On a Generalized $K^h$- Birecurrent Finsler Space

Abstract

In the present paper, a Finsler space whose curvature tensor $K^i_{jkh}$ satisfies $K^i_{jkh├Р\ell ├Рm}{\rm =\ }{\ a}_{\ell m}K^i_{jkh}+{\ b}_{\ell m}\left({\delta }^i_kg_{jh}-{\delta }^i_hg_{jk}\right)\ ,K^i_{jkh}\ne $ 0 , where ${\ a}_{\ell m}$ and ${\ b}_{\ell m}\ $are non-zero covariant tensor fields of second order called recurrence tensor fields, is introduced, such space is called as a generalized $K^h-$birecurrent Finsler space . The associate tensor $K_{jrkh}$ of Cartan's fourth curvature tensor $K^i_{jkh}$ , the torsion tensor $H^i_{kh}$ ,the deviation tensor $K^i_h$, the Ricci tensor $K_{jk}$, the vector $H_k$ and the scalar curvature $K$ of such space are non-vanishing. Under certain conditions, a generalized $K^h-$birecurrent Finsler space becomes Landsberg space . Some conditions have been pointed out which reduce a generalized $K^h-$birecurrent Finsler space $F_n(n>2)$ into Finsler space of scalar curvature.