On $m$-Kropina Metrics of Scalar Flag Curvature

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an $m$-Kropina metric ($m\ne -1$) of scalar flag curvature must be locally Minkowskian in dimension $n\ge 3$. For $m=-1$, we respectively characterize Kropina metrics which are of scalar flag curvature and locally projectively flat in dimension $n\ge 3$ by some equations, and obtain some principles and approaches of constructing non-trivial examples of Kropina metrics of scalar flag curvature.