Math @ Duke
Publications [#243403] of Robert Bryant
- Bryant, RL, Projectively flat finsler 2-spheres of constant curvature,
Selecta Mathematica, vol. 3 no. 2
pp. 161-203, Springer Nature [MR98i:53101], [dg-ga/9611010], [doi]
(last updated on 2019/05/24)
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
with Hilbert's Fourth Problem.
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.
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