On a class of two-dimensional singular Douglas metrics

In this paper, we characterize a class of singular Douglas [Formula: see text]-metrics [Formula: see text] determined by [Formula: see text] on a two-dimensional manifold [Formula: see text], where [Formula: see text], [Formula: see text] is a Riemannian metric and [Formula: see text] is a 1-form on [Formula: see text]. When [Formula: see text] and under certain conditions about [Formula: see text], we show that any such a metric is a Douglas metric if and only if it is a Finsler metric derived from a [Formula: see text]-transformation of an [Formula: see text]-Kropina metric. For such a special class of singular [Formula: see text]-metrics, we further completely determine their local structure when they are, respectively, Douglasian and locally projectively flat. In particular, we show that a Kropina metric is always Douglasian and a Douglas [Formula: see text]-Kropina metric with [Formula: see text] is locally Minkowskian on a two-dimensional manifold. The main results in this paper are quite different from the results in the higher dimension case.