The Studies Of Some Important Problems On The Flag Curvature And The Projective Ricci Curvature Of Finsler Metrics

In this paper,we study some problems on the conformal vector fields,the flag curvature and the projective Ricci curvature of Finsler metrics,and get some meaningful research results.Firstly,we study the conformal vector fields of the Kropina metrics and obtain the equivalent conditions that the conformal vector fields of the Kropina metrics satisfy.Based this,we determine completely the conformal vector fields of the Kropina metrics of weakly isotropic flag curvature via navigation data(h,W).Secondly,we study the Finsler metrics with some special flag curvature properties.We obtain a system of partial differential equations which the Finsler metrics of weakly isotropic flag curvature satisfy.Besides,we prove that the H-curvature vanishes when a Finsler metric is of constant mean Berwald curvature.Further,we discuss Finsler metrics of scalar flag curvature and of constant mean Berwald curvature.In this case,we find an identity that the flag curvature K satisfies and prove that K is actually a constant when n is greater than 2.Finally,we study the projective Ricci curvature in Finsler geometry.We obtain a comparison theorem on the projective Ricci curvature on a complete Finsler manifold.In addition,we characterize the relations between two projective Ricci curvatures for two conformally related Finsler metrics on a manifold M.Then we prove that projective Ricci curvature is invariant under homothetic transformations.