The Studies On Some Curvature Properties In Spray Geometry

In this paper,we study some problems on the projective Ricci curvature of a Spray,Finsler metrizability of a Spray and the Finsler metrics with weakly isotropic scalar curvature.In section 3,we study a class of Sprays constructed by Funk metric.Under the condition that the Spray is projective Ricci flat,we determine the corresponding volume form of manifold M.Furthermore,we determine the metrizability conditions of this kind of Sprays,and obtain the condition that the Sprays are of weakly projective Ricci curvature.We also give the classification results of the metrics if the corresponding metrics are non-projective Ricci flat Finsler metrics.In section 4 the focus is on the Randers metrics of weakly isotropic scalar curvature.We prove that the Randers metrics with weakly isotropic scalar curvature must be of isotropic S-curvature.Moreover,the conformal flat Randers metrics of weakly isotropic scalar curvature are completely classified.