The Studies On The Projective Ricci Curvature And The Related Problems In Finsler Geometry

In this paper,we study a new geometric quantity in Finsler geometry,that is,the projective Ricci curvature.We mainly study the projective invariance of projective Ricci curvature,projective Ricci flat Kropina metrics and projective Ricci flat Randers metrics.Firstly,we study the projective invariance of projective Ricci curvature.We characterize the relations between projective Ricci curvatures of two projective equivalent Finsler metrics.In particular,we show that,if F and F are pointwise projectively related Finsler metrics on a manifold with a fixed volume form,then their projective Ricci curvatures are equal.In other words,the projective Ricci curvature is a projective invariant in this case.Secondly,we study and characterize projective Ricci flat Kropina metrics.Further,as a natural application,we study and characterize projective Ricci flat Kropina metrics defined by a Riemannian metric and a Killing 1-form of constant length.We also characterize projective Ricci flat Kropina metrics with isotropic S-curvature.In this case,the metric is Ricci flat.Besides,as a joint work with other people,we also obtain the formula of the projective Ricci curvature for Randers metrics.Based on this,we characterize projective Ricci flat Randers metrics.