Rigidity results for geodesically reversible Finsler metrics

TL;DR

This paper investigates the rigidity of geodesically reversible Finsler metrics, providing new theorems that clarify the scarcity of such metrics on closed manifolds and addressing aspects of Hilbert's fourth problem.

Contribution

It establishes rigidity theorems for geodesically reversible Finsler metrics, advancing understanding of their structure and rarity on closed manifolds.

Findings

Rigidity theorems limit the existence of certain asymmetric Finsler metrics.

Partially solves cases of Hilbert's fourth problem for asymmetric metrics.

Clarifies the relationship between volume, area, and geodesic reversibility.

Abstract

A Finsler metric is geodesically reversible if geodesics remain geodesics after a change of orientation. Asymmetric norms on vector spaces and Funk metrics in the interior of convex bodies are examples of geodesically reversible metrics that are not necessarily sums of reversible metrics and closed 1-forms. However, there seem to be few such examples in closed manifolds. In this paper the theory of volumes and areas on Finsler spaces is applied to establish a number of rigidity theorems which partially explain this paucity of examples. These rigidity results settle some hitherto unsolved cases of Hilbert's fourth problem for asymmetric metrics.