On generalized for curvature Tensor pjkhof second order in finsler space

الملخص EN

In this present paper, we introduced a Finsler space F which Cartan's second curvature tensor Psatisfies the generalized birecurrence property with respect to Berwald's connection parameters Gin which given by the condition BmBmPjkh=amn Ph+bmn(Sh9jk-Sk9jn)-2 μm Br (ShCjkn-8kCjnn)y Pjkh #0, where BBm is Berwald' scovariant differential of second order with respect to x and x", successively, μmis non-zero covariant vector field, amn and bmn are non-zero recurrence vectors field of second order, such space is called as a generalized BP-Birecurrent space and denoted it briefly by GBP-BIR.

We have obtained Berwald' scovariant derivative of second order for the h(v)-torsion tensor Ph P-Ricci tensor Pik and the curvature vector Pk for Cartan's second curvature tensor Ph.

Also, we find some theorems of the associate curvature tensor Pijkh of the (hv)-curvature tensor Pkhand the associate tensor Pjkh of the v(hv)-torsion tensor Pin this space.

We also obtained the necessary and sufficient condition for Cartan's fourth curvature tensor Pjkh to be generalized birecurrent and the necessary and sufficient condition of Berwald's covariant derivative of second order for the h (v)-torsion tensor Hh, the R-Ricci tensor Rjk andthe deviation tensor H½, also the necessary and sufficient condition for the curvature vector R; and the deviation tensor H to be non-vanishing in this space.