On $m$-Kropina Finsler Metrics of Scalar Flag Curvature

TL;DR

This paper studies a special class of singular Finsler metrics called $m$-Kropina metrics, showing that most with scalar flag curvature are locally Minkowskian, and characterizes the scalar flag curvature and projective flatness for the case $m=-1$.

Contribution

It proves that $m$-Kropina metrics with scalar flag curvature are locally Minkowskian for $m eq -1$, and characterizes scalar flag curvature and projective flatness for $m=-1$ via PDEs.

Findings

$m$-Kropina metrics ($m eq -1$) of scalar flag curvature are locally Minkowskian in dimension ≥ 3.

Characterization of $m=-1$ Kropina metrics with scalar flag curvature using PDEs.

Development of methods to construct non-trivial Kropina metrics with 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.