  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.
