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Certain Generalized Birecurrent Tensors In 퐊

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This document presents findings related to generalized birecurrent tensors in a specific Finsler space characterized by certain curvature tensor conditions. It discusses the intricate relationships between various tensors, including Cartan's fourth curvature tensor and the Berwald curvature tensor, and derives multiple theorems related to their generalized birecurrent properties. The research contributes to the mathematical framework of Finsler geometry by detailing the conditions for tensors to be considered generalized birecurrent.

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International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 4 Issue 5 || June. 2016 || PP-39-44 www.ijmsi.org 39 | Page Certain Generalized Birecurrent Tensors In 𝐊 𝐡 − 𝐆𝐁𝐑– 𝐅𝐧. Fahmi Yaseen Abdo Qasem, Amani Mohammed Abdullah Hanballa Department of Mathematics , Faculty of Education-Aden, University of Aden, Khormaksar , Aden, Yemen Abstract: We presented a Finsler space 𝐹𝑛 whose Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 satisfies 𝐾𝑗𝑘 ℎ|ℓ |𝑚 𝑖 = 𝜆ℓ 𝐾𝑗𝑘 ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 , 𝐾𝑗𝑘 ℎ 𝑖 ≠ 0, where 𝜆ℓ and 𝑏ℓ𝑚 are non-zero covariant vector field and covariant tensor field of second order, respectively. such space is called as 𝐾ℎ –generalized birecurrent space and denoted briefly by 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 . In the present paper we shall obtain some generalized birecurrent tensor in an 𝐾ℎ − 𝐺𝐵𝑅 − 𝐹𝑛 . Keywords: Finsler space, 𝐾ℎ – Generalized birecurrent Finsler space, Ricci tensor. I. Introduction Let 𝐹𝑛 be An 𝑛-dimensional Finsler space equipped with the metric function a 𝐹 𝑥, 𝑦 satisfying the request conditions [7]. Cartan's second kind covariant differentiation form arbitrary vector field 𝑥 𝑖 with respect to 𝑥 𝑘 is given by [3],[4] 𝑋|𝑘 𝑖 ∶= 𝜕 𝑘 𝑋 𝑖 − 𝜕𝑟 𝑋 𝑖 𝐺 𝑘 𝑟 + 𝑋 𝑟 Γ 𝑟𝑘 ∗ 𝑖 . M. Motsumoto [5],[6] calls this derivative as ℎ − covariant derivative. The vector 𝑦 𝑖 and the metric tensor g𝑖𝑗 and its associate satisfies the following relations (1.1) 𝑎) 𝑦 |𝑘 𝑖 = 0 , b) 𝑔 𝑖𝑗 |𝑘 = 0 and c) 𝑔 |𝑘 𝑖𝑗 =0. The tensor 𝐶𝑖𝑗𝑘 is known as ℎ ℎ𝑣 - torsion tensor [5], it is positively homogeneous of degree −1 in 𝑦 𝑖 and symmetric in all its indices. By using Euler' s theorem on homogeneous properties, this tensor satisfies the following (1.2) 𝐶 𝑖𝑗𝑘 𝑦 𝑖 = 𝐶 𝑘𝑖𝑗 𝑦 𝑖 = 𝐶 𝑗𝑘𝑖 𝑦 𝑖 = 0. Also satisfies the following relation (1.3) 𝐶 𝑖𝑗𝑘 g𝑗𝑘 = 𝐶𝑖. The (𝑣)ℎ𝑣-torsion tensor 𝐶𝑗𝑘 𝑖 is the associate tensor of the (ℎ)ℎ𝑣-tensor 𝐶𝑖𝑗𝑘 and defined by (1.4) 𝐶𝑖𝑘 ℎ ∶= gℎ𝑗 𝐶𝑖𝑗𝑘 . The tensor 𝐶𝑖𝑘 ℎ is positively homogeneous of degree −1 in 𝑦 𝑖 and symmetric in its lower indices. The tensor 𝑃 𝑗𝑘 𝑖 is called the 𝑣(ℎ𝑣)-torsion tensor and given by (1.5) 𝑃 𝑗𝑘 𝑟 = 𝜕𝑗 Γℎ𝑘 ∗𝑟 𝑦ℎ = Γ 𝑗ℎ𝑘 ∗𝑟 𝑦 ℎ . Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 and the ℎ(𝑣)- torsion tensor 𝐻 𝑘ℎ 𝑖 are related by 𝐻 𝑗𝑘 ℎ 𝑖 𝑦 𝑗 = 𝐻 𝑘ℎ 𝑖 . The deviation tensor 𝐻 𝑘 𝑖 is positively homogeneous of degree two in 𝑦 𝑖 and satisfies (1.6) 𝐻ℎ𝑘 𝑖 𝑦 ℎ = 𝐻 𝑘 𝑖 . Cartan's fourth curvature tensor 𝐾 𝑗𝑘 ℎ 𝑖 satisfies the following identity known as Bianchi identity (1.7) 𝐾𝑗𝑘 ℎ|ℓ 𝑖 + 𝐾𝑗ℓ𝑘|ℎ 𝑖 + 𝐾 𝑗ℎℓ|𝑘 𝑖 + 𝑦 𝑟 ∂sΓ𝑗𝑘 ∗𝑖 𝐾 𝑟ℎℓ 𝑠 + ∂sΓ𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ 𝑠 + ∂sΓ𝑗ℎ ∗𝑖 𝐾 𝑟ℓ𝑘 𝑠 = 0. The associate tensor 𝐾 𝑖𝑗𝑘 ℎ of the curvature tensor 𝐾 𝑗𝑘 ℎ 𝑖 is given by (1.8) 𝐾 𝑖𝑗𝑘 ℎ ∶= g 𝑟𝑗 𝐾 𝑖𝑘ℎ 𝑟 . The tensor 𝐾 𝑖𝑗𝑘 ℎ also satisfies the condition (1.9) 𝐾ℎ𝑖𝑗𝑘 + 𝐾 𝑖ℎ𝑗𝑘 = −2 𝐶ℎ𝑖𝑟 𝐾 𝑠𝑗𝑘 𝑟 𝑦 𝑠 . The curvature tensor 𝐾 𝑗𝑘 ℎ 𝑖 satisfies the following relations too (1.10) 𝐾 𝑗𝑘 ℎ 𝑖 𝑦 𝑗 = 𝐻 𝑘ℎ 𝑖 ,
Certain Generalized Birecurrent… www.ijmsi.org 40 | Page (1.11) 𝐾 𝑗𝑘𝑖 𝑖 = 𝐾 𝑗𝑘 and (1.12) 𝐻 𝑗𝑘 ℎ 𝑖 − 𝐾 𝑗𝑘 ℎ 𝑖 = 𝑃 𝑗𝑘 |ℎ 𝑖 + 𝑃 𝑗𝑘 𝑟 𝑃 𝑟ℎ 𝑖 − ℎ 𝑘 ∗ . * − ℎ 𝑘 means the subtraction from the former term by interchange the indices ℎ and 𝑘. N. S. H. Hussien [4] and M. A. A. Ali [1] obtained some birecurrent tensors in a 𝐾ℎ – birecurrent Finsler space. II. Certain Generalized Birecurrent Tensors Let us consider an 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 characterized by the condition (2.1) 𝐾𝑗𝑘 ℎ|ℓ |𝑚 𝑖 = 𝜆ℓ 𝐾𝑗𝑘 ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 , 𝐾𝑗𝑘 ℎ 𝑖 ≠ 0 where 𝜆ℓ and 𝑏ℓ𝑚 = 𝜆ℓ|𝑚 are non-zero covariant vector fields and covariant tensor field of second order, respectively. The space and the tensor satisfying the condition (2.1) will be called 𝐾ℎ –generalized birecurrent space and ℎ–generalized birecurrent tensor, respectively. We shall denote them briefly by 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 and ℎ– 𝐺𝐵𝑅, respectively. Transvecting (2.1) by the metric tensor g𝑖𝑝 , using (1.8) and (1.1b), we get (2.2) 𝐾𝑗𝑝𝑘 ℎ|ℓ|𝑚 = 𝜆ℓ 𝐾𝑗𝑝𝑘 ℎ|𝑚 + 𝑏ℓ𝑚 𝐾𝑗𝑝𝑘 ℎ . Contracting the indices 𝑖 and ℎ in (2.2) and using (1.11), we get (2.3) 𝐾𝑗𝑘 |ℓ|𝑚 = 𝜆ℓ 𝐾𝑗𝑘 |𝑚 + 𝑏ℓ𝑚 𝐾𝑗𝑘 . Transvecting (2.3) by 𝑦 𝑘 and using (1.1a), we get (2.4) 𝐾𝑗 |ℓ|𝑚 = 𝜆ℓ 𝐾𝑗 |𝑚 + 𝑏ℓ𝑚 𝐾𝑗 . where 𝐾𝑗𝑘 𝑦 𝑘 = 𝐾𝑗 . Transvecting (2.1) by 𝑦 𝑗 , using (1.1a) and (1.10), we get (2.5) 𝐻𝑘ℎ|ℓ |𝑚 𝑖 = 𝜆ℓ 𝐻𝑘ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐻𝑘ℎ . 𝑖 Contracting the indices 𝑖 and ℎ in (2.5) and using (𝐻𝑘 = 𝐻𝑘𝑖 𝑖 ), we get (2.6) 𝐻𝑘|ℓ|𝑚 = 𝜆ℓ 𝐻𝑘|𝑚 + 𝑏ℓ𝑚 𝐻𝑘 . Differentiating (1.9) covariantly with respect to 𝑥ℓ in the sense of Cartan and using(1.10), we get (2.7) 𝐾ℎ𝑖𝑗𝑘 |ℓ + 𝐾𝑖ℎ𝑗𝑘 |ℓ = (−2𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|ℓ . Differentiating (2.7) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.8) 𝐾ℎ𝑖𝑗𝑘 |ℓ|𝑚 + 𝐾𝑖ℎ𝑗𝑘 |ℓ|𝑚 = (−2𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|ℓ|𝑚 . Using (2.2) in (2.8), we get (2.9) 𝜆ℓ(𝐾ℎ𝑖𝑗𝑘 |𝑚 + 𝐾𝑖ℎ𝑗𝑘 |𝑚 ) + 𝑏ℓ𝑚 𝐾ℎ𝑖𝑗𝑘 + 𝐾𝑖ℎ𝑗𝑘 = (−2𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|ℓ|𝑚 . Putting (1.9), (1.10) and (2.7) in (2.9), we get (2.10) (𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 ) . Transvecting (2.10) by gℎ𝑝 , using (1.1c) and (1.4), we get (2.11) (𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 ). Transvecting (2.11) by 𝑦 𝑗 , using (1.1a) and (1.6), we get (2.12) (𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 ). Transvecting (2.10) by gℎ𝑖 , using (1.1c) and (1.3), we get (2.13) (𝐶𝑟 𝐻𝑗𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑟 𝐻𝑗𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑟 𝐻𝑗𝑘 𝑟 ). Transvecting (2.13) by 𝑦 𝑗 , using (1.1a) and (1.6), we get (2.14) (𝐶𝑟 𝐻𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑟 𝐻𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑟 𝐻𝑘 𝑟 ). Contracting the indices 𝑝 and 𝑘 in (2.12), we get (2.15) (𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 ). Thus, we conclude Theorem 2.1. In 𝐾 ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors (𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 ),(𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 ), 𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 , (𝐶𝑟 𝐻𝑗𝑘 𝑟 ), (𝐶𝑟 𝐻𝑘 𝑟 ) and (𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 ) are all generalized birecurrent.
Certain Generalized Birecurrent… www.ijmsi.org 41 | Page We know the identity [7] (2.16) 𝐾𝑗 = 𝐻𝑗 − 𝐻𝑗 𝑖 𝐶𝑖 . Differentiating (2.16) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.17) 𝐾𝑗|ℓ = 𝐻𝑗 |ℓ − (𝐻𝑗 𝑖 𝐶𝑖)|ℓ. Differentiating (2.17) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.18) 𝐾𝑗|ℓ|𝑚 = 𝐻𝑗 |ℓ|𝑚 − (𝐻𝑗 𝑖 𝐶𝑖)|ℓ|𝑚 . Using (2.4) and (2.14) in (2.18), we get (2.19) 𝐾𝑗 |ℓ|𝑚 = 𝜆ℓ 𝐻𝑗 |𝑚 − (𝐻𝑗 𝑖 𝐶𝑖)|𝑚 + 𝑏ℓ𝑚 𝐻𝑗 − (𝐻𝑗 𝑖 𝐶𝑖) . Putting (2.16) and (2.17) in (2.19), we get (2.20) 𝐾𝑗 |ℓ|𝑚 = 𝜆ℓ 𝐾𝑗 |𝑚 + 𝑏ℓ𝑚 𝐾𝑗 . Thus, we conclude Theorem 2.2. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the vector 𝐾𝑗 is generalized birecurrent. Also, we have the identity [7] (2.21) 𝑅𝑗 = 𝐾𝑗 + 𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 . Differentiating (2.21) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.22) 𝑅𝑗|ℓ = 𝐾𝑗 |ℓ + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 )|ℓ . Differentiating (2.22) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.23) 𝑅𝑗|ℓ|𝑚 = 𝐾𝑗 |ℓ|𝑚 + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 )|ℓ|𝑚 . Using (2.15) and (2.20) in (2.23), we get (2.24) 𝑅𝑗|ℓ|𝑚 = 𝜆ℓ 𝐾𝑗|𝑚 + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 )|𝑚 + 𝑏ℓ𝑚 𝐾𝑗 + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 ) . Putting (2.21) and (2.22) in (2.24), we get 𝑅𝑗|ℓ|𝑚 = 𝜆𝑙 𝑅𝑗|𝑚 + 𝑏ℓ𝑚 𝑅𝑗 . Thus, we conclude Theorem 2.3. In 𝐾 ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the vector 𝑅𝑗 is generalized birecurrent. We have Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 , 𝑣 ℎ𝑣 – torsion tensor 𝑃𝑗𝑘 𝑖 and Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 are connected by the formula (1.12). Differentiating (1.12) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.25) 𝐻 𝑗𝑘 ℎ|ℓ 𝑖 − 𝐾 𝑗𝑘 ℎ|ℓ 𝑖 = (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 − ℎ 𝑘)|ℓ. Differentiating ( 2.25) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.26) 𝐻 𝑗𝑘 ℎ|ℓ|𝑚 𝑖 − 𝐾 𝑗𝑘 ℎ|ℓ|𝑚 𝑖 = (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 − ℎ 𝑘)|ℓ|𝑚 . Using (2.1) and if Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 is generalized birecurrent, (2.26) reduces to (2.27) 𝜆ℓ 𝐻 𝑗𝑘 ℎ|𝑚 𝑖 – 𝐾 𝑗𝑘 ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐻𝑗𝑘 ℎ 𝑖 – 𝐾𝑗𝑘 ℎ 𝑖 = (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘)|ℓ|𝑚 . Putting (1.12) and (2.25) in (2.27), we get (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘) ℓ 𝑚 = 𝜆ℓ(𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘)|𝑚 +𝑏ℓ𝑚 𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘 . Thus, we conclude Theorem 2.4. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensor 𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 − ℎ 𝑘 is generalized birecurrent [provided Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 is generalized birecurrent]. We know the curvature tensor 𝐾𝑖𝑗𝑘 ℎ satisfies [5] the identity (2.28) 𝐾ℎ𝑖𝑗𝑘 – 𝐾𝑗𝑘 ℎ𝑖 = 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 – 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 . Differentiating (2.28) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.29) 𝐾ℎ𝑖𝑗𝑘 |ℓ– 𝐾𝑗𝑘 ℎ𝑖|ℓ = (𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 − 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|ℓ. Differentiating (2.29) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.30) 𝐾ℎ𝑖𝑗𝑘 ℓ 𝑚 – 𝐾𝑗𝑘 ℎ𝑖 ℓ 𝑚 = (𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 − 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘
Certain Generalized Birecurrent… www.ijmsi.org 42 | Page − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|ℓ|𝑚 . Using (2.2) in (2.30), we get (2.31) 𝜆ℓ 𝐾ℎ𝑖𝑗𝑘 |ℓ − 𝐾𝑗𝑘 ℎ𝑖|ℓ + 𝑏ℓ𝑚 𝐾ℎ𝑖𝑗𝑘 − 𝐾𝑗𝑘 ℎ𝑖 = (𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 − 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|ℓ|𝑚 . Putting (2.28) and (2.29) in (2.31), we get (2.32) ( 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘– 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 ) ℓ 𝑚 = 𝜆ℓ( 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 – 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|𝑚 + 𝑏ℓ𝑚 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 – 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 Transvecting (2.32) by 𝑦 𝑗 , using (1.1a), (1.2) and (1.6), we get (2.33) (𝐻ℎ 𝑟 𝐶𝑟𝑖𝑘 – 𝐻𝑖 𝑟 𝐶𝑟ℎ𝑘 + 𝐻𝑘 𝑟 𝐶𝑟ℎ𝑖) ℓ 𝑚 = 𝜆ℓ(𝐻ℎ 𝑟 𝐶𝑟𝑖𝑘 – 𝐻𝑖 𝑟 𝐶𝑟ℎ𝑘 + 𝐻𝑘 𝑟 𝐶𝑟ℎ𝑖)|𝑚 +𝑏ℓ𝑚 (𝐻ℎ 𝑟 𝐶𝑟𝑖𝑘 – 𝐻𝑖 𝑟 𝐶𝑟ℎ𝑘 + 𝐻𝑘 𝑟 𝐶𝑟ℎ𝑖). Transvecting (2.33) by g 𝑝𝑟 , using (1.1c) and (1.4), we get (𝐻ℎ 𝑟 𝐶𝑖𝑘 𝑝 − 𝐻𝑖 𝑟 𝐶ℎ𝑘 𝑝 + 𝐻𝑘 𝑟 𝐶ℎ𝑖 𝑝 ) ℓ 𝑚 = 𝜆ℓ(𝐻ℎ 𝑟 𝐶𝑖𝑘 𝑝 − 𝐻𝑖 𝑟 𝐶ℎ𝑘 𝑝 + 𝐻𝑘 𝑟 𝐶ℎ𝑖 𝑝 )|𝑚 +𝑏ℓ𝑚 (𝐻ℎ 𝑟 𝐶𝑖𝑘 𝑝 − 𝐻𝑖 𝑟 𝐶ℎ𝑘 𝑝 + 𝐻𝑘 𝑟 𝐶ℎ𝑖 𝑝 ). Thus, we conclude Theorem 2.5. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟𝐶𝑟𝑗𝑘, 𝐻ℎ 𝑟𝐶𝑟𝑖𝑘−𝐻𝑖 𝑟𝐶𝑟ℎ𝑘+𝐻𝑘 𝑟𝐶𝑟ℎ𝑖 and (𝐻ℎ 𝑟𝐶𝑖𝑘𝑝− 𝐻𝑖 𝑟𝐶ℎ𝑘𝑝+ 𝐻𝑘 𝑟𝐶ℎ𝑖𝑝) are all generalized birecurrent.We have the identity [7] (2.34) 𝐾𝑖𝑗ℎ𝑘 + 𝐾𝑖𝑘𝑗 ℎ + 𝐾𝑖ℎ𝑘𝑗 = −2 𝑦 𝑟 𝐶𝑖𝑗𝑠 𝐾𝑟ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐾𝑟𝑗 ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐾𝑟𝑘𝑗 𝑠 . Using (1.10) in (2.34), we get (2.35) 𝐾𝑖𝑗ℎ𝑘 + 𝐾𝑖𝑘𝑗 ℎ + 𝐾𝑖ℎ𝑘𝑗 = −2 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 . Differentiating (2.35) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.36) 𝐾𝑖𝑗ℎ𝑘|ℓ + 𝐾𝑖𝑘𝑗 ℎ|ℓ + 𝐾𝑖ℎ𝑘𝑗 |ℓ = −2 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ . Differentiating (2.36) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.37) 𝐾𝑖𝑗ℎ𝑘|ℓ|𝑚 + 𝐾𝑖𝑘𝑗 ℎ|ℓ|𝑚 + 𝐾𝑖ℎ𝑘𝑗 |ℓ|𝑚 = −2 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 . Using (2.2), (2.35) and (2.36) in (2.37), we get (2.38) 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 = 𝜆ℓ 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |𝑚 +𝑏ℓ𝑚 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 . Transvecting (2.38) by 𝑦 𝑗 , using (1.1a), (1.2) and (1.6), we get (2.39) 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 ℓ 𝑚 = 𝜆ℓ 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 . Transvecting (2.39) by g 𝑝𝑟 , using (1.1c) and (1.4), we get 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 ℓ 𝑚 = 𝜆ℓ 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 . Thus, we conclude Theorem 2.6. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 , 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 − 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 and 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 are all generalized birecurrent. Differentiating (1.7) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.40) 𝐾𝑗𝑘 ℎ|ℓ|𝑚 𝑖 + 𝐾𝑗ℓ𝑘|ℎ|𝑚 𝑖 + 𝐾𝑗ℎℓ|𝑘|𝑚 𝑖 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 𝐾𝑟ℎℓ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 𝐾𝑟ℓ𝑘|𝑚 𝑠 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0. Using (2.1) in (2.40), we get (2.41) 𝜆ℓ 𝐾𝑗𝑘 ℎ|𝑚 𝑖 + 𝜆ℎ 𝐾𝑗ℓ𝑘|𝑚 𝑖 + 𝜆 𝑘 𝐾𝑗ℎℓ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 +𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 𝐾𝑟ℎℓ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 𝐾𝑟ℓ𝑘|𝑚 𝑠 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0.
Certain Generalized Birecurrent… www.ijmsi.org 43 | Page If Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 is recurrent, (2.41) becomes (2.42) 𝜆ℓ 𝜆 𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝜆ℎ 𝜆 𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝜆 𝑘 𝜆 𝑚 𝐾𝑗ℎℓ 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 + 𝜆 𝑚 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 𝐾𝑟ℓ𝑘 𝑠 + 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0 . Putting (1.7) in (2.42), we get 𝜆ℓ 𝜆 𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝜆ℎ 𝜆 𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝜆 𝑘 𝜆 𝑚 𝐾𝑗ℎℓ 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 +𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 +𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 −𝜆 𝑚 ( 𝐾𝑗𝑘 ℎ|ℓ 𝑖 + 𝐾𝑗ℓ𝑘|ℎ 𝑖 + 𝐾𝑗ℎℓ|𝑘 𝑖 )+ 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0. which can be written as (2.43) 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 + + 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0 Transvecting (2.43) by 𝑦 𝑗 , using (1.1a), (1.10) and (1.5), we get (2.44) 𝑏ℓ𝑚 𝐻𝑘ℎ 𝑖 + 𝑏ℎ𝑚 𝐻ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐻ℎℓ 𝑖 + 𝑃𝑠𝑘|𝑚 𝑖 𝐻ℎℓ 𝑠 +𝑃𝑠ℓ|𝑚 𝑖 𝐻𝑘ℎ 𝑠 + 𝑃𝑠ℎ|𝑚 𝑖 𝐻ℓ𝑘 𝑠 = 0 . Thus, we conclude Theorem 2.7. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , we have the identities (2.43) and (2.44) [provided Cartan fourth curvature tensor Kjkh i is recurrent]. We know that the associate tensor 𝑅𝑖𝑗𝑘 ℎ of Cartan's third curvature tensor 𝑅𝑗𝑘 ℎ 𝑖 satisfies the identity [7] (2.45) 𝑅𝑖𝑗ℎ𝑘 + 𝑅𝑖𝑘𝑗 ℎ + 𝑅𝑖ℎ𝑘𝑗 + 𝐶𝑖𝑗𝑠 𝐾𝑟ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐾𝑟𝑗 ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐾𝑟𝑘𝑗 𝑠 𝑦 𝑟 = 0. Using (1.11) in (2.45), we get (2.46) 𝑅𝑖𝑗ℎ𝑘 + 𝑅𝑖𝑘𝑗 ℎ + 𝑅𝑖ℎ𝑘𝑗 + 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 = 0. Differentiating (2.46) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.47) 𝑅𝑖𝑗ℎ𝑘|ℓ + 𝑅𝑖𝑘𝑗 ℎ|ℓ + 𝑅𝑖ℎ𝑘𝑗 |ℓ + 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ = 0. The associate tensor 𝐾𝑖𝑗𝑘 ℎ of Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 and the associate tensor 𝑅𝑖𝑗𝑘 ℎ of Cartan's third curvature tensor 𝑅𝑗𝑘 ℎ 𝑖 are connected by the identity [7] (2.48) 𝐾ℎ𝑖𝑗𝑘 − 𝐾𝑖ℎ𝑗𝑘 = 2𝑅ℎ𝑖𝑗𝑘 . Differentiating (2.48) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.49) 𝐾ℎ𝑖𝑗𝑘 |ℓ − 𝐾𝑖ℎ𝑗𝑘 |ℓ = 2𝑅ℎ𝑖𝑗𝑘 |ℓ. Differentiating (2.49) covariantly with respect to 𝑥 𝑚 in the sense of Cartan and using (2.2), we get (2.50) 𝜆ℓ(𝐾ℎ𝑖𝑗𝑘 |𝑚 − 𝐾𝑖ℎ𝑗𝑘 |𝑚 ) + 𝑏ℓ𝑚 (𝐾ℎ𝑖𝑗𝑘 − 𝐾𝑖ℎ𝑗𝑘 ) = 2𝑅ℎ𝑖𝑗𝑘 |ℓ|𝑚 . Putting (2.48) and (2.49) in (2. 50), we get (2.51) 𝑅ℎ𝑖𝑗𝑘 |ℓ|𝑚 = 𝜆ℓ 𝑅ℎ𝑖𝑗𝑘 |𝑚 + 𝑏ℓ𝑚 𝑅ℎ𝑖𝑗𝑘 . Differentiating (2.47) covariantly with respect to 𝑥 𝑚 in the sense of Cartan and using (2.51), we get (2.52) 𝜆ℓ 𝑅𝑖𝑗ℎ𝑘|𝑚 + 𝑅𝑖𝑘𝑗 ℎ|𝑚 + 𝑅𝑖ℎ𝑘𝑗 |𝑚 + 𝑏ℓ𝑚 𝑅𝑖𝑗ℎ𝑘 + 𝑅𝑖𝑘𝑗 ℎ + 𝑅𝑖ℎ𝑘𝑗 + 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 = 0. In view of (2.46) and putting (2.45) in (2.52), we get (2.53) 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 = 𝜆ℓ 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 . Transvecting (2.53) by 𝑦 𝑗 , using (1.1a), (1.2) and (1.6), we get (2.54) 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 |𝑙|𝑚 = 𝜆ℓ 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 . Transvecting (2.54) by g 𝑝𝑖 , using (1.1c) and (1.4), we get 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 |ℓ|𝑚 = 𝜆ℓ 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 . Thus, we conclude
Certain Generalized Birecurrent… www.ijmsi.org 44 | Page Theorem 2.8. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 , 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 − 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 and 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 are all generalized birecurrent. References [1.] Ali, M.A.A.: On 𝐾ℎ - birecurrent Finsler space, M.sc. Thesis, University of Aden, (Aden) (Yemen), (2014). [2.] Cartan, 𝐄.: Sur les espaces de Finsler, C.R. Aead, Sci.(Paris) 196, (1933), 582-586. [3.] Cartan, 𝐄.: Sur les espaces de Finsler, Actualite, (Paris),79, (1934); 2nd edit,(1971). [4.] Hussien, N.S.H.: On 𝐾ℎ –recurrent Finsler space, M.sc.Thesis, University of Aden, (Aden) (Yemen), (2014). [5.] Matsumoto, M.: On h-isotropic and 𝐶ℎ – recurrent Finsler, J. Math. Kyoto Univ . 11, (1971) ,1- 9. [6.] Matsumoto, M.: On Finsler spaces with curvature tensor of some special forms, Tensor N.S., 22, (1971) ,201- 209. [7.] Rund, H.: The differential geometry of Finsler space, Springer verlag, Berlin-Gottingen-Heidelberg,(1959); 2nd Edit. (in Russian), Nauka, (Moscow),(1981).

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Certain Generalized Birecurrent Tensors In 퐊

  • 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 4 Issue 5 || June. 2016 || PP-39-44 www.ijmsi.org 39 | Page Certain Generalized Birecurrent Tensors In 𝐊 𝐡 − 𝐆𝐁𝐑– 𝐅𝐧. Fahmi Yaseen Abdo Qasem, Amani Mohammed Abdullah Hanballa Department of Mathematics , Faculty of Education-Aden, University of Aden, Khormaksar , Aden, Yemen Abstract: We presented a Finsler space 𝐹𝑛 whose Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 satisfies 𝐾𝑗𝑘 ℎ|ℓ |𝑚 𝑖 = 𝜆ℓ 𝐾𝑗𝑘 ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 , 𝐾𝑗𝑘 ℎ 𝑖 ≠ 0, where 𝜆ℓ and 𝑏ℓ𝑚 are non-zero covariant vector field and covariant tensor field of second order, respectively. such space is called as 𝐾ℎ –generalized birecurrent space and denoted briefly by 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 . In the present paper we shall obtain some generalized birecurrent tensor in an 𝐾ℎ − 𝐺𝐵𝑅 − 𝐹𝑛 . Keywords: Finsler space, 𝐾ℎ – Generalized birecurrent Finsler space, Ricci tensor. I. Introduction Let 𝐹𝑛 be An 𝑛-dimensional Finsler space equipped with the metric function a 𝐹 𝑥, 𝑦 satisfying the request conditions [7]. Cartan's second kind covariant differentiation form arbitrary vector field 𝑥 𝑖 with respect to 𝑥 𝑘 is given by [3],[4] 𝑋|𝑘 𝑖 ∶= 𝜕 𝑘 𝑋 𝑖 − 𝜕𝑟 𝑋 𝑖 𝐺 𝑘 𝑟 + 𝑋 𝑟 Γ 𝑟𝑘 ∗ 𝑖 . M. Motsumoto [5],[6] calls this derivative as ℎ − covariant derivative. The vector 𝑦 𝑖 and the metric tensor g𝑖𝑗 and its associate satisfies the following relations (1.1) 𝑎) 𝑦 |𝑘 𝑖 = 0 , b) 𝑔 𝑖𝑗 |𝑘 = 0 and c) 𝑔 |𝑘 𝑖𝑗 =0. The tensor 𝐶𝑖𝑗𝑘 is known as ℎ ℎ𝑣 - torsion tensor [5], it is positively homogeneous of degree −1 in 𝑦 𝑖 and symmetric in all its indices. By using Euler' s theorem on homogeneous properties, this tensor satisfies the following (1.2) 𝐶 𝑖𝑗𝑘 𝑦 𝑖 = 𝐶 𝑘𝑖𝑗 𝑦 𝑖 = 𝐶 𝑗𝑘𝑖 𝑦 𝑖 = 0. Also satisfies the following relation (1.3) 𝐶 𝑖𝑗𝑘 g𝑗𝑘 = 𝐶𝑖. The (𝑣)ℎ𝑣-torsion tensor 𝐶𝑗𝑘 𝑖 is the associate tensor of the (ℎ)ℎ𝑣-tensor 𝐶𝑖𝑗𝑘 and defined by (1.4) 𝐶𝑖𝑘 ℎ ∶= gℎ𝑗 𝐶𝑖𝑗𝑘 . The tensor 𝐶𝑖𝑘 ℎ is positively homogeneous of degree −1 in 𝑦 𝑖 and symmetric in its lower indices. The tensor 𝑃 𝑗𝑘 𝑖 is called the 𝑣(ℎ𝑣)-torsion tensor and given by (1.5) 𝑃 𝑗𝑘 𝑟 = 𝜕𝑗 Γℎ𝑘 ∗𝑟 𝑦ℎ = Γ 𝑗ℎ𝑘 ∗𝑟 𝑦 ℎ . Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 and the ℎ(𝑣)- torsion tensor 𝐻 𝑘ℎ 𝑖 are related by 𝐻 𝑗𝑘 ℎ 𝑖 𝑦 𝑗 = 𝐻 𝑘ℎ 𝑖 . The deviation tensor 𝐻 𝑘 𝑖 is positively homogeneous of degree two in 𝑦 𝑖 and satisfies (1.6) 𝐻ℎ𝑘 𝑖 𝑦 ℎ = 𝐻 𝑘 𝑖 . Cartan's fourth curvature tensor 𝐾 𝑗𝑘 ℎ 𝑖 satisfies the following identity known as Bianchi identity (1.7) 𝐾𝑗𝑘 ℎ|ℓ 𝑖 + 𝐾𝑗ℓ𝑘|ℎ 𝑖 + 𝐾 𝑗ℎℓ|𝑘 𝑖 + 𝑦 𝑟 ∂sΓ𝑗𝑘 ∗𝑖 𝐾 𝑟ℎℓ 𝑠 + ∂sΓ𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ 𝑠 + ∂sΓ𝑗ℎ ∗𝑖 𝐾 𝑟ℓ𝑘 𝑠 = 0. The associate tensor 𝐾 𝑖𝑗𝑘 ℎ of the curvature tensor 𝐾 𝑗𝑘 ℎ 𝑖 is given by (1.8) 𝐾 𝑖𝑗𝑘 ℎ ∶= g 𝑟𝑗 𝐾 𝑖𝑘ℎ 𝑟 . The tensor 𝐾 𝑖𝑗𝑘 ℎ also satisfies the condition (1.9) 𝐾ℎ𝑖𝑗𝑘 + 𝐾 𝑖ℎ𝑗𝑘 = −2 𝐶ℎ𝑖𝑟 𝐾 𝑠𝑗𝑘 𝑟 𝑦 𝑠 . The curvature tensor 𝐾 𝑗𝑘 ℎ 𝑖 satisfies the following relations too (1.10) 𝐾 𝑗𝑘 ℎ 𝑖 𝑦 𝑗 = 𝐻 𝑘ℎ 𝑖 ,
  • 2. Certain Generalized Birecurrent… www.ijmsi.org 40 | Page (1.11) 𝐾 𝑗𝑘𝑖 𝑖 = 𝐾 𝑗𝑘 and (1.12) 𝐻 𝑗𝑘 ℎ 𝑖 − 𝐾 𝑗𝑘 ℎ 𝑖 = 𝑃 𝑗𝑘 |ℎ 𝑖 + 𝑃 𝑗𝑘 𝑟 𝑃 𝑟ℎ 𝑖 − ℎ 𝑘 ∗ . * − ℎ 𝑘 means the subtraction from the former term by interchange the indices ℎ and 𝑘. N. S. H. Hussien [4] and M. A. A. Ali [1] obtained some birecurrent tensors in a 𝐾ℎ – birecurrent Finsler space. II. Certain Generalized Birecurrent Tensors Let us consider an 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 characterized by the condition (2.1) 𝐾𝑗𝑘 ℎ|ℓ |𝑚 𝑖 = 𝜆ℓ 𝐾𝑗𝑘 ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 , 𝐾𝑗𝑘 ℎ 𝑖 ≠ 0 where 𝜆ℓ and 𝑏ℓ𝑚 = 𝜆ℓ|𝑚 are non-zero covariant vector fields and covariant tensor field of second order, respectively. The space and the tensor satisfying the condition (2.1) will be called 𝐾ℎ –generalized birecurrent space and ℎ–generalized birecurrent tensor, respectively. We shall denote them briefly by 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 and ℎ– 𝐺𝐵𝑅, respectively. Transvecting (2.1) by the metric tensor g𝑖𝑝 , using (1.8) and (1.1b), we get (2.2) 𝐾𝑗𝑝𝑘 ℎ|ℓ|𝑚 = 𝜆ℓ 𝐾𝑗𝑝𝑘 ℎ|𝑚 + 𝑏ℓ𝑚 𝐾𝑗𝑝𝑘 ℎ . Contracting the indices 𝑖 and ℎ in (2.2) and using (1.11), we get (2.3) 𝐾𝑗𝑘 |ℓ|𝑚 = 𝜆ℓ 𝐾𝑗𝑘 |𝑚 + 𝑏ℓ𝑚 𝐾𝑗𝑘 . Transvecting (2.3) by 𝑦 𝑘 and using (1.1a), we get (2.4) 𝐾𝑗 |ℓ|𝑚 = 𝜆ℓ 𝐾𝑗 |𝑚 + 𝑏ℓ𝑚 𝐾𝑗 . where 𝐾𝑗𝑘 𝑦 𝑘 = 𝐾𝑗 . Transvecting (2.1) by 𝑦 𝑗 , using (1.1a) and (1.10), we get (2.5) 𝐻𝑘ℎ|ℓ |𝑚 𝑖 = 𝜆ℓ 𝐻𝑘ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐻𝑘ℎ . 𝑖 Contracting the indices 𝑖 and ℎ in (2.5) and using (𝐻𝑘 = 𝐻𝑘𝑖 𝑖 ), we get (2.6) 𝐻𝑘|ℓ|𝑚 = 𝜆ℓ 𝐻𝑘|𝑚 + 𝑏ℓ𝑚 𝐻𝑘 . Differentiating (1.9) covariantly with respect to 𝑥ℓ in the sense of Cartan and using(1.10), we get (2.7) 𝐾ℎ𝑖𝑗𝑘 |ℓ + 𝐾𝑖ℎ𝑗𝑘 |ℓ = (−2𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|ℓ . Differentiating (2.7) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.8) 𝐾ℎ𝑖𝑗𝑘 |ℓ|𝑚 + 𝐾𝑖ℎ𝑗𝑘 |ℓ|𝑚 = (−2𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|ℓ|𝑚 . Using (2.2) in (2.8), we get (2.9) 𝜆ℓ(𝐾ℎ𝑖𝑗𝑘 |𝑚 + 𝐾𝑖ℎ𝑗𝑘 |𝑚 ) + 𝑏ℓ𝑚 𝐾ℎ𝑖𝑗𝑘 + 𝐾𝑖ℎ𝑗𝑘 = (−2𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|ℓ|𝑚 . Putting (1.9), (1.10) and (2.7) in (2.9), we get (2.10) (𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 ) . Transvecting (2.10) by gℎ𝑝 , using (1.1c) and (1.4), we get (2.11) (𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 ). Transvecting (2.11) by 𝑦 𝑗 , using (1.1a) and (1.6), we get (2.12) (𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 ). Transvecting (2.10) by gℎ𝑖 , using (1.1c) and (1.3), we get (2.13) (𝐶𝑟 𝐻𝑗𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑟 𝐻𝑗𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑟 𝐻𝑗𝑘 𝑟 ). Transvecting (2.13) by 𝑦 𝑗 , using (1.1a) and (1.6), we get (2.14) (𝐶𝑟 𝐻𝑘 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑟 𝐻𝑘 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑟 𝐻𝑘 𝑟 ). Contracting the indices 𝑝 and 𝑘 in (2.12), we get (2.15) (𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 ) ℓ 𝑚 = 𝜆ℓ(𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 )|𝑚 + 𝑏ℓ𝑚 (𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 ). Thus, we conclude Theorem 2.1. In 𝐾 ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors (𝐶ℎ𝑖𝑟 𝐻𝑗𝑘 𝑟 ),(𝐶𝑖𝑟 𝑝 𝐻𝑗𝑘 𝑟 ), 𝐶𝑖𝑟 𝑝 𝐻𝑘 𝑟 , (𝐶𝑟 𝐻𝑗𝑘 𝑟 ), (𝐶𝑟 𝐻𝑘 𝑟 ) and (𝐶𝑖𝑟 𝑝 𝐻𝑝 𝑟 ) are all generalized birecurrent.
  • 3. Certain Generalized Birecurrent… www.ijmsi.org 41 | Page We know the identity [7] (2.16) 𝐾𝑗 = 𝐻𝑗 − 𝐻𝑗 𝑖 𝐶𝑖 . Differentiating (2.16) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.17) 𝐾𝑗|ℓ = 𝐻𝑗 |ℓ − (𝐻𝑗 𝑖 𝐶𝑖)|ℓ. Differentiating (2.17) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.18) 𝐾𝑗|ℓ|𝑚 = 𝐻𝑗 |ℓ|𝑚 − (𝐻𝑗 𝑖 𝐶𝑖)|ℓ|𝑚 . Using (2.4) and (2.14) in (2.18), we get (2.19) 𝐾𝑗 |ℓ|𝑚 = 𝜆ℓ 𝐻𝑗 |𝑚 − (𝐻𝑗 𝑖 𝐶𝑖)|𝑚 + 𝑏ℓ𝑚 𝐻𝑗 − (𝐻𝑗 𝑖 𝐶𝑖) . Putting (2.16) and (2.17) in (2.19), we get (2.20) 𝐾𝑗 |ℓ|𝑚 = 𝜆ℓ 𝐾𝑗 |𝑚 + 𝑏ℓ𝑚 𝐾𝑗 . Thus, we conclude Theorem 2.2. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the vector 𝐾𝑗 is generalized birecurrent. Also, we have the identity [7] (2.21) 𝑅𝑗 = 𝐾𝑗 + 𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 . Differentiating (2.21) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.22) 𝑅𝑗|ℓ = 𝐾𝑗 |ℓ + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 )|ℓ . Differentiating (2.22) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.23) 𝑅𝑗|ℓ|𝑚 = 𝐾𝑗 |ℓ|𝑚 + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 )|ℓ|𝑚 . Using (2.15) and (2.20) in (2.23), we get (2.24) 𝑅𝑗|ℓ|𝑚 = 𝜆ℓ 𝐾𝑗|𝑚 + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 )|𝑚 + 𝑏ℓ𝑚 𝐾𝑗 + (𝐶𝑗𝑟 𝑖 𝐻𝑖 𝑟 ) . Putting (2.21) and (2.22) in (2.24), we get 𝑅𝑗|ℓ|𝑚 = 𝜆𝑙 𝑅𝑗|𝑚 + 𝑏ℓ𝑚 𝑅𝑗 . Thus, we conclude Theorem 2.3. In 𝐾 ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the vector 𝑅𝑗 is generalized birecurrent. We have Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 , 𝑣 ℎ𝑣 – torsion tensor 𝑃𝑗𝑘 𝑖 and Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 are connected by the formula (1.12). Differentiating (1.12) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.25) 𝐻 𝑗𝑘 ℎ|ℓ 𝑖 − 𝐾 𝑗𝑘 ℎ|ℓ 𝑖 = (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 − ℎ 𝑘)|ℓ. Differentiating ( 2.25) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.26) 𝐻 𝑗𝑘 ℎ|ℓ|𝑚 𝑖 − 𝐾 𝑗𝑘 ℎ|ℓ|𝑚 𝑖 = (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 − ℎ 𝑘)|ℓ|𝑚 . Using (2.1) and if Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 is generalized birecurrent, (2.26) reduces to (2.27) 𝜆ℓ 𝐻 𝑗𝑘 ℎ|𝑚 𝑖 – 𝐾 𝑗𝑘 ℎ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐻𝑗𝑘 ℎ 𝑖 – 𝐾𝑗𝑘 ℎ 𝑖 = (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘)|ℓ|𝑚 . Putting (1.12) and (2.25) in (2.27), we get (𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘) ℓ 𝑚 = 𝜆ℓ(𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘)|𝑚 +𝑏ℓ𝑚 𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 – ℎ 𝑘 . Thus, we conclude Theorem 2.4. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensor 𝑃𝑗𝑘 |ℎ 𝑖 + 𝑃𝑗𝑘 𝑟 𝑃𝑟ℎ 𝑖 − ℎ 𝑘 is generalized birecurrent [provided Berwald curvature tensor 𝐻𝑗𝑘 ℎ 𝑖 is generalized birecurrent]. We know the curvature tensor 𝐾𝑖𝑗𝑘 ℎ satisfies [5] the identity (2.28) 𝐾ℎ𝑖𝑗𝑘 – 𝐾𝑗𝑘 ℎ𝑖 = 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 – 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 . Differentiating (2.28) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.29) 𝐾ℎ𝑖𝑗𝑘 |ℓ– 𝐾𝑗𝑘 ℎ𝑖|ℓ = (𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 − 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|ℓ. Differentiating (2.29) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.30) 𝐾ℎ𝑖𝑗𝑘 ℓ 𝑚 – 𝐾𝑗𝑘 ℎ𝑖 ℓ 𝑚 = (𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 − 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘
  • 4. Certain Generalized Birecurrent… www.ijmsi.org 42 | Page − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|ℓ|𝑚 . Using (2.2) in (2.30), we get (2.31) 𝜆ℓ 𝐾ℎ𝑖𝑗𝑘 |ℓ − 𝐾𝑗𝑘 ℎ𝑖|ℓ + 𝑏ℓ𝑚 𝐾ℎ𝑖𝑗𝑘 − 𝐾𝑗𝑘 ℎ𝑖 = (𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 − 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|ℓ|𝑚 . Putting (2.28) and (2.29) in (2.31), we get (2.32) ( 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘– 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 ) ℓ 𝑚 = 𝜆ℓ( 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 – 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 )|𝑚 + 𝑏ℓ𝑚 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 – 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 – 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟 𝐶𝑟𝑗𝑘 Transvecting (2.32) by 𝑦 𝑗 , using (1.1a), (1.2) and (1.6), we get (2.33) (𝐻ℎ 𝑟 𝐶𝑟𝑖𝑘 – 𝐻𝑖 𝑟 𝐶𝑟ℎ𝑘 + 𝐻𝑘 𝑟 𝐶𝑟ℎ𝑖) ℓ 𝑚 = 𝜆ℓ(𝐻ℎ 𝑟 𝐶𝑟𝑖𝑘 – 𝐻𝑖 𝑟 𝐶𝑟ℎ𝑘 + 𝐻𝑘 𝑟 𝐶𝑟ℎ𝑖)|𝑚 +𝑏ℓ𝑚 (𝐻ℎ 𝑟 𝐶𝑟𝑖𝑘 – 𝐻𝑖 𝑟 𝐶𝑟ℎ𝑘 + 𝐻𝑘 𝑟 𝐶𝑟ℎ𝑖). Transvecting (2.33) by g 𝑝𝑟 , using (1.1c) and (1.4), we get (𝐻ℎ 𝑟 𝐶𝑖𝑘 𝑝 − 𝐻𝑖 𝑟 𝐶ℎ𝑘 𝑝 + 𝐻𝑘 𝑟 𝐶ℎ𝑖 𝑝 ) ℓ 𝑚 = 𝜆ℓ(𝐻ℎ 𝑟 𝐶𝑖𝑘 𝑝 − 𝐻𝑖 𝑟 𝐶ℎ𝑘 𝑝 + 𝐻𝑘 𝑟 𝐶ℎ𝑖 𝑝 )|𝑚 +𝑏ℓ𝑚 (𝐻ℎ 𝑟 𝐶𝑖𝑘 𝑝 − 𝐻𝑖 𝑟 𝐶ℎ𝑘 𝑝 + 𝐻𝑘 𝑟 𝐶ℎ𝑖 𝑝 ). Thus, we conclude Theorem 2.5. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors 𝐻ℎ𝑗 𝑟 𝐶𝑟𝑖𝑘 – 𝐻ℎ𝑘 𝑟 𝐶𝑟𝑖𝑗 + 𝐻𝑖𝑘 𝑟 𝐶𝑟ℎ𝑗 − 𝐻𝑖𝑗 𝑟 𝐶𝑟ℎ𝑘 − 𝐻𝑗𝑘 𝑟 𝐶𝑟ℎ𝑖 + 𝐻ℎ𝑖 𝑟𝐶𝑟𝑗𝑘, 𝐻ℎ 𝑟𝐶𝑟𝑖𝑘−𝐻𝑖 𝑟𝐶𝑟ℎ𝑘+𝐻𝑘 𝑟𝐶𝑟ℎ𝑖 and (𝐻ℎ 𝑟𝐶𝑖𝑘𝑝− 𝐻𝑖 𝑟𝐶ℎ𝑘𝑝+ 𝐻𝑘 𝑟𝐶ℎ𝑖𝑝) are all generalized birecurrent.We have the identity [7] (2.34) 𝐾𝑖𝑗ℎ𝑘 + 𝐾𝑖𝑘𝑗 ℎ + 𝐾𝑖ℎ𝑘𝑗 = −2 𝑦 𝑟 𝐶𝑖𝑗𝑠 𝐾𝑟ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐾𝑟𝑗 ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐾𝑟𝑘𝑗 𝑠 . Using (1.10) in (2.34), we get (2.35) 𝐾𝑖𝑗ℎ𝑘 + 𝐾𝑖𝑘𝑗 ℎ + 𝐾𝑖ℎ𝑘𝑗 = −2 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 . Differentiating (2.35) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.36) 𝐾𝑖𝑗ℎ𝑘|ℓ + 𝐾𝑖𝑘𝑗 ℎ|ℓ + 𝐾𝑖ℎ𝑘𝑗 |ℓ = −2 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ . Differentiating (2.36) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.37) 𝐾𝑖𝑗ℎ𝑘|ℓ|𝑚 + 𝐾𝑖𝑘𝑗 ℎ|ℓ|𝑚 + 𝐾𝑖ℎ𝑘𝑗 |ℓ|𝑚 = −2 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 . Using (2.2), (2.35) and (2.36) in (2.37), we get (2.38) 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 = 𝜆ℓ 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |𝑚 +𝑏ℓ𝑚 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 . Transvecting (2.38) by 𝑦 𝑗 , using (1.1a), (1.2) and (1.6), we get (2.39) 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 ℓ 𝑚 = 𝜆ℓ 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 . Transvecting (2.39) by g 𝑝𝑟 , using (1.1c) and (1.4), we get 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 ℓ 𝑚 = 𝜆ℓ 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 . Thus, we conclude Theorem 2.6. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 , 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 − 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 and 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 are all generalized birecurrent. Differentiating (1.7) covariantly with respect to 𝑥 𝑚 in the sense of Cartan, we get (2.40) 𝐾𝑗𝑘 ℎ|ℓ|𝑚 𝑖 + 𝐾𝑗ℓ𝑘|ℎ|𝑚 𝑖 + 𝐾𝑗ℎℓ|𝑘|𝑚 𝑖 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 𝐾𝑟ℎℓ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 𝐾𝑟ℓ𝑘|𝑚 𝑠 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0. Using (2.1) in (2.40), we get (2.41) 𝜆ℓ 𝐾𝑗𝑘 ℎ|𝑚 𝑖 + 𝜆ℎ 𝐾𝑗ℓ𝑘|𝑚 𝑖 + 𝜆 𝑘 𝐾𝑗ℎℓ|𝑚 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 +𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 𝐾𝑟ℎℓ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ|𝑚 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 𝐾𝑟ℓ𝑘|𝑚 𝑠 +𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0.
  • 5. Certain Generalized Birecurrent… www.ijmsi.org 43 | Page If Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 is recurrent, (2.41) becomes (2.42) 𝜆ℓ 𝜆 𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝜆ℎ 𝜆 𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝜆 𝑘 𝜆 𝑚 𝐾𝑗ℎℓ 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 + 𝜆 𝑚 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 𝐾𝑟ℓ𝑘 𝑠 + 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0 . Putting (1.7) in (2.42), we get 𝜆ℓ 𝜆 𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝜆ℎ 𝜆 𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝜆 𝑘 𝜆 𝑚 𝐾𝑗ℎℓ 𝑖 + 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 +𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 +𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 −𝜆 𝑚 ( 𝐾𝑗𝑘 ℎ|ℓ 𝑖 + 𝐾𝑗ℓ𝑘|ℎ 𝑖 + 𝐾𝑗ℎℓ|𝑘 𝑖 )+ 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0. which can be written as (2.43) 𝑏ℓ𝑚 𝐾𝑗𝑘 ℎ 𝑖 + 𝑏ℎ𝑚 𝐾𝑗ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐾𝑗ℎℓ 𝑖 + + 𝑦 𝑟 𝜕𝑠 𝛤𝑗𝑘 ∗𝑖 |𝑚 𝐾𝑟ℎℓ 𝑠 + 𝜕𝑠 𝛤𝑗ℓ ∗𝑖 |𝑚 𝐾𝑟𝑘ℎ 𝑠 + 𝜕𝑠 𝛤𝑗ℎ ∗𝑖 |𝑚 𝐾𝑟ℓ𝑘 𝑠 = 0 Transvecting (2.43) by 𝑦 𝑗 , using (1.1a), (1.10) and (1.5), we get (2.44) 𝑏ℓ𝑚 𝐻𝑘ℎ 𝑖 + 𝑏ℎ𝑚 𝐻ℓ𝑘 𝑖 + 𝑏 𝑘𝑚 𝐻ℎℓ 𝑖 + 𝑃𝑠𝑘|𝑚 𝑖 𝐻ℎℓ 𝑠 +𝑃𝑠ℓ|𝑚 𝑖 𝐻𝑘ℎ 𝑠 + 𝑃𝑠ℎ|𝑚 𝑖 𝐻ℓ𝑘 𝑠 = 0 . Thus, we conclude Theorem 2.7. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , we have the identities (2.43) and (2.44) [provided Cartan fourth curvature tensor Kjkh i is recurrent]. We know that the associate tensor 𝑅𝑖𝑗𝑘 ℎ of Cartan's third curvature tensor 𝑅𝑗𝑘 ℎ 𝑖 satisfies the identity [7] (2.45) 𝑅𝑖𝑗ℎ𝑘 + 𝑅𝑖𝑘𝑗 ℎ + 𝑅𝑖ℎ𝑘𝑗 + 𝐶𝑖𝑗𝑠 𝐾𝑟ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐾𝑟𝑗 ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐾𝑟𝑘𝑗 𝑠 𝑦 𝑟 = 0. Using (1.11) in (2.45), we get (2.46) 𝑅𝑖𝑗ℎ𝑘 + 𝑅𝑖𝑘𝑗 ℎ + 𝑅𝑖ℎ𝑘𝑗 + 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 = 0. Differentiating (2.46) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.47) 𝑅𝑖𝑗ℎ𝑘|ℓ + 𝑅𝑖𝑘𝑗 ℎ|ℓ + 𝑅𝑖ℎ𝑘𝑗 |ℓ + 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ = 0. The associate tensor 𝐾𝑖𝑗𝑘 ℎ of Cartan's fourth curvature tensor 𝐾𝑗𝑘 ℎ 𝑖 and the associate tensor 𝑅𝑖𝑗𝑘 ℎ of Cartan's third curvature tensor 𝑅𝑗𝑘 ℎ 𝑖 are connected by the identity [7] (2.48) 𝐾ℎ𝑖𝑗𝑘 − 𝐾𝑖ℎ𝑗𝑘 = 2𝑅ℎ𝑖𝑗𝑘 . Differentiating (2.48) covariantly with respect to 𝑥ℓ in the sense of Cartan, we get (2.49) 𝐾ℎ𝑖𝑗𝑘 |ℓ − 𝐾𝑖ℎ𝑗𝑘 |ℓ = 2𝑅ℎ𝑖𝑗𝑘 |ℓ. Differentiating (2.49) covariantly with respect to 𝑥 𝑚 in the sense of Cartan and using (2.2), we get (2.50) 𝜆ℓ(𝐾ℎ𝑖𝑗𝑘 |𝑚 − 𝐾𝑖ℎ𝑗𝑘 |𝑚 ) + 𝑏ℓ𝑚 (𝐾ℎ𝑖𝑗𝑘 − 𝐾𝑖ℎ𝑗𝑘 ) = 2𝑅ℎ𝑖𝑗𝑘 |ℓ|𝑚 . Putting (2.48) and (2.49) in (2. 50), we get (2.51) 𝑅ℎ𝑖𝑗𝑘 |ℓ|𝑚 = 𝜆ℓ 𝑅ℎ𝑖𝑗𝑘 |𝑚 + 𝑏ℓ𝑚 𝑅ℎ𝑖𝑗𝑘 . Differentiating (2.47) covariantly with respect to 𝑥 𝑚 in the sense of Cartan and using (2.51), we get (2.52) 𝜆ℓ 𝑅𝑖𝑗ℎ𝑘|𝑚 + 𝑅𝑖𝑘𝑗 ℎ|𝑚 + 𝑅𝑖ℎ𝑘𝑗 |𝑚 + 𝑏ℓ𝑚 𝑅𝑖𝑗ℎ𝑘 + 𝑅𝑖𝑘𝑗 ℎ + 𝑅𝑖ℎ𝑘𝑗 + 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 = 0. In view of (2.46) and putting (2.45) in (2.52), we get (2.53) 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |ℓ|𝑚 = 𝜆ℓ 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 . Transvecting (2.53) by 𝑦 𝑗 , using (1.1a), (1.2) and (1.6), we get (2.54) 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 |𝑙|𝑚 = 𝜆ℓ 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 – 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 . Transvecting (2.54) by g 𝑝𝑖 , using (1.1c) and (1.4), we get 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 |ℓ|𝑚 = 𝜆ℓ 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 |𝑚 + 𝑏ℓ𝑚 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 . Thus, we conclude
  • 6. Certain Generalized Birecurrent… www.ijmsi.org 44 | Page Theorem 2.8. In 𝐾ℎ – 𝐺𝐵𝑅– 𝐹𝑛 , the tensors 𝐶𝑖𝑗𝑠 𝐻ℎ𝑘 𝑠 + 𝐶𝑖𝑘𝑠 𝐻𝑗ℎ 𝑠 + 𝐶𝑖ℎ𝑠 𝐻𝑘𝑗 𝑠 , 𝐶𝑖𝑘𝑠 𝐻ℎ 𝑠 − 𝐶𝑖ℎ𝑠 𝐻𝑘 𝑠 and 𝐶𝑘𝑠 𝑝 𝐻ℎ 𝑠 − 𝐶ℎ𝑠 𝑝 𝐻𝑘 𝑠 are all generalized birecurrent. References [1.] Ali, M.A.A.: On 𝐾ℎ - birecurrent Finsler space, M.sc. Thesis, University of Aden, (Aden) (Yemen), (2014). [2.] Cartan, 𝐄.: Sur les espaces de Finsler, C.R. Aead, Sci.(Paris) 196, (1933), 582-586. [3.] Cartan, 𝐄.: Sur les espaces de Finsler, Actualite, (Paris),79, (1934); 2nd edit,(1971). [4.] Hussien, N.S.H.: On 𝐾ℎ –recurrent Finsler space, M.sc.Thesis, University of Aden, (Aden) (Yemen), (2014). [5.] Matsumoto, M.: On h-isotropic and 𝐶ℎ – recurrent Finsler, J. Math. Kyoto Univ . 11, (1971) ,1- 9. [6.] Matsumoto, M.: On Finsler spaces with curvature tensor of some special forms, Tensor N.S., 22, (1971) ,201- 209. [7.] Rund, H.: The differential geometry of Finsler space, Springer verlag, Berlin-Gottingen-Heidelberg,(1959); 2nd Edit. (in Russian), Nauka, (Moscow),(1981).