On Some Relations of $R$-Projective Curvature Tensor in Recurrent Finsler Space

Adel. M. Al-Qashbari; S. Saleh; Ismail Ibedou

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On Some Relations of $R$ -Projective Curvature Tensor in Recurrent Finsler Space

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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

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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.