Properties of the riemannian curvature of $(\alpha,\beta)$-metrics

In this paper, we discuss some important properties of the Riemannian curvature of $(\alpha,\beta)$-metrics. When the dimension of the manifold is greater than 2, we classify Randers metrics of weakly isotropic flag curvature (that is, Randers metrics of scalar flag curvature with isotropic $S$-curvature). Further, we characterize $(\alpha,\beta)$-metrics of scalar flag curvature with isotropic $S$-curvature. We also characterize Einstein $(\alpha,\beta)$-metrics and determine completely the local structure of Ricci-flat Douglas $(\alpha,\beta)$-metrics when the dimension $\dim M\geq 3$.