On a Class of Two-Dimensional Singular Douglas and Projectively flat Finsler Metrics

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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 two-dimensional singular Finsler metrics defined by a Riemann metric α and 1-form β, and we characterize those which are Douglasian or locally projectively flat by some equations. It shows that the main class induced is an m-Kropina metric plus a linear part on β. For this class, the local structure of Douglasian or (in part) projectively flat case is determined, and in particular we show that a Kropina metric is always Douglasian and a Douglas m-Kropina metric with m≠ -1 is locally Minkowskian. It indicates that the two-dimensional case is quite different from the higher dimensional ones.