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

In this paper, we study a class of two-dimensional Finsler metrics defined by a Riemannian metric $\alpha$ and a 1-form $\beta$. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that $\beta$ is always closed for those metrics in higher dimensions is no longer true in two dimensional case. Further, we determine the local structures of two-dimensional $(\alpha,\beta)$-metrics which are Douglassian, and some families of examples are given for projectively flat classes with $\beta$ being not closed.