The Studies On Ricci Curvature And Projective Equivalence In Finsler Geometry

The Ricci curvature in Finsler geometry is the natural extension of the Ricci curvature in Riemannian geometry and plays an important role in Finsler geometry.In recent years,the study on the Ricci curvature has gained more and more attentions.In this paper,we mainly study some relationships of two projective equivalent metrics under certain conditions on Ricci curvature and study projective Ricci flat Finsler metrics.Firstly,we study two projective equivalent metrics under certain conditions on Ricci curvature and scalar curvature.In this case,we prove that any pointwise C-projective change from a Berwald space (M,(?))to a Riemann space (M,F) is trivial.In particular,under the same conditions,we prove that any pointwise projective change from a Riemann space to another Riemann space is trivial.Secondly,we study the projective Ricci curvature in Finsler geometry.We characterize the geometrical properties and structure of projective Ricci flat Randers metrics.As a natural application,we characterize projective Ricci flat Randers metrics with isotropic S-curvature.In this case,the metrics are acturally weak Einstein Finsler metrics.Finally,as a joint work with others,we study and characterize projective Ricci flat Kropina metrics.