Analysis for Cartan’s Second Curvature Tensor in Finsler Space

: The decomposition of curvature tensors have been studied by the Finslerian geometrics. The aim of the present paper is to three decomposable of Cartan’s second curvature tensor i jkh P to prove that Cartan’s second curvature tensor i jkh P in affinely connected space is symmetric in first and second indices of their decomposable.


I. INTRODUCTION
The analysis for Cartan's fourth curvature tensor in Finsler space discussed by Qasem and Nasr [6].The decomposition of Berwald curvature tensor i jkh H and Cartan's fourth curvature tensor i jkh K for some spaces in sense of Berwald and Cartan were studied by Pandey [11].The decomposition of Cartan's third curvature tensor i jkh R and Cartan's fourth curvature tensor i jkh K equipped with non − symmetric connection in Finsler space were discussed by Mishra et al. [10] and Al−Qashbari [3], respectively.The decomposition of normal projective curvature tensor in Finsler space was discussed by Qasem and Saleem [5].Hit [12] introduced Berwald curvature tensor which be decomposable and obtained several results.Al_Qufail [8] studied decomposability of curvature tensors in nonsymmetric recurrent Finsler space, Nor [2] introduced the decomposability of Cartan's forth curvature tensor i jkh K in Finsler space.Also, Bisht and Neg [9] studied decomposition of normal projective curvature tensor fields in Finsler manifolds.In this paper, we find the condition for Cartan's second curvature tensor i jkh P to be symmetric of their decomposable.
II. PRELIMINARIES In this section, we introduce some conditions and definitions which are needed in this paper.
Let n F be an -n dimensional Finsler space equipped with the metric function   , F x y satisfying the request conditions [4,7] tensor is a mixed tensor of the type (1,3), i.e. of rank 4, it may be written as product of contravariant (or covariant) vector and a tensor of rank 3 , i.e. covariant tensor of the type (0,3) {or mixed tensor of the type (1,2)} as following [5 , 6] (2.8) a Or in the second case as product of two tensors each them of rank 2, i.e. mixed tensors of the type (1,1) and covariant tensor of the type (0,2) as following [5,6] (2.9) a) Here, in this paper, we discuss the possible forms in three decomposable of the tensor, two decompositions for the first case (the other are similar) and one decomposition for the second case (the other are similar).Clearly, from all several possibilities, we study the possibilities which are given by (2.8a), (2.8b) and (2.9a).

III. MAIN RESULTS
In this section, several theorems have been established and proved.Let Cartan's second curvature tensor i jkh P is decomposable in the form (2.8a).Transvecting (2.8a) by j y and using (2.4), we get    Transvecting (2.9a) by j y and using (2.4), we get

IV. CONCLUSION
The possibilities of decomposition for Cartan's second curvature tensor i jkh P have been studied.We obtained that Cartan's second curvature tensor i jkh P in affinely connected space is symmetric in first and second indices of their decomposable.
.In view of (2.1) and (2.4), then eq.(3.3) can be written previous , the equations (3.2a) and (3.2b) lead to the   v hv  torsion tensor i kh P is vanishing.
previous, the equations (3.5a) and (3.5b) lead to the   v hv  torsion tensor i kh P is vanishing.
previous, the equations (3.7a) and (3.7b) lead to the   v hv  torsion tensor i kh P is vanishing.Then, by using this fact and (2.7) in (2.6), we deduce decomposition tensors field, then in affinely connected space, the curvature tensor i jkh P is symmetric in first and second indices.