On non-positively curved homogeneous Finsler metrics

Abstract

In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. Our second main result is to show that every homogeneous (α,β)-metric with non-positive flag curvature and almost isotropic S-curvature is Riemannian or locally Minkowskian.