Riemannian metrics having the same geodesics

with Berwald metrics

with Berwald metrics

(Submitted on 31 Oct 2008)

Abstract: In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F.

Further, we investigate geodesic equivalence of Berwald metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwald 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 a corollary, we obtain that geodesic equivalence of an essentially Berwald and a Riemannian metrics is always affine equivalence provided both metrics are complete.

the link is http://arxiv.org/abs/0811.0031