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

TL;DR

This paper proves that non-trivial Funk functions do not exist for Finsler spaces with non-zero scalar flag curvature, answering a question posed by Zhongmin Shen.

Contribution

It establishes a non-existence result for Funk functions in a specific class of Finsler spaces, advancing understanding of their geometric properties.

Findings

Funk functions do not exist for Finsler spaces with non-vanishing scalar flag curvature.

The result addresses a question posed by Zhongmin Shen.

The proof clarifies the structure of geodesic sprays in these Finsler spaces.

Abstract

In his book "Differential Geometry of Spray and Finsler spaces", page 177, Zhongmin Shen asks "wether or not there always exist non-trivial Funk functions on a spray space". In this note, we will prove that the answer is negative for the geodesic spray of a finslerian function of non-vanishing scalar flag curvature.