A note on non-negatively curved Berwald spaces

Abstract

In this note it is shown that Berwald spaces admitting the same norm-preserving torsion-free affine connection have the same (weighted) Ricci curvatures. Combing this with Szab\'o's Berwald metrization theorem one can apply the Cheeger-Gromoll splitting theorem in order to get a full structure theorem for Berwald spaces of non-negative Ricci curvature. Furthermore, if none of the factor is a symmetric space one obtains an explicit expression of Finsler norm of the resulting product. By the general structure theorem one can apply the soul theorem to the factor in case of non-negative flag curvature to obtain a compact totally geodesics, totally convex submanifolds whose normal bundle is diffeomorphic to the whole space. In the end we given applications to the structure of Berwald-Einstein manifolds and non-negatively curved Berwald spaces of large volume growth.