On Berwald Spaces with non-Zero Flag Curvature

TL;DR

This paper proves that Berwald manifolds with non-zero flag curvature are Riemannian, extending rigidity theorems, and classifies various isotropic Berwald manifolds, showing they are either Riemannian, Minkowskian, or Berwaldian.

Contribution

It extends rigidity results to Berwald spaces with non-zero flag curvature and classifies isotropic Berwald manifolds under various curvature conditions.

Findings

Berwald manifolds with non-zero flag curvature are Riemannian

Positively curved isotropic Berwald manifolds are Riemannian or Minkowskian

Homogeneous isotropic Berwald metrics are Riemannian, Minkowskian, or Berwaldian

Abstract

We prove that every Berwald manifold with non-zero flag curvature is Riemannian. This result provides an extension of Numata and Szabo's rigidity theorems. We show that every positively curved constant isotropic Berwald manifold is Riemannian or locally Minkowskian. Then, we prove that every compact strictly positive (or negative) isotropic Berwald manifold reduces to a Berwald manifold. Finally, we prove that every homogeneous isotropic Berwald metric is either locally Minkowskian, or Riemannian, or Berwald metric or Berwald-Randers metric generalizing result previously only known in the case of Randers metric.