On Douglas general $(\alpha ,\beta )$-metrics

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general $(\alpha ,\beta )$-metrics, which are defined by a Riemannian metric $\alpha =\sqrt{a_{ij}(x)y^i{y}^j}$ and a $1$-form $\beta =b_i(x)y^i$. We obtain the differential equations that characterizes these metrics with vanishing Douglas curvature. By solving the equivalent PDEs, the metrics in this class are totally determined. Then many new Douglas metrics are constructed.