Rigidity of Busemann convex Finsler metrics

TL;DR

This paper establishes a rigidity result for Busemann convex Finsler metrics, showing they are essentially Riemannian with nonpositive curvature, and provides new characterizations of Berwald metrics.

Contribution

It proves that Busemann convex Finsler metrics are affinely equivalent to nonpositively curved Riemannian metrics and introduces new characterizations of Berwald metrics.

Findings

Busemann convex Finsler metrics are equivalent to nonpositively curved Riemannian metrics.

In dimension 2, such metrics are either Riemannian or locally isometric to a normed plane.

New characterizations of Berwald metrics via linear parallel transport are obtained.

Abstract

We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane. In the course of the proof we obtain new characterizations of Berwald metrics in terms of the so-called linear parallel transport.