Some Identities for Generalized Curvature Tensors in 𝓑-Recurrent Finsler Space

Abstract

The purpose of the present paper is to consider and study a certain identities for some generalized curvature tensors in ℬ-recurrent Finsler space 𝐹𝑛 in which Cartan’s second curvature tensor 𝑃𝑗𝑘ℎ𝑖 satisfies the generalized of recurrence condition with respect to Berwald’s connection parameters G𝑘ℎ𝑖 which given by the condition ℬ𝑚𝑃𝑗𝑘ℎ𝑖=𝜆𝑚 𝑃𝑗𝑘ℎ𝑖+𝜇𝑚( 𝛿ℎ𝑖 𝑔𝑗𝑘− 𝛿𝑘𝑖 𝑔𝑗ℎ), where ℬ𝑚 is covariant derivative of first order (Berwald’s covariant differential operator ) with respect to 𝑥𝑚, it’s called a generalized ℬP-recurrent space. We shall denote it briefly by 𝐺ℬ𝑃-𝑅𝐹𝑛. We have obtained Berwald’s covariant derivative of first order for the h(v)-torsion tensor 𝑃𝑘ℎ 𝑖, the deviation tensor 𝑃ℎ 𝑖 and the covariant derivative of the tensor 𝐻𝑘𝑝.ℎ (in the sense of Berwald), also we find some theorems of the R-Ricci tensor 𝑅𝑗𝑘 and the curvature vector 𝑅𝑗in our space. We obtained the necessary and sufficient condition for Berwald’s covariant derivative of Weyl’s projective curvature tensor 𝑊𝑗𝑘ℎ𝑖 and its torsion tensor 𝑊𝑘ℎ𝑖 in our space. Also, we have proved that in 𝐺ℬ𝑃-𝑅𝐹𝑛, Cartan’s second curvature tensor 𝑃𝑗𝑘ℎ𝑖 and the v(hv)-torsion tensor 𝑃𝑘ℎ𝑖 for 𝑛=4 .

Keywords

• Finsler space
• Cartan’s second curvature tensor 𝑃𝑗𝑘ℎ𝑖
• Generalized ℬ𝑃-recurrent space
• Weyl’s projective curvature tensor 𝑊𝑗𝑘ℎ𝑖
• Cartan’s fourth curvature tensor 𝑅𝑗𝑘ℎ𝑖

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