Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature (Q282868)

A \textit{Funk function} is a projective deformation of a homogeneous system of second-order ordinary differential equations (SODE) that preserves the curvature (Jacobi endomorphism) of the given SODE. For a given flat SODE there are many corresponding Funk functions and each of them will projectively deform the given SODE into a Finsler metrizable one, solution to Hilbert's fourth problem, after Theorem 10.3.5 of \textit{Z. Shen}'s book [Differential geometry of spray and Finsler spaces. Dordrecht: Kluwer (2001; Zbl 1009.53004)].NEWLINENEWLINEThe existence of non-trivial Funk functions for the general case of a non-flat SODE is an open problem from the same reference, page 177. The main result of this paper, namely Theorem 3.1, proves that there are no non-trivial Funk functions in the case of non-vanishing scalar flag curvature. In particular, there are no Funk functions for an isotropic spray \(S\) of non-vanishing Ricci curvature that will projectively deform \(S\) into a Finsler metrizable spray.