%0 Journal Article
%T On a Class of Two-Dimensional Singular Douglas and Projectively flat Finsler Metrics
%A Guojun Yang
%J Mathematics
%D 2013
%I arXiv
%X 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 $\alpha$ and 1-form $\beta$, 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 $\beta$. 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 $m\ne -1$ is locally Minkowskian. It indicates that the two-dimensional case is quite different from the higher dimensional ones.
%U http://arxiv.org/abs/1302.4120v1