Properties of the Riemannian Curvature of (α, β)-Metrics

Abstract

In this paper, we discuss some important properties of the Riemannian curvature of (α, β)-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 (α, β)-metrics of scalar flag curvature with isotropic S-curvature. We also characterize Einstein (α, β)-metrics and determine completely the local structure of Ricci-flat Douglas (α, β)-metrics when the dimension dim M ≥ 3.