The main objective of this paper is to find the necessary and sufficient condition of a given Finsler metric to be Einstein in order to classify the Einstein Finsler metrics on a compact manifold. The considered Einstein Finsler metric in the study describes all different kinds of Einstein metrics which are point wise projective to the given one. This study has resulted in the following theorem that needs the proof of three prepositions. Let F be a Finsler metric tric (n >2) Projectively related to an Einstein non-protectively flat Finsler metric F- , then F is Einstein if and only if F = l F- where l is a constant. A Schur type lemma is also proved.

In this paper, we find necessary and sufficient conditions underwhich the infinite series metric and Randers metric on a manifold M of dimension n >3 be Projectively related.

In this paper, we study pointwise Projectively related Finsler gradient Ricci solitons. We obtain an equation that characterizes the relationship between two pointwise Projectively related Finsler gradient Ricci solitons. Further, if two Finsler gradient Ricci solitons (M, F, dV ˜ F˜) and (M, F, dVF ) satisfy F˜,k = µ ∂[F˜2 ] ∂yk and some extra conditions, where “, ” denotes the horizontal covariant derivative with respect to F, we characterize their relationships along the geodesics. In particular, if two Finsler gradient Ricci solitons are both complete, then (M, F, dVF ) is expanding or shrinking and (M, F, dV ˜ F˜) is shrinking.

In this paper, we consider some special Finsler spaces obtained via Randers-β change. First, we find the fundamental metric tensor and Cartan tensor of these metrics. Next, we establish a general formula for the inverse tensor of fundamental metric tensors of these metrics. Finally, we find the necessary and sufficient conditions under which these metrics are Projectively flat, and we give examples to support our results.

In this paper we highlight the connection between certain classes of regular subgroups of the affine group AGLn (F), F a field, and associative nilpotent F-algebras of dimension n. We also describe how the classification of projective congruence classes of square matrices is equivalent to the classification of regular subgroups of particular shape.

In this paper we have taken the n−power (α, β)-metric and obtained the condition for Projectively flatness and further find the the some special cases.

In this paper, we consider the square metric of the form $F=(\alpha+\beta)^2/\alpha$ and study the change of square metrics of the $m$-th root Finsler metrics. We also investigate and study the necessary and sufficient conditions for this type of Finsler metrics to be locally Projectively flat and locally dually flat.

In this paper, we consider the generalized Kropina conformal change of m-th root metric and for this, prove a necessary and su, cient condition of locally Projectively atness. Also we proved a necessary and su, cient condition for the generalized Kropina conformal change of m-th root metric is locally dually at.

The pullback approach to global Finsler geometry is adopted. Some new types of special Finsler spaces are introduced and investigated, namely, Ricci, generalized Ricci, Projectively recurrent and m-Projectively recurrent Finsler spaces. The properties of these special Finsler spaces are studied and the relations between them are singled out.

In this paper‎, ‎we study the Kropina transformation of exponential (α,β)-metric F=α\exp(s),  s:=β/α‎. ‎We characterize the conditions under which this class of (α,β)-metric is locally Projectively flat‎, ‎locally dually flat‎, ‎and Douglas metric‎. ‎Based on‎, ‎we show that the Kropina transformation of an exponential (α,β)-metric is locally Projectively flat‎, ‎locally dually flat and Douglas metric if and only if the exponential (α,β)-metric is locally Projectively flat‎, ‎locally dually flat and Douglas metric‎, ‎respectively.