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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.

The Sa’di researchers have often assumed the tales of the Golestan to be “real or semi-real incidents” and, then, interpreted the tales in line with this assumption. For instance, they have considered the internal genre of the tale of “Faqih’s Extremely Ugly Daughter’s Marriage to the Blind Man” to be based on a true daily story about the “importance of woman’s beauty in marriage”. The consequence of this certain type of reading is lack of convergence in the existing interpretations and accusing Sa’di of contradictory and even futile words. The writers of this research paper, however, believe the tale to be a “symbolic-metaphorical allegory” and its subject to be “a criticism of political behaviors of those faqihs (jurisprudents) serving the men of power”. The body of this tale is composed of tangible and experimental structured concepts with a long metaphorical history in the Persian literature which are used to form a secondary concept. The correspondence between the parallel lines of “semantic field of each word” and its “metaphorical field” reveal the network of partial implications and finally the writer’s intent. Therefore, “the faqih’s extremely ugly daughter” is a metaphor for the “daughter of imposition” or “the faqih’s extremely meaningless words”, “reluctance to marriage” is a metaphor for people’s reluctance to read the faqih’s works, and “the blind son-in-law” is a metaphor for a reader who has no common sense.

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

projective Ricci curvature. Then we characterize isotropic projective Ricci curvature of Randers metrics. we also show that Randers metrics are PRic-reversible if and only if they are PRic-quadratic../files/site1/files/0Abstract2.pdf

The collection of all projective vector fields on a Finsler space (M,F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by p(M,F) and is the Lie algebra of the projective group P(M,F). The projective algebra p(M,F=α+β) of a Randers space is characterized as a certain Lie subalgebra of the projective algebra p(M,α). Certain subgroups of the projective group P(M,F) and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the projective algebra of Randers metrics constant flag curvature on a compact n-manifold either equals n(n+2) or at most is n(n+1)/2.