On Study Generalized $\mathcal{B}\boldsymbol{R}$-Recurrent Finsler Space

Abstract

In this paper, we defined the generalized $\mathcal{B}R-$recurrent space which characterized by the following condition \[{ \mathcal{B}}_mR^i_{jkh}= {\lambda }_mR^i_{jkh}+ {\mu }_m\left({\delta }^i_jg_{kh}- {\delta }^i_kg_{jh}\right), {R}^i_{jkh} \neq 0, \] where ${\mathcal{B}}_m$ is Berwald's covariant differential operator with respect to $x^m, {\lambda }_m$ and ${\mu}_m$ are known as recurrence vectors. The purpose of the present paper to obtain the necessary and sufficient condition for (i) Berwald curvature tensor $H^i_{jkh}$, its associative $H_{jpkh}$ and Cartan's fourth curvature tensor to be generalized recurrent, (ii) the tensor ($H_{hk}-H_{kh}$) and $H-$Ricci tensor $H_{kh}$ are to be non-vanishing and (iii) the torsion tensor $K^i_{jk}$, the deviation tensor $K^i_{h}$, $K-$Ricci tensor $K_{jk}$, the curvature vectors $K_k$, $R_j$ and the curvature scalar $H$ to behave as recurrent. Also to study the covariant vectors ${\lambda}_{m}$ and ${\mu}_{m}$.