On Generalized H-Birecurrent Finsler Space

Abstract

In this paper, we introduced a Finsler space for which the h-curvature tensor $H^i_{jkh}$ ( curvature tensor of Berwald ) satisfies the condition ${\mathcal{B}}_m{\mathcal{B}}_n H^i_{jkh}=a_{mn}{ H}^i_{jkh}+b_{mn}({\delta }^i_jg_{kh}-{\delta }^i_kg_{jh})-2{ y}^r{\mu }_n{\mathcal{B}}_r({\delta }^i_jC_{khm}-{\delta }^i_kC_{jhm})$ , $ H^i_{jkh}\neq 0$, $C_{jhm} $ is (h) hv-torsion tensor, where ${\mathcal{B}}_m{\mathcal{B}}_n$ is Berwald's covariant differential operator of the second order with respect to $x^n$ and ${ x}^m$, successively, ${\mathcal{B}}_r$ 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 and ${\mu }_n$ is non-zero covariant vector field. We called this space \textit{a generalized H-birecurrent space}. The aim of this paper is to develop some properties of a generalized H-birecurrent space by obtaining Berwald's covariant derivative of the second order for the (h)v-torsion tensor $H^i_{jk}$ and the deviation tensor $H^i_j$ . The non-vanishing of Ricci tensor $H_{kh}$, the curvature vector $H_k$ and the curvature scalar $H$ are investigated. Different results regarding the covariant tensors field $a_{mn}$ and $b_{mn}$ have been established. Some conditions have been pointed out which reduce a generalized H-birecurrent space $F_n\left(n>2\right)$ into Landsberg space. We obtained an identity for a generalized H-birecurrent space. The conditions which reduce a generalized H-birecurrent space $F_n\left(n>2\right)$ in to a space of curvature scalar are given.