Abstract

In this paper, we introduced a Finsler space for which the h - curvature tensor \(H_{jkh}^{i}\) (curvature tensor of Berwald) satisfies the condition
\(β_{l} β_{m} β_{n} H_{jkh}^{i}=c_{lmn} H_{jkh}^{i}+d_{lmn} (δ_{k}^{i} g_{jh}-δ_{h}^{i} g_{jk} )-2y^{r} b_{mn} β_{r} (δ_{k}^{i} C_{jhl}-δ_{h}^{i} C_{jkl})\)
                           \(-2y^{r} w_{ln} β_{r} (δ_{k}^{i} C_{jhm}-δ_{h}^{i} C_{jkm})-2y^{r} μ_{n} β_{l} β_{r} (δ_{k}^{i} C_{jhm}-δ_{h}^{i} C_{jkm}), H_{jkh}^{i}=0,\)
where \(C_{jkm}\) is (h) hv - tortion tensor, \(β_{l} β_{m} β_{n}\) is Berwald's covariant differential operator of the third order with respect to \(x^{n}\), \(x^{m}\) and \(x^{l}\), successively, \(β_{l} β_{r}\) is Berwald's covariant differential operator of the second order with respect to \(x^{l}\) and \(x^{r}\), successively, \(β_{r}\) is Berwald's covariant differential operator of the first order with respect to \(x^{r}\), \(c_{lmn}\) and \(d_{lmn}\) are non – zero covariant tensors field of third order, \(b_{mn}\) and \(w_{ln}\) are non – zero covariant tensors field of second order and \(μ_{l}\) is non – zero covariant vector field. We called this space a generalized \(βH\) – trirecurrent space. The aim of this paper is to develop some properties of a generalized \(βH\) – trirecurrent space by obtaining Berwald's covariant derivative of the third order for the (h)v – torsion tensor \(H_{kh}^{i}\) and the deviation tensor \(H_{k}^{i}\) , the curvature vector \(H_{k}\) and the scalar curvature \(H\) are investigated.