|
| On Some Relations of $R$-Projective Curvature Tensor in Recurrent Finsler Space |
| |
| View Full Text View/Add Comment Download reader |
| KeyWord:$n$-dimensional Finsler space$\ F_n$, generalized $\mathcal{B}R$-$3rd$ recurrent spaces, employing Berwald's third order covariant derivative, $R^i_{jkh}$ Cartan's third curvature tensor |
| Author Name | Affiliation | | Adel. M. Al-Qashbari | Department of Mathematics and Department of Engineering, University of Aden and University of Science and Technology, Aden, Yemen | | S. Saleh | Department of Mathematics, Hodeidah University, Hodeidah, Yemen
Department of Computer Science, Cihan University-Erbil, Erbil, Iraq | | Ismail Ibedou | Department of Mathematics, Benha University, Benha, Egypt |
|
| Hits: 10 |
| Download times: 13 |
| Abstract: |
| In this paper, we present a novel class of relations and investigate the connection between the R-projective curvature tensor and other tensors of Finsler space $F_n$. This space is characterized by the property for Cartan's the third curvature tensor $R^i_{jkh}$ which satisfies the certain relationship with given covariant vectors field, as follows:
\[{\mathcal{B}}_n{{{\mathcal{B}}_m\mathcal{B}}_lR}^i_{jkh}={a_{lmn}R}^i_{jkh}+b_{lmn}({\delta }^i_hg_{jk}-{\delta }^i_k\ g_{jh})-2[c_{lm}{\mathcal{B}}_r({\delta }^i_hC_{jkn}-{\delta }^i_kC_{jhn})y^r\]
\[\mathrm{+}d_{ln}{\mathcal{B}}_r({\delta }^i_hC_{jkm}-{\delta }^i_kC_{jhm}){y}^r\mathrm{+}{\mu }_{l\ }{\mathcal{B}}_n{\mathcal{B}}_r({\delta }^i_hC_{jkm}-{\delta }^i_kC_{jhm})y^r], \]
where $R^i_{jkh}\neq 0$ and ${\mathcal{B}}_n{\mathcal{B}}_m{\mathcal{B}}_l$ is the Berwald's third order covariant derivative with respect to $x^l$, $x^m$ and $x^n$ respectively. The quantities $a_{lmn}={\mathcal{B}}_nu_{lm}+u_{lm}\ {\lambda }_{n\ }$, $b_{lmn}={\mathcal{B}}_nv_{lm}+{{\ u}_{lm}\ \mu }_n$, $c_{lm\ }=v_{lm}$, and $d_{ln}={\mathcal{B}}_n{\mu }_l\ $ are non-zero covariant vector fields. We define this space a generalized$\ \mathcal{B}R$-$3rd$ recurrent space and denote it briefly by$ G\mathcal{B}R$-$3RF_n$. This paper aims to derive the third-order Berwald covariant derivatives of the torsion tensor $H^i_{kh}$ and the deviation tensor $H^i_h$. Additionally, it demonstrates that the curvature vector $K_j$, the curvature vector $H_k$ , and the curvature scalar $H$ are all non-vanishing within the considered space. We have some relations between Cartan's third curvature tensor $R^i_{jkh}$ and some tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal third-order Berwald covariant derivatives with their lower-order counterparts. |
|
|
|
