ar X iv : d g - ga / 9 61 10 10 v 1 2 5 N ov 1 99 6 PROJECTIVELY FLAT FINSLER 2 - SPHERES OF CONSTANT

After recalling the structure equations of Finsler structures on surfaces, I define a notion of ‘generalized Finsler structure’ as a way of micro-localizing 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. 0. Introduction Hilbert’s Fourth Problem was entitled “Problem of the straight line as the shortest distance between two points”. It concerned, in its most general formulation, the problem of characterizing the not-necessarily-symmetric distance functions d that could be defined on (convex) subsets U ⊂ R so that the lines were geodesics, i.e., so that d(x, z) ≤ d(x, y) + d(y, z) with equality if and only if x, y, and z are collinear, with y lying on the segment joining x to z. Hilbert’s reason for considering non-symmetric distances was that interesting non-symmetric examples had already been discovered by Minkowski. He was also aware that notions of length of curves defined in many calculus of variations problems leads naturally to non-symmetric distance functions. For some interesting examples of physical relevance, see Carathéodory’s book [Cara]. The Fourth Problem can be regarded as a fundamental example of the inverse problem in the calculus of variations. That is, given that the straight lines are the extremals (i.e., geodesics) of a first order Lagrangian for oriented curves in the 1991 Mathematics Subject Classification. 53C60, 53A20, 58G30.