The Conformal Relations Of(α, β)-metrics And Properties Of Douglas Curvatures

In this paper, we study a very important class of Finsler metrics—— ??),(? metrics in Finsler geometry. We first study the conformal invariances for two important classes of locally dually flat ??),(? metrics. Then we get the sufficient and necessary conditions for a ??),(? metric to be a generalized Douglas-Weyl metric.Firstly, we first study the conformal invariances for locally dually flat Randers metrics and square metrics respectively. By using the conditions that characterize a locally dually flat Randers metric and comparing the geodesic coefficients and 1-forms of two conformally related Randers metrics, we obtain the property of a conformal transformation which makes the quality of locally dually flat Randers metrics remains unchanged. We conclude that the conformal transformation between two locally dually flat Randers metrics must be a homothety. In addition, we found that the similar conclusion holds for two conformally related locally dually flat square metrics, that is the conformal transformation between two locally dually flat square metrics must be a homothety.Secondly, inspired by the conditions which characterize generalized Douglas-Weyl Randers metrics, we study generalized Douglas-Weyl ??),(? metrics. Under the condition that the metric is of isotropic S-curvature, we obtain the conditions that a regular ??),(?metric of non-Randers type to be a generalized Douglas-Weyl metric.