On a Class of Generalized Berwald Manifolds

The class of generalized Berwald metrics contains the class of Berwald metrics. In this paper, we characterize two-dimensional generalized Berwald $(\alpha, \beta)$-metrics with vanishing S-curvature. Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be a two-dimensional generalized Berwald $(\alpha,\beta)$-metric on a manifold $M$. Suppose that $F$ has vanishing S-curvature. We show that one of the following holds: (i) if $F$ is a regular metric, then it reduces to a Riemannian metric of isotropic sectional curvature or a locally Minkowskian metric; (ii) if $F$ is an almost regular metric that is not Riemannian nor locally Minkowskian, then we find the explicit form of $\phi=\phi(s)$ which obtains a generalized Berwald metric that is neither a Berwald nor Landsberg nor a Douglas metric. This provides a generalization of Szab\'{o} rigidity theorem for the class of $(\alpha,\beta)$-metrics. In the following, we prove that left invariant Finsler surfaces with vanishing S-curvature must be Riemannian surfaces of constant sectional curvature. Finally, we construct a family of odd-dimensional generalized Berwald Randers metrics.