Higher rank rigidity for Berwald spaces

TL;DR

This paper extends the higher rank rigidity theorem to Berwald spaces, showing that under certain conditions, such spaces are either locally symmetric or Minkowski, broadening the understanding of geometric rigidity in Finsler geometry.

Contribution

It generalizes the higher rank rigidity theorem from Riemannian to Berwald Finsler spaces, identifying conditions under which these spaces are locally symmetric or Minkowski.

Findings

Berwald spaces with rank ≥ 2 and irreducible universal cover are locally symmetric or Minkowski.

The theorem applies to complete, connected spaces with finite volume and bounded nonpositive flag curvature.

The result broadens the scope of rigidity theorems in Finsler geometry.

Abstract

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded nonpositive flag curvature with rank at least $2$ whose universal cover is irreducible, is a locally symmetric space or a locally Minkowski space.