On generalized for curvature Tensor \(P_{jkh}^i\) of second order in Finsler space

Abstract

In this present paper, we introduced a Finsler space \(F_n\) which Cartan’s second curvature tensor \(P_{jkh}^i\) satisfies the generalized birecurrence property with respect to Berwald’s connection parameters \(G_{kh}^i\) which given by the condition\(B_n B_m P_{jkh}^i = a_{mn} P_{jkh}^i + b_{mn} ( δ_h^i g_{jk} - δ_k^i g_jh ) - 2 μ_m B_r (δ_h^i C_{jkn} - δ_k^i C_{jhn} ) y^r ,P_jkh^i≠0,\)where \(B_n B_m\) is Berwald’ scovariant differential of second order with respect to \(x^m\) and \(x^n\), successively, \(μ_m\) is non-zero covariant vector field, \(a_{mn}\) and \(b_{mn}\) are non-zero recurrence vectors field of second order, such space is called as a generalized \(BP\)-Birecurrent space and denoted it briefly by \(GBP - BIRF_n\). We have obtained Berwald’ scovariant derivative of second order for the h(v)-torsion tensor \(P_{kh}^i\), P-Ricci tensor \(P_{jk}\) and the curvature vector \(P_k\) for Cartan’s second curvature tensor \(P_{jkh}^i\). Also, we find some theorems of the associate curvature tensor \(P_{ijkh}\) of the (hv)-curvature tensor \(P_{jkh}^i\) and the associate tensor \(P_{jkh}\) of the v(hv)-torsion tensor \(P_{kh}^i\) in this space. We also obtained the necessary and sufficient condition for Cartan’s fourth curvature tensor \(P_{jkh}^i\) to be generalized birecurrent and the necessary and sufficient condition of Berwald’s covariant derivative of second order for the h(v)-torsion tensor \(H_{kh}^i\), the R-Ricci tensor \(R_{jk}\) andthe deviation tensor \(H_h^i\), also the necessary and sufficient condition for the curvature vector \(R_j\) and the deviation tensor \(H_j^i\) to be non-vanishing in this space.