On a generalized βK-Birecurrent Finsler space

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 Bn Bm are Berwald's covariant differential operator of the second order with respect to xm and xn, successively, Br is Berwald's covariant differential operator of the first order with respect to xr, amn and bmn are non-zero covariant tensors field of second order called recurrence tensorsfield and μn 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 Hjk, Kjk, the curvature vector Hk 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 Fn (n>2) into Finsler space of curvature scalar.