On a Class of Singular Douglas and Projectively flat Finsler Metrics

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric $\alpha$ and 1-form $\beta$ and characterize those which are respectively Douglasian and locally projectively flat in dimension $n\ge 3$ by some equations. Our study shows that the main class induced is an $m$-Kropina metric plus a linear part on $\beta$. For this class with $m\ne -1$, the local structure of projectively flat case is determined, and it is proved that a Douglas $m$-Kropina metric must be Berwaldian and a projectively flat $m$-Kropina metric must be locally Minkowskian. It indicates that the singular case is quite different from the regular one.