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 $\kappa$.
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