On projectively flat Kropina metrics

TL;DR

This paper characterizes when Kropina metrics are projectively flat, showing that under certain conditions they are locally Minkowskian, thus advancing understanding of their geometric properties.

Contribution

It provides a necessary and sufficient condition for projectively flat Kropina metrics with constant curvature to be locally Minkowskian.

Findings

A characteristic condition for projectively flat Kropina metrics is established.

Kropina metrics with constant curvature and unit norm are projectively flat iff they are locally Minkowskian.

Abstract

In this paper, a characteristic condition of the projectively flat Kropina metric is given. By it, we prove that a Kropina metric $F=\alpha^2/\beta$ with constant curvature $K$ and $\|\beta\|_{\alpha}=1$ is projectively flat if and only if $F$ is locally Minkowskian.