Abstract:
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 $\dim M\geq 3$.
