We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane. In the course of the proof we obtain new characterizations of Berwald metrics in terms of the so-called linear parallel transport.
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Sergei Ivanov, Alexander Lytchak, Rigidity of Busemann convex Finsler metrics. Comment. Math. Helv. 94 (2019), no. 4, pp. 855–868DOI 10.4171/CMH/476