Riemannian metrics having common geodesics with Berwald metrics

Abstract

In Theorem 1, we generalize the results of Szabo for metrics that are not necessary strictly convex: we show that for every metric F there always exists a Riemannian metric affine equivalent to F. As an application we show (Corollary 3) that every projectively flat metric is a Minkowski metric; this statement is a Berwald version of Hilbert's 4th problem. Further, we investigate geodesic equivalence of metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially metric admits a Riemannian metric that is (nontrivially) geodesically equivalent to it. The system of PDE is linear and of Cauchy-Frobenius type, i.e., the derivatives of unknown functions are explicit expressions of the unknown functions. As an application (Corollary 2), we obtain that geodesic equivalence of an essentially metric and a Riemannian metric is always affine equivalence provided both metrics are complete.