Metrizable isotropic second-order differential equations and Hilbert's fourth problem

Abstract

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. The proof of Theorem 3.1 provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. One condition of Theorem 3.1, regarding the regularity of the sought after Finsler function, can be relaxed. By relaxing this condition, we provide examples of sprays that are metrizable by conic pseudo-Finsler functions as well as degenerate Finsler functions. Hilbert's fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert's fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the conditions of [11, Theorem 4.1] and Theorem 3.1 to test when the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided by the proofs of [11, Theorem 4.1] and Theorem 3.1 to construct solutions to Hilbert's fourth problem.