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.

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.

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.

In this paper, we investigate a FINSLER space characterized by a cubic changed infinite series METRIC given by F(γ, β)=β2/(β-γ_. We derive the fundamental tensors necessary to describe the geoMETRIC properties of this FINSLER space. Additionally, we determine the conditions under which this FINSLER space with the cubic modified infinite series METRIC can be simplified into special types of FINSLER spaces, including quasi-C-reducible, semi-C-reducible, C-reducible, and C2-like FINSLER spaces, based on its various forms of the Cartan tensor.

A class of FINSLER METRICs are (α,β) -METRICs which has many applications in sciences and  technology. One of the problems in FINSLER geometry is finding FINSLER METRICs with constant flag curvature. In this paper after some preliminaries in (α,β) -METRICs, we find a sufficient condition for a FINSLER METRIC of type F= α φp(α/β) to be a space of scalar curvature. We also find a necessary and sufficient condition for a FINSLER METRIC of type F= (α+β) 2 / α to be of constant flag curvature. The method of this paper may be useful for developing the navigation problem.  

The scope of this paper is twofold. On the one hand, we will investigate the reversible geodesics of a FINSLER space endowed with the deformed newly introduced (, ,, )-METRIC F, (, ,, ) = , 2 + , 2(a + 1) ,+ , ,(1) where ,is a real parameter with j, j < 2 p a + 1 and a 2 􀀀,1 4,+1
, , and on the other hand, we will investigate the T-tensor for this METRIC.

We discuss the invariant properties of curvatures effected by the Matsumoto’s C or L-process. We find equivalent conditions on curvatures by comparing the difference between the corresponding curvatures of closely related connections. As an application, Matsumoto’s L-process on Randers manifold is studied. Shen’s connection can not be obtained by using Matsumoto’s processes from other connections. This leads us to two new processes which we call Shen’s C and L-processes. We study the invariant properties of curvatures under Shen’s processes.

In this paper we study the geoMETRIC properties of FINSLER Σ-spaces . we prove that Infinite series Σ-spaces are Riemannian.

In order to extend the sphere theorem for FINSLER METRICs, the concept of reversibil-ity introduced by H-B. Rademacher for a compact FINSLER manifold. In this paper, weextend this notion to the general FINSLER manifolds. Then we find an upper bound forthe reversibility of some important spherically symMETRIC FINSLER METRICs. Furthermore,we introduce the concept of sub-reversibility for a general FINSLER manifold and obtain anon-zero lower bound for this new quantity.