Some properties for Weyl’s projective Curvature Tensor of Generalized W-Birecurrent in Finsler Space

In this paper, we defined a Finsler space Fn for which Weyl’s projective curvature tensor Wjkh i satisfies the generalized-birecurrence condition with respect to Cartan’s connection parameters Gkh ∗i , given by the condition W jkh׀l׀m i = αlmWjkh i + βlm (δh i gjk − δk i gjh), where ׀l ׀m is h-covariant derivative of second order ( Cartan’s second kind covariant differential operator ) with respect to x and x, successively, αlm and βlm are non-null covariant vectors field and such space is called as a generalized W-birecurrent space and denoted briefly by G W-BRFn . We have obtained the h-covariant derivative of the second order for Wely’s projective torsion tensor Wkh i , Wely’s projective deviation tensor Wh i and Weyl’s projective curvature tensor Wjkh i and some tensors are birecurrent in our space. We have obtained the necessary and sufficient condition for Cartan’s third curvature tensor Rjkh i , the associate curvature tensor Rjpkh to be generalized birecurrent, the necessary and sufficient condition of h-covariant derivative of second order for the h(v)-torsion tensor Hkh i , the associate torsion tensor Hkp.h and the deviation tensor Hh i has been obtained in our space.

Let us consider an n-dimensional Finsler space equipped with the metric function F satisfying the requisite conditions [19]. Let consider the components of the corresponding metric tensor , Cartan's connection parameters Γ * and Berwald's connection parameters . These are symmetric in their lower indices .
The vectors and satisfy the following relations [19]: The h-covariant derivative of second order for an arbitrary vector field with respect to and , successively ,we get Taking skew-symmetric part with respect to the indices k and j , we get the commutation formula for h-covariant differentiation as follows [19]: The tensor as defined above is called Cartan's fourth curvature tensor. The metric tensor and the vector are covariant constant with respect to the above process.
The process of h-covariant differentiation , with respect to , commute with partial differentiation with respect to for arbitrary vector filed , according to [19] The quantities ℎ and ℎ form the components of tensors and they are called h-curvature tensor of Berwald (Berwald curvature tensor) and h(v)-torsion tensor, respectively and are defined as follow [19]: a) ℎ ≔ ℎ + ℎ + ℎ − ℎ/ and b) ℎ ∶= ℎ + ℎ − ℎ⁄ . They are also related by [19] ( and c) =̇ . These tensors were constructed initially by means of the tensor ℎ , called the deviation tensor, given by .

The tensor
ℎ is known as projective curvature tensor (generalized Wely's projective curvature tensor ), the tensor is known as projective torsion tensor ( Wely's torsion tensor ) and the tensor is known as projective deviation tensor ( Wely's deviation tensor ) are defined by The tensors ℎ , and are satisfying the following identities [19] (1.14) a) The projective curvature tensor ℎ is skew-symmetric in its indices k and h. Cartan's third curvature tensor ℎ and the R-Ricci tensor are respectively given by [19] * The indices , , , … assume positive integral values from 1 to n . ** −j k ⁄ means the subtraction from the former term by interchanging the indices k and j . Univ. Aden J. Nat. and Appl. Sc. Vol. 23 No.1 -April 2019

A Generalized -Birecurrent Space
A Finsler space for which Weyl's projective curvature tensor ℎ satisfies the recurrence property with respect to Cartan's coefficient connection parameters Γ * which is characterized by the condition [16] (2.1) where ‫|‬ is h-covariant derivative of first order ( Cartan's second kind covariant differential operator ) with respect to , the quantities and are non-null covariant vectors field. We shall call such space as a generalized ℎ -recurrent space and we shall denote it briefly by ℎ -. Taking the h-covariant derivative for (2.1) with respect to and using (1.5a), we get