Funk functions and projective deformations of sprays and Finsler spaces of scalar flag curvature

TL;DR

This paper investigates whether projectively related Finsler metrics of scalar flag curvature can share the same Riemann curvature tensor, concluding negatively for non-vanishing scalar flag curvature and exploring spray representations.

Contribution

It provides a negative answer to Shen's question for non-vanishing scalar flag curvature and constructs sprays that cannot be metrized to share the same Riemann curvature tensor.

Findings

Negative answer for non-vanishing scalar flag curvature

Construction of sprays not metrizable to share Riemann curvature

Relation to Hilbert's Fourth Problem

Abstract

In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature tensor, [14, page 184]. In this paper, we provide an answer to this question, within the class of Finsler metrics of scalar flag curvature. In Theorem 3.1, we show that the answer is negative, for non-vanishing scalar flag curvature. The answer is known to be positive when the scalar flag curvature vanishes, [12, 14] and this positive answer is related to the existence of many solutions to Hilbert's Fourth Problem. As a generalisation of this problem, we can ask if it is possible for a given spray, with non-vanishing scalar flag curvature, to represent, after reparametrisation, the geodesic spray of a Finsler metric. In Proposition 3.3, we show how to construct sprays whose projective class does not contain any Finsler metrizable spray with the same Riemann curvature tensor.