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 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 Douglasian, and some families of examples are given for projectively flat classes with beta being not closed.