On m-Kropina Finsler Metrics of Scalar Flag Curvature

Abstract

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. We show that an m-Kropina metric ( $$m\ne -1$$ ) of scalar flag curvature must be locally Minkowskian in dimension $$n\ge 3$$ . We characterize by some PDEs a Kropina metric ( $$m=-1$$ ) which is respectively of scalar flag curvature and locally projectively flat in dimension $$n\ge 3$$ , and obtain some principles and approaches of constructing non-trivial examples of Kropina metrics of scalar flag curvature.