(untitled) (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, we present a novel class of relations and investigate the connection between the R-projective curvature tensor and other tensors of Finsler space F_n. This space is characterized by the property for Cartan’s the third curvature tensor R^i_jkh which satisfies the certain relationship with given covariant vectors field, as follows: where R^i_jkh eq 0 and B_nB_mB_l is the Berwald's third order covariant derivative with respect to x^l, x^m and x^n respectively. The quantities a_lmn= B_nu_lm + u_lm λ_n, b_lmn = B_nv_lm+u_lm μ_n, c_lm = v_lm, and d_ln =B_nμ_l are non-zero covariant vector fields. We define this space a generalized BR-3rd recurrent space and denote it briefly by GBR-3RF_n. This paper aims to derive the third-order Berwald covariant derivatives of the torsion tensor H^i_kh and the deviation tensor H^i_h. Additionally, it demonstrates that the curvature vector K_j, the curvature vector H_k, and the curvature scalar H are all non-vanishing within the considered space. We have some relations between Cartan’s third curvature tensor R^i_jkh and some tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal third-order Berwald covariant derivatives with their lower-order counterparts.