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The Ricci curvature is introduced by spray on $M^n$. Sprays are deformed to projective sprays with a volume form $dV$ on $M^n$. The projective Ricci curvature is defined as the expression of Ricci curvature with sprays. With this paper, we use the new notion that is called weakly isotropic projective Ricci curvature. We have introduced the idea of weakly isotropic projective Ricci curvature in \cite{GRS21Square}. Then we study and characterize $m$-th root metrics of weakly isotropic projective Ricci curvature. We obtain that every $m$-th root metric of weakly isotropic projective Ricci curvature is projectively Ricci-flat.

Randers metrics are the most important class of Finsler metrics which is defined by a Riemannian metric and a 1-form as . In this paper, the concept of geodesic circle preserve transformations in Finslerian space is studied and the weak Einstein Randers metrics have been investigated. Further we prove this condition for Randers metric of weak isotropic flag curvature and weak isotropic Berwald curvature. ./files/site1/files/%D8%B5%D8%A7%D8%A8%D8%B1%D8%B9%D9%84%DB%8C.pdf

In 2016, Prof. Shen et al. studied one of the most important warped structures of Finsler metrics and investigated Einstein type of the metrics. In this paper, we first compute the E-curvature of the metrics and characterize Finsler warped product metrics with isotropic E-curvature. ./files/site1/files/%DA%AF%D8%A8%D8%B1%D8%A7%D9%86%DB%8C.pdf

An important class of Finsler metric is named Kropina metrics which is defined by Riemannian metric α and 1-form β which have many applications in physic, magnetic field and dynamic systems. In this paper, conformal transformations of χ-curvature and H-curvature of Kropina metrics are studied and the conditions that preserve this quantities are investigated. Also it is shown that in the special cases the conformal transformations reduced to homothetic transformations. ./files/site1/files/62/14Abstract.pdf