Rigidity of Busemann convex Finsler metrics

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.

arXiv commentv3: one of the open questions posed in the previous version turned out to have simple counter-examples; Section 6 is rewritten and now describes these examples; the corresponding part of the introduction is changed