R. Miron initiated the study of L-duality in Lagrange and Finsler Spaces in 1987. The concrete L-duals of the Randers metric, Kropina metric, Matsumoto metric, exponential metric, as well as a few more unique (α, β)-metrics, are really just an of the remarkable results obtained. The importance of L-duality, however, is basically limited to finding the dual of a few key Finsler functions. In this paper, we find L-Dual of a Finsler space with a special (α, β)-metric F = (α+β) 3 α2, where α is a Riemannian metric and β is a differential one form.

There are several non-Riemannian curvatures in Finsler geometry which show the complexity of Finsler geometry with respect to Riemannian geometry. Among these quantities, the Cartan and mean Cartan torsion have very important and brilliant positions. In this paper, we find the necessary and sufficient condition under which a 4-dimensional Finsler manifold is C-reducible. Also, we find the necessary and sufficient condition under which an aribtrary 4-dimensional Finsler manifold has vanishing I¯-curvature.

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.

Here, the concept of electric capacity on Finsler Spaces is introduced and the fundamental conformal invariant property is proved, i.e. the capacity of a compact set on a connected non-compact Finsler manifold is conformal invariant. This work enables mathematicians and theoretical physicists to become more familiar with the global Finsler geometry and one of its new applications.

In this paper, we have defined the concept of the R-complex Finsler space with an arctangent (α, β)-metric  F = α + ε β + β  tan-1( β / α).  For this metric, we have obtained the fundamental metric tensor fields gijand gi¯j as well as their determinants and inverse tensor fields. Further, some properties of non-Hermitian R-complex Finsler Spaces with this metric have been described.

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.

The paper considers the symmetries of Finsler Spaces. We give some conditions about globally symmetric Finsler Spaces. Then we prove that, each of these Spaces can be written as a coset space of a Lie group with an invariant Finsler metric.Finally, we prove that such a space must be Berwaldian.