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 α and 1-form β   and characterize those which are respectively Douglasian and locally projectively flat in dimension n⩾3n⩾3 by some equations. Our study shows that the main class induced is an m-Kropina metric plus a linear part on β  . For this class with m≠−1m≠−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.