In this paper, we consider invariant (α, β)-metrics and describe all geodesic vectors and investigate the set of all homogeneous geodesics on left invariant Hypercomplex four dimensional simply connected Lie groups. Also, we study the conditions for the Douglas and Berwald type of (α, β)-metrics on the left invariant Hypercomplex four dimensional simply connected Lie groups.

In this paper, we consider invariant infinite series  (α, β)--metrics. Then we describe all geodesic vectors of this spaces on the left invariant Hypercomplex four dimensional simply connected Lie groups.

Let G be a Lie group equipped with a left-invariant Randers metric F. Suppose that Fv and Fc denote the vertical and complete lift of F on TG, respectively. We give the necessary and su, cient conditions under which Fv and Fc are generalized Douglas-Weyl metrics. Then, we characterize all 2-step nilpotent Lie groups G such that their tangent Lie groups (TG,Fc) are generalized Douglas-Weyl Randers metrics.

When Einstein was thinking about the theory of general relativity based on the elimination of especial relativity constraints (especially the geometric relationship of space and time), he understood the first limitation of especial relativity is ignoring changes over time. Because in especial relativity, only the curvature of the space was considered. Therefore, tensor calculations should be to explain it. For this purpose, he obtained a combination of Ricci tensor and Ricci scalar through Bianchi identity, which its covariant derivative is zero and is known as Einstein tensor. The main purpose of this paper is to classify Einstein left-invariant metrics of signature (2, 2) on four-dimensional Lie groups. Recently, four-dimensional Lie groups equipped with left-invariant metric of neutral signature has been investigated extensively and a complete list of them has been presented. Now we study Einstein structures on these Lie groups. Then some geometric properties of these spaces, such as Ricci-flatness will be considered.

The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to de ne the integral of linear representation and extend this concept to special linear representation for next using.