Strong rigidity of constant curvature Finsler manifolds

TL;DR

This paper extends a classical rigidity theorem to Finsler geometry, showing that complete connected Finsler manifolds with positive constant flag curvature are isometrically homeomorphic to spheres with specific Finsler metrics.

Contribution

It generalizes the Obata-Tanno's theorem to Finsler manifolds, establishing a rigidity result for positive constant flag curvature cases.

Findings

Complete connected Finsler manifolds with positive constant flag curvature are sphere-like.

Such Finsler manifolds are isometrically homeomorphic to spheres with particular Finsler metrics.

The extension of classical rigidity results to Finsler geometry is achieved.

Abstract

Here, an extension of the Obata-Tanno's theorem to Finsler geometry is established and the following rigidity result is obtained; Every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an $n$-sphere equipped with a certain Finsler metric, and vise versa.