A Schur type lemma for the Mean Berwald curvature in Finsler geometry

TL;DR

This paper investigates the mean Berwald curvature in Finsler geometry, showing that isotropy along fibers implies constancy of the Berwald scalar curvature, thus extending classical results with a Schur type lemma.

Contribution

It establishes a Schur type lemma for the mean Berwald curvature, linking isotropy to scalar curvature constancy in Finsler geometry.

Findings

Mean Berwald curvature isotropic along fibers implies constant Berwald scalar curvature

Analysis of fiber geometry in the tangent sphere bundle

Extension of classical Schur lemma to Finsler setting

Abstract

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.