Abstract

In the present paper, we introduceda Finsler space whose Cartan's fourth curvature tensor \(K_{jkh}^i\) satisfies the condition \(B_n B_m K_{jkh}^i=a_{mn} K_{jkh}^i+b_{mn} (δ_k^i g_{jh}-δ_h^i g_{jk} )- 2y^r μ_n B_r (δ_k^i C_{jhm}-δ_h^i C_{jkm})\), where B_n B_m are Berwald's covariant differential operator of the second order with respect to x^m and xⁿ, successively, B_r is Berwald's covariant differential operator of the first order with respect to x^r, a_mn and b_mn are non-zero covariant tensors field of second order called recurrence tensorsfield and μ_n is non-zero covariant vector field, such space is called as a generalized βK-birecurrent space. The aim of this paper is to prove that thecurvature tensor \(H_{jkh}^i\) satisfies the generalized birecurrence property. We proved that Ricci tensors H_jk, K_jk, the curvature vector H_k and the curvature scalarHof such space are non-vanishing andunder certain conditions, a generalized βK-birecurrent space becomes Landsberg space. Also, some conditions have been pointed out which reduce a generalized βK-birecurrent space F_n (n>2) into Finsler space of curvature scalar.