FINSLER METRICS WITH BUSEMANN CURVATURE BOUNDS

Abstract

We prove that a Finsler metric has Busemann curvature bounded above (below, respectively) by $\kappa$ if and only if it is the Berwald metric with flag curvature bounded above (below, respectively) by $\kappa$. Combining this with Szabó’s Berwald metrization theorem, we can obtain that such a Finsler metric is affinely equivalent to a Riemannian metric with sectional curvature bounded above (below, respectively) by $\kappa$.