Publications [#243403] of Robert Bryant

Papers Published

1. Bryant, RL, Projectively flat Finsler 2-spheres of constant curvature, Selecta Mathematica, vol. 3 no. 2 (June, 1997), pp. 161-203, Springer Science and Business Media LLC [MR98i:53101], [dg-ga/9611010], [doi]
   (last updated on 2025/03/31)

   Author's Comments:
   This paper might be regarded as a sequel to the previous one. In it, I prove that, up to diffeomorphism, there is a 2-parameter family of Finsler metrics on the standard 2-sphere whose geodesics are the great circles and whose Finsler-Gauss curvature is identically 1. Explicit formulas for these Finsler metrics are established and it is shown that the only symmetric Finsler metrics with this property are the known Riemannian ones. The introduction contains a discussion of the relation of these results with Hilbert's Fourth Problem.

   Abstract:
   After recalling the structure equations of Finsler structures on surfaces, I define a notion of "generalized Finsler structure" as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of "generalized path geometry" analogous to that of "generalized Finsler structure." I use these ideas to study the geometry of Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature K and whose geodesic path geometry is projectively flat, i.e., locally equivalent to that of straight lines in the plane. I show that, modulo diffeomorphism, there is a 2-parameter family of projectively flat Finsler structures on the sphere whose Finsler-Gauss curvature K is identically 1. © Birkhäuser Verlag, 1997.