The Studies On Some Important Conformal Properties Of (α,β)-metrics

Conformal geometry is an important part of Finsler geometry. In recent years, thestudy on the conformal properties of Finsler metrics has gained more and more attentions.In this paper, we mainly study some important curvature properties of conformally flat(α,β)-metrics and conformally Berwald Kropina metrics.Firstly, we study conformally flat (α,β)-metrics F=φα (α/β), here is a Riemannmetric, β is a1-form on manifold. Through computation we find that the horizontalcovariant derivative of with respect to satisfies one equation, by using which weprove that conformally flat weak Landsberg (α,β)-metrics must be Riemann metrics orlocally Minkowski metrics, and so do conformally flat (α,β)-metrics with relativelyisotropic mean Landsberg curvature when φ=φ(s)is of special expression.Secondly, based on the navigation representation of Randers metric, we study anotherspecial metric——Kropina metric with navigation representation, through which we obtainrespectively equivalent conditions of Berwald Kropina metric and conformally BerwaldKropina metric. At last, we characterize the local structure of conformally BerwaldKropina metric with weakly isotropic flag curvature.