert Busemann

   Pages 41-55

5. Geodesic

   • Herbert Busemann

   Pages 56-90

6. Motions

   • Herbert Busemann

   Pages 91-100

7. Observations on Method and Content

   • Herbert Busemann

   Pages 101-105

8. Back Matter

   Pages 106-112

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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.

• Differentialgeometrie
• Riemannian geometry
• Synthetic Differential Geometry
• differential geometry
• geometry

• University of Southern California, Los Angeles, USA

  Herbert Busemann

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