The notion of quasi-EINSTEIN metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces. Quasi-EINSTEIN METRICS serve also as solution to the Ricci flow equation. Here, the Riemannian metric is replaced by a Hessian matrix derived from a FINSLER structure and a quasi-EINSTEIN FINSLER metric is defined. In compact case, it is proved that the quasi-EINSTEIN METRICS are solutions to the FINSLERian Ricci flow and conversely, certain form of solutions to the FINSLERian Ricci flow are quasi-EINSTEIN FINSLER METRICS.

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 some (a, b)- METRICS such as generalized Kropina, Matsumoto and F=(a+b)2/a METRICS, and obtain necessary and sufficient conditions for them to be EINSTEIN METRICS when b is a constant Killing form. Then we prove with this assumption that the mentioned EINSTEIN METRICS must be Riemannian or Ricci flat.

In this paper, we study generalized Douglas-Weyl FINSLER METRICS. We find some conditions under which the class of generalized Douglas-Weyl (a, b)-metric with vanishing S-curvature reduce to the class of Berwald METRICS.

In this paper, we study the class of C 3-like FINSLER METRICS which contains the class of semi-C-reducible FINSLER metric. We find a condition on C 3-like METRICS under which the notions of Landsberg curvature and mean Landsberg curvature are equivalent.

In Riemannian manifolds, there exists a canonical Riemannian metric on the product of them (Lee in Riemannian geometry an introduction to curvature, Springer, New York, 1997). But at the product of FINSLER manifolds, the canonical FINSLER metric has not been defined. In this paper, we are going to study the product of FINSLER manifolds and give a canonical FINSLER metric on it.

In this paper, we study a class of FINSLER METRICS which contains the class of P-reducible and general relatively isotropic Landsberg METRICS, as special cases. We prove that on a compact FINSLER manifold, this class of METRICS is nothing other than Randers METRICS. Finally, we study this class of FINSLER METRICS with scalar flag curvature and find a condition under which these METRICS reduce to Randers metric.

Equality of hh -curvatures of the Berwald and Cartan connections leads to a new class of FINSLER METRICS, so-called BC-generalized Landsberg METRICS. Here, we prove that every BC-generalized Landsberg metric of scalar flag curvature with dimension greater than two is of constant flag curvature.

In this paper, we study projective Randers change and C-conformal change of P-reducible METRICS. Then we show that every P-reducible generalized Landsberg metric of dimension n > 2 must be a Landsberg metric. This implies that on Randers manifolds the notions of generalized Landsberg metric and Berwald metric are equivalent.