Projectively flat Finsler 2-spheres of constant curvature

Abstract

After recalling the 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 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 of Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature K and whose geodesic path 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.