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 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 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 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.

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 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.

Recent studies show that warped product manifolds are useful in differential geometry as well as in physics. The goal of this paper is to study on some projective invariants of a special product manifold with Finsler METRICs arising from warped products. Firstly, we consider the class of weakly Douglas METRICs, weaker notion of Douglas METRICs, introduced by Atashafrouz, Najafi and Tayebi in [4]. We prove that every Finsler warped product manifold Mn (n ≥ 3) is weakly Douglas if and only if it is Douglas. Finally, under a certian condition, we show that a class of Finsler warped product METRIC is locally PROJECTIVELY FLAT if and only if it is of scalar flag curvature