Projectively flat Finsler 2-spheres of constant curvature

TL;DR

This paper classifies special Finsler structures on the 2-sphere with constant curvature and geodesics as great circles, revealing a 2-parameter family with only one symmetric case, related to Hilbert's Fourth Problem.

Contribution

It provides a complete classification of projectively flat Finsler 2-spheres with constant curvature, identifying a 2-parameter family and highlighting the unique symmetric case.

Findings

Two-parameter family of Finsler structures classified

Only one structure is homogeneous or symmetric

Connections with Hilbert's Fourth Problem discussed

Abstract

I classify the Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature and whose geodesics are the great circles. Modulo diffeomorphism, there is a 2-parameter family of such Finsler structures, only one of which is homogeneous or symmetric, namely the Riemannian one. I discuss the history of the problem and its relation with Hilbert's Fourth Problem and the calculus of variations.