On a Generalized βH – Trirecurrent Finsler Space

In this paper, we introduced a Finsler space for which the h − curvature tensor Hjkh i (curvature tensor of Berwald) satisfies the condition βlβmβnHjkh i = clmnHjkh i + dlmn(δk i gjh − δh i gjk) − 2y bmnβr(δk i Cjhl − δh i Cjkl) −2ywlnβr(δk i Cjhm − δh i Cjkm) − 2y μnβlβr(δk i Cjhm − δh i Cjkm),Hjkh i = 0, where Cjkm is (h) hv − tortion tensor, βlβmβn is Berwald's covariant differential operator of the third order with respect to x, x and x, successively, βl βr is Berwald's covariant differential operator of the second order with respect to x and x, successively, βr is Berwald's covariant differential operator of the first order with respect to x, clmn and dlmn are non – zero covariant tensors field of third order, bmn and wln are non – zero covariant tensors field of second order and μl is non – zero covariant vector field. We called this space a generalized βH – trirecurrent space. The aim of this paper is to develop some properties of a generalized βH – trirecurrent space by obtaining Berwald's covariant derivative of the third order for the (h)v – torsion tensor Hkh i and the deviation tensor Hk i , the curvature vector Hk and the scalar curvature H are investigated.


Abstract
In this paper, we introduced a Finsler space for which the h − curvature tensor ℎ (curvature tensor of Berwald) satisfies the condition is (h) hv − tortion tensor, ℓ is Berwald's covariant differential operator of the third order with respect to , and ℓ , successively, ℓ is Berwald's covariant differential operator of the second order with respect to ℓ and , successively, is Berwald's covariant differential operator of the first order with respect to , ℓ and ℓ are nonzero covariant tensors field of third order, and ℓ are nonzero covariant tensors field of second order and ℓ is nonzero covariant vector field. We called this space a generalized trirecurrent space. The aim of this paper is to develop some properties of a generalized trirecurrent space by obtaining Berwald's covariant derivative of the third order for the (h)vtorsion tensor ℎ and the deviation tensor , the curvature vector and the scalar curvature H are investigated.

Introduction
Pandey P.N., Saxena S.S. and Goswami A. [3] introduced and studied a generalized Hrecurrent Finsler space. F.Y.A.Qasem [4] introduced and discussed generalized H− birecurrent curvature tensor and W.H.A. Hadi [1] studied the generalizedbirecurrent for some tensors and studied some special spaces in this space.
Let Fn be an n − dimensional Finsler spaces equipped with the metric function F satisfies conditions [5], the tensor is positively homogeneous of degree −1 in and symmetric in all its indices and is called (h)hv − torsion tensor [2]. According to Euler's theorem on homogeneous functions, this tensor satisfies the following: of an arbitrary tensor field with respect to is given by [5] (1.2) ℬ : = − (̇) + − . Berwald's covariant derivative of vanish as identically [5], i.e. (1.3) ℬ = 0. In view of (1.2), the second covariant derivative of an arbitrary vector field with respect to ℎ in the sense of Berwald [5], ℬ ℎ ℬ = (ℬ ℎ ) − ( ℬ ℎ ) − (ℬ ) ℎ + (ℬ ℎ ) . Using (1.4) and taking skew − symmetric part, with respect to the indices k and h, we get the commutation formula for Berwald's covariant differentiation as follows [5]: Univ. Aden J. Nat. and Appl. Sc. Vol. 23 No.2-October 2019 Remark 1.1. −ℎ/ means the subtraction from the former term by interchanging the indices h and k.
The tensors ℎ and ℎ , as defined above, are called h − curvature tensor (h − curvature tensor of Berwald) and h(hv) − torsion tensor are positively homogeneous of degree zero and one in , respectively. Berwald construcated the tensors ℎ and ℎ from the tensor called by him as deviation tensor , according to The h(hv)torsion tensor and the deviation tensor satisfy the following [5]: The H -Ricci tensor , the curvature vector and the scalar curvature satisfy the following [5]:
Taking the covariant derivative of third order with respect to , and ℓ , successively, for the equation (2.10), we get In view of (1.8c) , the above equation can be written as Thus , we may conclude Theorem 2.5. In G -TR Fn , Berwald torsion tensor ℎ is generalizedtrirecurrent tensor.