The Studies On The Conformal Invariances In Finsler Geometry

In this paper,we study some important conformal invariances in Finsler geometry,in which the locally dually flat??,??-metrics,the conformally flat??,??-metrics and the projective Ricci curvature are involved.Firstly,we prove that,if a Finsler metric F is conformally related to a locally dually flat regular??,??-metric F,that is,F=e^s?x?F,then Fis also a locally dually flat regular??,??-metric if and only if the conformal transformation is a homothety.Further,in the case with singularity,we prove that any transformation between two locally dually flat general Kropina metrics must be a homothety.Secondly,we study the rigidity properties of conformally flat??,??-metrics.Based on the characterrization of projective Ricci flat Randers metrics,we prove that conformally flat and projective Ricci flat Randers metrics must be Minkowskian.Besides,under certain conditions,we prove that conformally flat and locally dually flat??,??-metrics must be Minkowskian.Finally,under the condition that?is a closed and conformal 1-form with respect to?,we prove that conformal flat??,??-metrics must be Minkowskian.