On a class of two-dimensional Douglas and projectively flat Finsler metrics

We study a class of two-dimensional Finsler metrics defined by a Riemannian metric α and a 1-form β. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that β 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 (α, β)-metrics which are Douglasian, and some families of examples are given for projectively flat classes with β being not closed.