On general (alpha,beta)-metrics with vanishing Douglas curvature

Abstract

In this paper, we study a class of Finsler metrics called general $(\alpha ,\beta )$-metrics, which are defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$. We find an equation which is necessary and sufficient condition for such Finsler metric to be a Douglas metric. By solving this equation, we obtain all of general $(\alpha ,\beta )$-metrics with vanishing Douglas curvature under certain condition. Many new non-trivial examples are explicitly constructed.