52 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ On Study Kh Generalized Birecurrent Affinely Connected Space Fahmi Yaseen Abdo Qasema*, Amani Mohammed Abdulla Hanballab aDepartment of Mathematics , Faculty of Education-Aden, University of Aden, Khormaksar , Aden, Yemen bDepartment of Mathematics , Community College, Dar-Saad , Aden, Yemen a,bEmail: Fahmiyassen1@gmail.com, ahanballa@yahoo.com Abstract In the present paper we introduce a πΎπΎβ„Ž – generalized birecurrent space which characterized by the condition πΎπΎπ‘—π‘—π‘—π‘—β„Ž|β„“ |π‘šπ‘š 𝑖𝑖 = πœ†πœ†β„“ πΎπΎπ‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 , πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– β‰  0, where πœ†πœ†β„“ and π‘π‘β„“π‘šπ‘š are non-zero covariant vector fields and covariant tensor field of second order, respectively. This space satisfies the condition of affinely connected space called πΎπΎβ„Ž – generalized birecurrent affinely connected space. Keywords: Finsler space; πΎπΎβ„Ž βˆ’ Generalized birecurrent space; Ricci tensor. 1. Introduction H. D. Pande and B. Single [4] discussed the recurrence property in an affinely connected space. P. K. Dwivedi [7] worked out the role of π‘ƒπ‘ƒβˆ— – reducible space in affinely connected space. A. A. M. Saleem [2] obtained some results when the πΆπΆβ„Ž – generalized birecurrent and πΆπΆβ„Ž – special generalized birecurrent are affinely connected spaces. A. A. A. Muhib [1] obtained some results when π‘…π‘…β„Ž – generalized trirecurrent and π‘…π‘…β„Ž – special generalized trirecurrent are affinely connected spaces. M. A. A. Ali [5] obtained certain identities in a πΎπΎβ„Ž – birecurrent affinely connected spaces. N. S. H. Hussien [6] obtained certain identities in a πΎπΎβ„Ž – recurrent affinely connected spaces. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ mailto:Fahmiyassen1@gmail.com mailto:ahanballa@yahoo.com 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 53 Let 𝐹𝐹𝑛𝑛 be an 𝑛𝑛-dimensional Finsler space equipped with the metric function a 𝐹𝐹(π‘₯π‘₯,𝑦𝑦) satisfying the request conditions [3]. The vectors 𝑦𝑦𝑖𝑖 , 𝑦𝑦𝑖𝑖 and the metric tensor g𝑖𝑖𝑗𝑗 satisfies the following relations (1.1) a) 𝑦𝑦 |𝑗𝑗 𝑖𝑖 = 0 and b) g 𝑖𝑖𝑗𝑗|𝑗𝑗 = 0 , The β„Ž βˆ’ covariant differentiation with respect to π‘₯π‘₯ 𝑗𝑗, commute with the partial differentiation with respect to 𝑦𝑦𝑗𝑗 according to (1.2) a) οΏ½Μ‡οΏ½πœ•π‘—π‘—οΏ½π‘‹π‘‹ |𝑗𝑗 𝑖𝑖 οΏ½ βˆ’ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘‹π‘‹ 𝑖𝑖� |𝑗𝑗 = π‘‹π‘‹π‘Ÿπ‘ŸοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—Ξ“ π‘Ÿπ‘Ÿπ‘—π‘— βˆ— 𝑖𝑖 οΏ½ βˆ’ οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ‘‹π‘‹ 𝑖𝑖� 𝑃𝑃 𝑗𝑗𝑗𝑗 π‘Ÿπ‘Ÿ , where (1.2) b) 𝑃𝑃𝑗𝑗𝑗𝑗 π‘Ÿπ‘Ÿ ∢= οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—Ξ“ β„Žπ‘—π‘— βˆ—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦β„Ž = Ξ“ π‘—π‘—β„Žπ‘—π‘— βˆ—π‘Ÿπ‘Ÿ 𝑦𝑦 β„Ž The tensor 𝐾𝐾 π‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑖𝑖 is called Cartan's fourth curvature tensor which is skew-symmetric in its last two lower indices π‘˜π‘˜ and β„Ž , i. e. (1.3) 𝐾𝐾 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = βˆ’ 𝐾𝐾 π‘—π‘—β„Žπ‘—π‘— 𝑖𝑖 . The associate tensor 𝐾𝐾 π‘–π‘–π‘—π‘—π‘—π‘—β„Ž of the curvature tensor 𝐾𝐾 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 is given by (1.4) 𝐾𝐾 π‘–π‘–π‘—π‘—π‘—π‘—β„Ž ∢= g π‘Ÿπ‘Ÿπ‘—π‘— 𝐾𝐾 π‘–π‘–π‘—π‘—β„Ž π‘Ÿπ‘Ÿ . The Ricci tensor 𝐾𝐾 𝑗𝑗𝑗𝑗 of the curvature tensor 𝐾𝐾 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 is given by (1.5) 𝐾𝐾 𝑗𝑗𝑗𝑗𝑖𝑖 𝑖𝑖 = 𝐾𝐾 𝑗𝑗𝑗𝑗. The curvature tensor 𝐾𝐾 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 satisfies the following relations too (1.6) 𝐾𝐾 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 𝑦𝑦 𝑗𝑗 = π»π»π‘—π‘—β„Ž 𝑖𝑖 and (1.7) π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 βˆ’ πΎπΎπ‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = 𝑃𝑃𝑗𝑗𝑗𝑗|β„Ž 𝑖𝑖 + 𝑃𝑃𝑗𝑗𝑗𝑗 π‘Ÿπ‘Ÿ 𝑃𝑃 π‘Ÿπ‘Ÿβ„Ž 𝑖𝑖 βˆ’ β„Ž π‘˜π‘˜β„ . Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies the relation (1.8) π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 𝑦𝑦 𝑗𝑗 = π»π»π‘—π‘—β„Ž 𝑖𝑖 and 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 54 (1.9) π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = οΏ½Μ‡οΏ½πœ•π‘—π‘—π»π»π‘—π‘—β„Ž 𝑖𝑖 , where 𝐻𝐻 π‘—π‘—β„Ž 𝑖𝑖 called β„Ž(𝑣𝑣) βˆ’ torsion tensor. Also, satisfies bianchi identity (1.10) a) π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 + π»π»β„Žπ‘—π‘—π‘—π‘— 𝑖𝑖 + π»π»π‘—π‘—β„Žπ‘—π‘— 𝑖𝑖 = 0 and it is skew- symmetric in its last two lower indices, i.e. (1.10) 𝑏𝑏) π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = βˆ’ π»π»π‘—π‘—β„Žπ‘—π‘— 𝑖𝑖 . The deviation tensor 𝐻𝐻 𝑗𝑗 𝑖𝑖 is positively homogeneous of degree two in 𝑦𝑦 𝑖𝑖 and satisfies (1.11) π»π»β„Žπ‘—π‘— 𝑖𝑖 𝑦𝑦 β„Ž = 𝐻𝐻𝑗𝑗 𝑖𝑖 , (1.12) 𝐻𝐻𝑗𝑗𝑗𝑗 = 𝐻𝐻𝑗𝑗𝑗𝑗𝑖𝑖 𝑖𝑖 , (1.13) 𝐻𝐻𝑗𝑗 = 𝐻𝐻𝑗𝑗𝑖𝑖 𝑖𝑖 , and (1.14) 𝐻𝐻 = 1 π‘›π‘›βˆ’1 𝐻𝐻𝑖𝑖𝑖𝑖 , where 𝐻𝐻𝑗𝑗𝑗𝑗 and 𝐻𝐻 are called β„Ž-Ricci tensor and curvature scalar, respectively. Since contraction of the indices does not affect the homogeneity in 𝑦𝑦𝑖𝑖 , hence the tensors π»π»π‘Ÿπ‘Ÿπ‘—π‘— , π»π»π‘Ÿπ‘Ÿ and the scalar 𝐻𝐻 are also homogeneous of degree zero, one and two in 𝑦𝑦𝑖𝑖 , respectively. The associate tensor π»π»π‘–π‘–π‘—π‘—π‘—π‘—β„Ž of Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 is given by (1.15) π»π»π‘–π‘–π‘—π‘—π‘—π‘—β„Ž = gπ‘Ÿπ‘Ÿπ‘—π‘— π»π»π‘–π‘–π‘—π‘—β„Žπ‘Ÿπ‘Ÿ . The contraction of the indices 𝑖𝑖 and 𝑗𝑗 in (1.10a) and by using (1.12) and the skew-symmetric property of the curvature tensor 𝐻𝐻 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 in the last two lower indices, shows that the β„Ž βˆ’ Ricci tensor satisfies (1.16) π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘Ÿπ‘Ÿ = π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž . The tensor π»π»π‘—π‘—β„Ž.𝑗𝑗 defined by (1.17) π»π»π‘—π‘—β„Ž.𝑗𝑗 ∢= gπ‘–π‘–β„Žπ»π»π‘—π‘—π‘—π‘— 𝑖𝑖 . Cartan's fourth curvature tensor πΎπΎπ‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 satisfies the following identity known as Bianchi identity 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 55 (1.18) πΎπΎπ‘—π‘—π‘—π‘—β„Ž|β„“ 𝑖𝑖 + 𝐾𝐾𝑗𝑗ℓ𝑗𝑗|β„Ž 𝑖𝑖 + πΎπΎπ‘—π‘—β„Žβ„“|𝑗𝑗 𝑖𝑖 + 𝑦𝑦 π‘Ÿπ‘ŸοΏ½οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘ π›€π›€ 𝑗𝑗𝑗𝑗 βˆ—π‘–π‘–οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“ 𝑠𝑠 + οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘ π›€π›€ 𝑗𝑗ℓ βˆ—π‘–π‘–οΏ½πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘ π‘  + οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘ π›€π›€ π‘—π‘—β„Ž βˆ—π‘–π‘–οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘— 𝑠𝑠 οΏ½ = 0. A Finsler space whose Berwald's connection parameter 𝐺𝐺𝑗𝑗𝑗𝑗𝑖𝑖 is independent of 𝑦𝑦𝑖𝑖 is called an affinely connected space (Berwald space ). Thus, an affinely connected space has some properties as follows: (1.19) πΊπΊπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– = 0 and (1.20) 𝐢𝐢𝑖𝑖𝑗𝑗𝑗𝑗|β„Ž= 0 . The connection parameters Ξ“ 𝑗𝑗𝑗𝑗 βˆ—π‘–π‘– of Cartan and 𝐺𝐺𝑗𝑗𝑗𝑗𝑖𝑖 of Berwald coincide in an affinely connected space and they are independent of the direction arguments [3], i.e. (1.21) πΊπΊπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– = οΏ½Μ‡οΏ½πœ•π‘—π‘—πΊπΊπ‘—π‘—β„Žπ‘–π‘– = 0 and (1.22) οΏ½Μ‡οΏ½πœ•π‘—π‘—Ξ“ π‘—π‘—β„Ž βˆ—π‘–π‘– = 0. N. S. H. Hussein [3] introduce the 𝐾𝐾 β„Žβ€“recurrent space which characterized by the condition (1.23) πΎπΎπ‘—π‘—π‘—π‘—β„Ž|β„“ 𝑖𝑖 = πœ†πœ†β„“ πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– , πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– β‰  0, where the covariant vector field πœ†πœ†β„“ being the recurrence vector field. 2. An 𝑲𝑲 𝒉𝒉 βˆ’ Generalized Birecurrent Space Let us consider a Finsler space 𝐹𝐹𝑛𝑛 whose Cartan's fourth curvature tensor πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies the condition (2.1) πΎπΎπ‘—π‘—π‘—π‘—β„Ž|β„“|π‘šπ‘š 𝑖𝑖 = πœ†πœ†β„“πΎπΎπ‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 , πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– β‰  0, where πœ†πœ†β„“ and π‘π‘β„“π‘šπ‘š are non-zero covariant vector field and covariant tensor field of second order, respectively. The space satisfying the condition (2.1) will be called 𝐾𝐾 β„Žβ€“generalized birecurrent space. We shall denote it briefly by 𝐾𝐾 β„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“πΉπΉπ‘›π‘› . Transvecting (2.1) by 𝑦𝑦𝑗𝑗, using (1.1a) and (1.6), we get (2.2) π»π»π‘—π‘—β„Ž|β„“|π‘šπ‘š 𝑖𝑖 = πœ†πœ†β„“π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—β„Ž 𝑖𝑖 . 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 56 Theorem 2.1. In 𝐾𝐾 β„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“πΉπΉπ‘›π‘›, the β„Ž(𝑣𝑣)– torsion tensor π»π»π‘—π‘—β„Žπ‘–π‘– is generalized birecurrent. Differentiating (2.2) partially with respect to 𝑦𝑦𝑗𝑗 and using (1.9), we get (2.3) οΏ½Μ‡οΏ½πœ•π‘—π‘—π»π»π‘—π‘—β„Ž|β„“|π‘šπ‘š 𝑖𝑖 = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“οΏ½Μ‡οΏ½πœ•π‘—π‘—οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 οΏ½ + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘— π‘π‘β„“π‘šπ‘šοΏ½π»π»π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– . Using the commutation formula exhibited by (1.2a) for (π»π»π‘—π‘—β„Ž|β„“ 𝑖𝑖 ) and (π»π»π‘—π‘—β„Ž 𝑖𝑖 ) in (2.3) and using (1.9), we get (2.4) { οΏ½Μ‡οΏ½πœ•π‘—π‘—(π»π»π‘—π‘—β„Ž|β„“ 𝑖𝑖 ) }|π‘šπ‘š + π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘ŸοΏ½π»π»π‘—π‘—β„Ž|β„“ 𝑖𝑖 οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“ οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ π»π»π‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ [ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ ] + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘β„“π‘šπ‘š οΏ½π»π»π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– . Again applying the commutation formula exhibited by (1.2a) for (π»π»π‘—π‘—β„Ž 𝑖𝑖 ) in (2.4) and using (1.9), we get { π»π»π‘—π‘—π‘—π‘—β„Ž|β„“ 𝑖𝑖 + π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿβ„“ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žβ„“ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—β„“π‘Ÿπ‘Ÿ }|π‘šπ‘š + π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ { π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž|β„“ 𝑖𝑖 + π»π»π‘—π‘—β„Žπ‘ π‘  οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘ π‘ β„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½ βˆ’ 𝐻𝐻𝑗𝑗𝑠𝑠𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ β„Žβ„“ βˆ—π‘ π‘ οΏ½ βˆ’ π»π»π‘ π‘ π‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  } π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“ οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ π»π»π‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ [ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ ] + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘— π‘π‘β„“π‘šπ‘š οΏ½π»π»π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– which can be written as (2.5) π»π»π‘—π‘—π‘—π‘—β„Ž|β„“|π‘šπ‘š 𝑖𝑖 + οΏ½π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿβ„“ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žβ„“ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—β„“π‘Ÿπ‘Ÿ οΏ½|π‘šπ‘š + π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž|β„“ 𝑖𝑖 π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’π»π»π‘—π‘—β„Žπ‘ π‘  οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ β„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ β„Žβ„“ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ π‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ = πœ†πœ†β„“ π»π»π‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ πœ†πœ†β„“ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘β„“π‘šπ‘š οΏ½π»π»π‘—π‘—β„Žπ‘–π‘– . This shows that 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 57 π»π»π‘—π‘—π‘—π‘—β„Ž|β„“|π‘šπ‘š 𝑖𝑖 = πœ†πœ†β„“ π»π»π‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– if and only if (2.6) οΏ½ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿβ„“ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žβ„“ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—β„“π‘Ÿπ‘Ÿ οΏ½|π‘šπ‘š + π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž|β„“ 𝑖𝑖 π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘—π‘—β„Žπ‘ π‘  οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ β„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ β„Žβ„“ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ π‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ πœ†πœ†β„“ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’πœ†πœ†β„“ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘–π‘– + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘— π‘π‘β„“π‘šπ‘š οΏ½π»π»π‘—π‘—β„Žπ‘–π‘– . Thus, we conclude Theorem 2.2. In 𝐾𝐾 β„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“πΉπΉπ‘›π‘›, Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 is generalized birecurrent if and only if (2.6) holds good. Transvecting (2.5) by g𝑖𝑖𝑖𝑖, using (1.1b) and (1.15), we get (2.7) π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž|β„“|π‘šπ‘š + gπ‘–π‘–π‘–π‘–οΏ½οΏ½π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿβ„“ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žβ„“ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—β„“π‘Ÿπ‘Ÿ οΏ½|π‘šπ‘š +π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž|β„“ 𝑖𝑖 π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘—π‘—β„Žπ‘ π‘  οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ β„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ β„Žβ„“ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ π‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ οΏ½ = οΏ½ πœ†πœ†β„“ π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž|π‘šπ‘š + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž οΏ½ + gπ‘–π‘–π‘–π‘–οΏ½οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“ οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ πœ†πœ†β„“ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†π‘™π‘™ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘— π‘π‘β„“π‘šπ‘šοΏ½π»π»π‘—π‘—β„Žπ‘–π‘– οΏ½. This shows that π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž|β„“|π‘šπ‘š=πœ†πœ†β„“ π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž|π‘šπ‘š + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž if and only if (2.8) gπ‘–π‘–π‘–π‘–οΏ½οΏ½π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿβ„“ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žβ„“ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—β„“π‘Ÿπ‘Ÿ οΏ½|π‘šπ‘š +π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 58 π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž|β„“ 𝑖𝑖 π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘—π‘—β„Žπ‘ π‘  οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ β„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ β„Žβ„“ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ π‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ οΏ½ = gπ‘–π‘–π‘–π‘–οΏ½οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†π‘™π‘™ οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†π‘™π‘™ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ πœ†πœ†π‘™π‘™ π»π»π‘Ÿπ‘Ÿβ„Žπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’πœ†πœ†π‘™π‘™ π»π»π‘—π‘—π‘Ÿπ‘Ÿπ‘–π‘– οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†π‘™π‘™ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘–π‘– π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘™π‘™π‘šπ‘šοΏ½π»π»π‘—π‘—β„Žπ‘–π‘– οΏ½. Thus, we conclude Theorem 2.3. In 𝐾𝐾 β„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“πΉπΉπ‘›π‘›, the associate tensor π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž of Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– is generalized birecurrent if and only if (2.8) holds good. Contracting the indices 𝑖𝑖 and β„Ž in (2.5), using (1.12) and (1.13), we get (2.9) 𝐻𝐻𝑗𝑗𝑗𝑗|β„“|π‘šπ‘š + οΏ½π»π»π‘—π‘—π‘–π‘–π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿβ„“ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘ŸοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑖𝑖ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½οΏ½ |π‘šπ‘š +𝐻𝐻𝑗𝑗𝑖𝑖|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿ|β„“ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘–π‘–π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ 𝐻𝐻𝑗𝑗|π‘Ÿπ‘ŸοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—|β„“ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’ 𝐻𝐻𝑗𝑗𝑖𝑖𝑠𝑠 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑖𝑖ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑠𝑠𝑗𝑗 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ = οΏ½πœ†πœ†β„“ 𝐻𝐻𝑗𝑗𝑗𝑗|π‘šπ‘š + π‘π‘β„“π‘šπ‘š 𝐻𝐻𝑗𝑗𝑗𝑗 οΏ½ + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“οΏ½π»π»π‘—π‘—|π‘šπ‘š + πœ†πœ†β„“π»π»π‘—π‘—π‘–π‘–π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’πœ†πœ†β„“π»π»π‘Ÿπ‘ŸοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘–π‘–π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘Ÿπ‘Ÿπ‘—π‘— π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘β„“π‘šπ‘š �𝐻𝐻𝑗𝑗. This shows that 𝐻𝐻𝑗𝑗𝑗𝑗|β„“|π‘šπ‘š=πœ†πœ†β„“π»π»π‘—π‘—π‘—π‘—|π‘šπ‘š + π‘π‘β„“π‘šπ‘š 𝐻𝐻𝑗𝑗𝑗𝑗 if and only if (2.10) οΏ½π»π»π‘—π‘—π‘–π‘–π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿβ„“ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘ŸοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑖𝑖ℓ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ 𝑗𝑗ℓ βˆ—π‘Ÿπ‘ŸοΏ½οΏ½ |π‘šπ‘š +𝐻𝐻𝑗𝑗𝑖𝑖|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿ|β„“ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘–π‘–π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ 𝐻𝐻𝑗𝑗|π‘Ÿπ‘ŸοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—|β„“ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’ 𝐻𝐻𝑗𝑗𝑖𝑖𝑠𝑠 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑖𝑖ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑠𝑠𝑗𝑗 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“οΏ½π»π»π‘—π‘—|π‘šπ‘š + πœ†πœ†β„“π»π»π‘—π‘—π‘–π‘–π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘Ÿπ‘ŸοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ πœ†πœ†β„“π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π›€π›€ π‘–π‘–π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘Ÿπ‘Ÿπ‘—π‘— π‘ƒπ‘ƒπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿ + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘β„“π‘šπ‘š �𝐻𝐻𝑗𝑗 . Theorem 2.4. In 𝐾𝐾 β„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“πΉπΉπ‘›π‘›, 𝐾𝐾–Ricci tensor 𝐻𝐻𝑗𝑗𝑗𝑗 in sense of Cartan is generalized birecurrent if and only if (2.10) holds good. 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 59 Contracting the indices 𝑖𝑖 and 𝑗𝑗 in (2.5) and using (1.16), we get (2.11) (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž)|β„“|π‘šπ‘š + οΏ½π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘Ÿπ‘Ÿπ‘™π‘™ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ 𝑗𝑗𝑙𝑙 βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„Žπ‘™π‘™ βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑖𝑖 π‘ƒπ‘ƒπ‘–π‘–β„“π‘Ÿπ‘Ÿ οΏ½|π‘šπ‘š + π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž|β„“ 𝑖𝑖 π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘—π‘—β„Žπ‘ π‘  οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ β„Ž 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ β„Žβ„“ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ π‘—π‘—β„Ž 𝑖𝑖 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ = πœ†πœ†β„“ ( π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž )|π‘šπ‘š + π‘π‘β„“π‘šπ‘š (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž) + οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–πœ†πœ†β„“οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘Ÿπ‘Ÿβ„Ž 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’πœ†πœ†β„“π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑖𝑖 π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π‘π‘β„“π‘šπ‘š οΏ½π»π»π‘—π‘—β„Ž 𝑖𝑖 . This shows that (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž)|β„“|π‘šπ‘š = πœ†πœ†β„“(π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž)|π‘šπ‘š + π‘π‘β„“π‘šπ‘š (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž) if and only if (2.12) οΏ½π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘Ÿπ‘Ÿπ‘™π‘™ βˆ—π‘–π‘–οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ 𝑗𝑗𝑙𝑙 βˆ—π‘Ÿπ‘ŸοΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„Žπ‘™π‘™ βˆ—π‘Ÿπ‘ŸοΏ½βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑖𝑖 π‘ƒπ‘ƒπ‘–π‘–β„“π‘Ÿπ‘Ÿ οΏ½|π‘šπ‘š +π»π»π‘—π‘—β„Ž|β„“ π‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’ π»π»π‘Ÿπ‘Ÿβ„Ž|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘—π‘—π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—π‘Ÿπ‘Ÿ|β„“ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ π»π»π‘—π‘—β„Ž|π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„“π‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž|β„“ 𝑖𝑖 π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘—π‘—β„Žπ‘ π‘  οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑠𝑠ℓ βˆ—π‘–π‘–οΏ½ π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ β„Ž 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ 𝑗𝑗ℓ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝐻𝐻𝑗𝑗𝑠𝑠 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘Ÿπ‘Ÿπ›€π›€ β„Žβ„“ βˆ—π‘ π‘ οΏ½π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + π»π»π‘ π‘ π‘—π‘—β„Ž 𝑖𝑖 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβ„“π‘ π‘  π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ οΏ½ = π‘π‘β„“π‘šπ‘š (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž) + {οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–πœ†πœ†β„“ οΏ½π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„“ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ π‘Ÿπ‘Ÿπ‘šπ‘š βˆ—π‘–π‘– οΏ½ βˆ’πœ†πœ†β„“π»π»π‘—π‘—π‘Ÿπ‘Ÿ 𝑖𝑖 οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π›€π›€ β„Žπ‘šπ‘š βˆ—π‘Ÿπ‘Ÿ οΏ½ βˆ’ πœ†πœ†β„“ π»π»π‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑖𝑖 π‘ƒπ‘ƒπ‘–π‘–π‘šπ‘šπ‘Ÿπ‘Ÿ + οΏ½οΏ½Μ‡οΏ½πœ•π‘–π‘–π‘π‘β„“π‘šπ‘š οΏ½π»π»π‘—π‘—β„Ž 𝑖𝑖 . Thus, we conclude Theorem 2.5. In 𝐾𝐾 β„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“πΉπΉπ‘›π‘›, the tensor (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž) is generalized birecurrent if and only if (2.12) holds good. Differentiating (1.18) covariantly with respect to π‘₯π‘₯π‘šπ‘š in the sense of Cartan and using (1.1a), we get (2.13) πΎπΎπ‘—π‘—π‘—π‘—β„Ž|β„“|π‘šπ‘š 𝑖𝑖 + 𝐾𝐾𝑗𝑗ℓ𝑗𝑗|β„Ž|π‘šπ‘š 𝑖𝑖 + πΎπΎπ‘—π‘—β„Žβ„“|𝑗𝑗|π‘šπ‘š 𝑖𝑖 +𝑦𝑦 π‘Ÿπ‘ŸοΏ½οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑗𝑗 βˆ—π‘–π‘–οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“|π‘šπ‘š 𝑠𝑠 + οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘ π›€π›€π‘—π‘—β„“ βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Ž|π‘šπ‘š 𝑠𝑠 + οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘ π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘—|π‘šπ‘š 𝑠𝑠 οΏ½+𝑦𝑦 π‘Ÿπ‘Ÿ οΏ½οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘ π›€π›€π‘—π‘—π‘—π‘— βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“ 𝑠𝑠 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 60 +οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗ℓ βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑠𝑠 + οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘  π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘—π‘ π‘  οΏ½ = 0. Using (2.1) in (2.13), we get (2.14) πœ†πœ†β„“πΎπΎπ‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„ŽπΎπΎπ‘—π‘—β„“π‘—π‘—|π‘šπ‘š 𝑖𝑖 + πœ†πœ†π‘—π‘— πΎπΎπ‘—π‘—β„Žβ„“|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„Žπ‘šπ‘šπΎπΎπ‘—π‘—β„“π‘—π‘—π‘–π‘– + π‘π‘π‘—π‘—π‘šπ‘šπΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– +𝑦𝑦 π‘Ÿπ‘ŸοΏ½οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑗𝑗 βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“|π‘šπ‘š 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗ℓ βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Ž|π‘šπ‘š 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘—|π‘šπ‘š 𝑠𝑠 οΏ½ +𝑦𝑦 π‘Ÿπ‘Ÿ οΏ½οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑗𝑗 βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“ 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗ℓ βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Žπ‘ π‘  + οΏ½οΏ½Μ‡οΏ½πœ•π‘ π‘  π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘— 𝑠𝑠 οΏ½ = 0. If Cartan's fourth curvature tensor πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– is recurrent which is given by (1.23), (2.14) becomes (2.15) πœ†πœ†β„“ πœ†πœ†π‘šπ‘šπΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + πœ†πœ†β„Ž πœ†πœ†π‘šπ‘šπΎπΎπ‘—π‘—β„“π‘—π‘—π‘–π‘– + πœ†πœ†π‘—π‘— πœ†πœ†π‘šπ‘šπΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– + π‘π‘β„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„Žπ‘šπ‘šπΎπΎπ‘—π‘—β„“π‘—π‘—π‘–π‘– + π‘π‘π‘—π‘—π‘šπ‘šπΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– + πœ†πœ†π‘šπ‘š 𝑦𝑦 π‘Ÿπ‘ŸοΏ½οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑗𝑗 βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“ 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗ℓ βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘– οΏ½πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘— 𝑠𝑠 οΏ½ + 𝑦𝑦 π‘Ÿπ‘Ÿ οΏ½οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑗𝑗 βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“ 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗ℓ βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘— 𝑠𝑠 οΏ½ = 0 Putting (1.18) in (2.15), we get πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– πœ†πœ†β„“ πœ†πœ†π‘šπ‘šπΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + πœ†πœ†β„Ž πœ†πœ†π‘šπ‘šπΎπΎπ‘—π‘—β„“π‘—π‘—π‘–π‘– + πœ†πœ†π‘—π‘— πœ†πœ†π‘šπ‘šπΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– + π‘π‘β„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„Žπ‘šπ‘šπΎπΎπ‘—π‘—β„“π‘—π‘—π‘–π‘– + π‘π‘π‘—π‘—π‘šπ‘šπΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– βˆ’ πœ†πœ†π‘šπ‘š( πΎπΎπ‘—π‘—π‘—π‘—β„Ž|β„“ 𝑖𝑖 + 𝐾𝐾𝑗𝑗ℓ𝑗𝑗|β„Ž 𝑖𝑖 + πΎπΎπ‘—π‘—β„Žβ„“|𝑗𝑗 𝑖𝑖 ) + 𝑦𝑦 π‘Ÿπ‘Ÿ οΏ½οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑗𝑗 βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„Žπ‘™π‘™π‘ π‘  + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑙𝑙 βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑠𝑠 +οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿπ‘™π‘™π‘—π‘— 𝑠𝑠 οΏ½ = 0 which can be written as (2.16) π‘π‘β„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„Žπ‘šπ‘šπΎπΎπ‘—π‘—β„“π‘—π‘—π‘–π‘– + π‘π‘π‘—π‘—π‘šπ‘šπΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– + 𝑦𝑦 π‘Ÿπ‘Ÿ οΏ½οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗𝑗𝑗 βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„Žβ„“π‘ π‘  + +οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  𝛀𝛀𝑗𝑗ℓ βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿπ‘—π‘—β„Ž 𝑠𝑠 + οΏ½ οΏ½Μ‡οΏ½πœ•π‘ π‘  π›€π›€π‘—π‘—β„Ž βˆ—π‘–π‘–οΏ½|π‘šπ‘š πΎπΎπ‘Ÿπ‘Ÿβ„“π‘—π‘— 𝑠𝑠 οΏ½ = 0. Transvecting (2.16) by 𝑦𝑦 𝑗𝑗, using (1.1a), (1.6) and (1.2b), we get π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„Žπ‘šπ‘š 𝐻𝐻ℓ𝑗𝑗𝑖𝑖 + π‘π‘π‘—π‘—π‘šπ‘š π»π»β„Žβ„“π‘–π‘– + 𝑃𝑃𝑠𝑠𝑗𝑗|π‘šπ‘š 𝑖𝑖 π»π»β„Žβ„“π‘ π‘  +𝑃𝑃𝑠𝑠ℓ|π‘šπ‘š 𝑖𝑖 π»π»π‘—π‘—β„Žπ‘ π‘  + π‘ƒπ‘ƒπ‘ π‘ β„Ž|π‘šπ‘š 𝑖𝑖 𝐻𝐻ℓ𝑗𝑗𝑠𝑠 = 0 . 3. 𝑲𝑲𝒉𝒉– Generalized Birecurrent Affinely Connected Space Let us consider an affinely connected or Berwald's space which is characterized by any one of the equivalent conditions (1.19), (1.20), (1.21) and (1.22). 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 61 Definition 3.1. The πΎπΎβ„Žβ€“ generalized birecurrent space is called πΎπΎβ„Žβ€“ generalized birecurrent affinely connected if it satisfies any one of the conditions (1.19), (1.20), (1.21) and (1.22) and denoted briefly by πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space. Let us consider πΎπΎβ„Ž βˆ’ 𝐺𝐺𝐺𝐺𝑅𝑅 βˆ’ affinely connected space. If οΏ½Μ‡οΏ½πœ•π‘—π‘— πœ†πœ†β„“ = 0, οΏ½Μ‡οΏ½πœ•π‘—π‘— π‘π‘β„“π‘šπ‘š = 0 and in view of the conditions (1.2b) and (1.22), the equation (2.5) reduces to (1.3) π»π»π‘—π‘—π‘—π‘—β„Ž|β„“|π‘šπ‘š 𝑖𝑖 = πœ†πœ†β„“ π»π»π‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 . Thus, we conclude Theorem 3.1. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space, if the directional derivative of covariant vector field and covariant tensor of second order are vanish, then Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 is generalized bireurrent. If οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“ = 0, οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘β„“π‘šπ‘š = 0 and in view of the conditions (1.2b) and (1.22), the equation (2.7) reduces to π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž|β„“|π‘šπ‘š = πœ†πœ†β„“ π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž|π‘šπ‘š + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž. Thus, we conclude Theorem 3.2. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space, if the directional derivative of covariant vector field and covariant tensor of second order are vanish, then the associate tensor π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž of Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 is generalized bireurrent. If οΏ½Μ‡οΏ½πœ•π‘—π‘—πœ†πœ†β„“ = 0, οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘β„“π‘šπ‘š = 0 and in view of the conditions (1.2b) and (1.22), the equation (2.9) reduces to 𝐻𝐻𝑗𝑗𝑗𝑗|β„“|π‘šπ‘š = πœ†πœ†β„“ 𝐻𝐻𝑗𝑗𝑗𝑗|π‘šπ‘š + π‘π‘β„“π‘šπ‘š 𝐻𝐻𝑗𝑗𝑗𝑗 . Thus, we conclude Theorem 3.3. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space, if the directional derivative of covariant vector field and covariant tensor of second order are vanish, then the Ricci tensor 𝐻𝐻𝑗𝑗𝑗𝑗 in sense of Berwald is generalized bireurrent. If οΏ½Μ‡οΏ½πœ•π‘—π‘— πœ†πœ†β„“ = 0, οΏ½Μ‡οΏ½πœ•π‘—π‘— π‘π‘β„“π‘šπ‘š = 0 and in view of the conditions (1.2b) and (1.22), the equation (2.11) reduces to (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž)|β„“|π‘šπ‘š = πœ†πœ†β„“ ( π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž )|π‘šπ‘š + π‘π‘β„“π‘šπ‘š ( π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž ). Thus, we conclude 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 62 Theorem 3.4. In πΎπΎβ„Ž βˆ’ 𝐺𝐺𝐺𝐺𝑅𝑅 βˆ’affinely connected space, if the directional derivative of covariant vector field and covariant tensor of second order are vanish, then the tensor (π»π»β„Žπ‘—π‘— βˆ’ π»π»π‘—π‘—β„Ž) is generalized bireurrent. Now, transvecting (3.1) by 𝑦𝑦𝑗𝑗, using (1.1a) and (1.8), we get (3.2) π»π»π‘—π‘—β„Ž|β„“ |π‘šπ‘š 𝑖𝑖 = πœ†πœ†β„“ π»π»π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š π»π»π‘—π‘—β„Ž 𝑖𝑖 . Transvecting (3.2) by 𝑦𝑦𝑗𝑗 , using (1.1a) and (1.11), we get (3.3) π»π»β„Ž|β„“ |π‘šπ‘š 𝑖𝑖 = πœ†πœ†β„“ π»π»β„Ž|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š π»π»β„Ž 𝑖𝑖 . Contracting the indices 𝑖𝑖 and β„Ž in (3.2) and using (1.13), we get 𝐻𝐻𝑗𝑗|β„“|π‘šπ‘š = πœ†πœ†β„“ 𝐻𝐻𝑗𝑗|π‘šπ‘š + π‘π‘β„“π‘šπ‘š 𝐻𝐻𝑗𝑗 Contracting the indices 𝑖𝑖 and β„Ž in (3.3) and using (1.14), we get 𝐻𝐻|β„“|π‘šπ‘š = πœ†πœ†β„“ 𝐻𝐻|π‘šπ‘š + π‘π‘β„“π‘šπ‘š 𝐻𝐻. Transvecting (3.2) by g𝑖𝑖𝑖𝑖 , using (1.1b) and (1.17), we get 𝐻𝐻𝑗𝑗𝑖𝑖.β„Ž|β„“|π‘šπ‘š = πœ†πœ†β„“ 𝐻𝐻𝑗𝑗𝑖𝑖.β„Ž|π‘šπ‘š + π‘π‘β„“π‘šπ‘š 𝐻𝐻𝑗𝑗𝑖𝑖.β„Ž. Thus, we conclude Theorem 3.5. In πΎπΎβ„Ž βˆ’ 𝐺𝐺𝐺𝐺𝑅𝑅 βˆ’affinely connected space, if the directional derivative of covariant vector field and covariant tensor of second order are vanish, then the β„Ž(𝑣𝑣)– torsion tensor π»π»π‘—π‘—β„Ž 𝑖𝑖 , the deviation tensor π»π»β„Ž 𝑖𝑖 , the curvature vector 𝐻𝐻𝑗𝑗 , the curvature scalar 𝐻𝐻 and the tensor 𝐻𝐻𝑗𝑗𝑖𝑖.β„Ž are all generalized birecurrent. In view of (1.22), the equation (2.14) can be written as (3.4) πœ†πœ†β„“ πΎπΎπ‘—π‘—π‘—π‘—β„Ž|π‘šπ‘š 𝑖𝑖 + πœ†πœ†β„ŽπΎπΎπ‘—π‘—β„“π‘—π‘—|π‘šπ‘š 𝑖𝑖 + πœ†πœ†π‘—π‘—πΎπΎπ‘—π‘—β„Žβ„“|π‘šπ‘š 𝑖𝑖 + π‘π‘β„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘β„Žπ‘šπ‘š 𝐾𝐾𝑗𝑗ℓ𝑗𝑗𝑖𝑖 + π‘π‘π‘—π‘—π‘šπ‘š πΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– = 0. In view of (1.23), (3.4) reduces to ( πœ†πœ†β„“ πœ†πœ†π‘šπ‘š + π‘π‘π‘™π‘™π‘šπ‘š) πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + (πœ†πœ†β„Ž πœ†πœ†π‘šπ‘š+ π‘π‘β„Žπ‘šπ‘š ) 𝐾𝐾𝑗𝑗ℓ𝑗𝑗𝑖𝑖 + (πœ†πœ†π‘—π‘— πœ†πœ†π‘šπ‘š+ π‘π‘π‘—π‘—π‘šπ‘š) πΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– = 0 which can be written as (3.5) π‘Žπ‘Žβ„“π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘Žπ‘Žβ„Žπ‘šπ‘š 𝐾𝐾𝑗𝑗ℓ𝑗𝑗𝑖𝑖 + π‘Žπ‘Žπ‘—π‘—π‘šπ‘š πΎπΎπ‘—π‘—β„Žβ„“π‘–π‘– = 0 , where π‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘šπ‘š = πœ†πœ†π‘Ÿπ‘Ÿ πœ†πœ†π‘šπ‘š+ π‘π‘π‘Ÿπ‘Ÿπ‘šπ‘š is covariant tensor field of second order. 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 63 Transvecting (3.5) by 𝑦𝑦𝑗𝑗, using (1.1a) and (1.6), we get (3.6) π‘Žπ‘Žβ„“π‘šπ‘š π»π»π‘—π‘—β„Žπ‘–π‘– + π‘Žπ‘Žβ„Žπ‘šπ‘š 𝐻𝐻ℓ𝑗𝑗𝑖𝑖 + π‘Žπ‘Žπ‘—π‘—π‘šπ‘š π»π»β„Žβ„“π‘–π‘– = 0. Thus, we conclude Theorem 3.6. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“ affinely connected space, the identities (3.5) and (3.6) are hold good. Contracting the indices 𝑖𝑖 and β„“ in (3.5), using (1.3) and (1.5), we get (3.7) π‘Žπ‘Žπ‘–π‘–π‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 βˆ’ π‘Žπ‘Žβ„Žπ‘šπ‘š 𝐾𝐾𝑗𝑗𝑗𝑗 + π‘Žπ‘Žπ‘—π‘—π‘šπ‘šπΎπΎπ‘—π‘—β„Ž = 0 which can be written as (3.8) πΎπΎπ‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = οΏ½ π‘Žπ‘Žβ„Žπ‘šπ‘š πΎπΎπ‘—π‘—π‘—π‘—βˆ’ π‘Žπ‘Žπ‘—π‘—π‘šπ‘š πΎπΎπ‘—π‘—β„ŽοΏ½ π‘Žπ‘Žπ‘π‘π‘šπ‘š . Thus, we conclude Theorem 3.7. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space, Cartan's fourth curvature tensor πΎπΎπ‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 is defined by (3.8). In view of (1.2b) and (1.22), (1.7) reduces to (3.9) 𝐻𝐻 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = 𝐾𝐾 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 . Putting (3.9) in (3.5), we get (3.10) π‘Žπ‘Žβ„“π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘Žπ‘Žβ„Žπ‘šπ‘š 𝐻𝐻𝑗𝑗ℓ𝑗𝑗𝑖𝑖 + π‘Žπ‘Žπ‘—π‘—π‘šπ‘š π»π»π‘—π‘—β„Žβ„“π‘–π‘– = 0. Thus, we conclude Theorem 3.8. In πΎπΎβ„Ž βˆ’ 𝐺𝐺𝐺𝐺𝑅𝑅 βˆ’ affinely connected space, Berwald curvature tensor 𝐻𝐻 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 coincide with Cartan's fourth curvature tensor 𝐾𝐾 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 and the identity (3.10) holds good. Contracting the indices 𝑖𝑖 and β„“ in (3.10), using (1.12), (1.10b) and the skew-symmetric property of Berwald curvature tensor 𝐻𝐻 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 in it's last two lower indices, we get π‘Žπ‘Žπ‘–π‘–π‘šπ‘š π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 βˆ’ π‘Žπ‘Žβ„Žπ‘šπ‘š 𝐻𝐻𝑗𝑗𝑗𝑗 + π‘Žπ‘Žπ‘—π‘—π‘šπ‘š π»π»π‘—π‘—β„Ž = 0. which can be written as (3.11) π»π»π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = οΏ½ π‘Žπ‘Žβ„Žπ‘šπ‘š π»π»π‘—π‘—π‘—π‘—βˆ’ π‘Žπ‘Žπ‘—π‘—π‘šπ‘š π»π»π‘—π‘—β„ŽοΏ½ π‘Žπ‘Žπ‘π‘π‘šπ‘š . 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 64 Thus, we conclude Theorem 3.9. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space, Berwald curvature tensor is defined by (3.11). Contracting the indices 𝑖𝑖 and β„Ž in (3.9), using (1.12) and (1.5), we get 𝐻𝐻𝑗𝑗𝑗𝑗 = 𝐾𝐾𝑗𝑗𝑗𝑗 . Thus, we conclude Theorem 3.10. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space, Ricci tensor 𝐻𝐻𝑗𝑗𝑗𝑗 in sense of Berwald coincide with Ricci tensor 𝐾𝐾𝑗𝑗𝑗𝑗 of Cartan's fourth curvature. Transvecting (3.9) by g𝑖𝑖𝑖𝑖 , using (1.1b), (1.15) and (1.4), we get π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž = πΎπΎπ‘—π‘—π‘–π‘–π‘—π‘—β„Ž . Thus, we conclude Theorem 3.11. In πΎπΎβ„Žβ€“πΊπΊπΊπΊπ‘…π‘…β€“affinely connected space, the associate curvature tensor π»π»π‘—π‘—π‘–π‘–π‘—π‘—β„Ž of Berwald curvature tensor coincide with the associate curvature tensor πΎπΎπ‘—π‘—π‘–π‘–π‘—π‘—β„Ž of Cartan's fourth curvature tensor. 4. Conclusion (3.1) The Kh– generalized birecurrent space is called Kh– generalized birecurrent affinely connected if it satisfies any one of the conditions (1.19), (1.20), (1.21) and (1.22). (3.2) In Kh– GBR–affinely connected space, if the directional derivative of covariant vector field and covariant tensor of second order are vanish, then Berwald curvature tensor Hjkh i is generalized bireurrent. (3.3) In Kh βˆ’ GBR βˆ’affinely connected space, if the directional derivative of covariant vector field a and covariant tensor of second order are vanish, then the h(v)– torsion tensor Hkh i , the deviation tensor Hh i , the curvature vector Hk, the curvature scalar H and the tensor Hkp.h are all generalized birecurrent. (3.4) In Kh– GBR–affinely connected space, Cartan's fourth curvature tensor Kjkh p is defined by (3.8). (3.5) In Kh– GBR–affinely connected space, Berwald curvature tensor H jkh i coincide with Cartan's fourth curvature tensor K jkh i and the identity (3.10) holds good. (3.6) In Kh– GBR–affinely connected space, Ricci tensor Hjk in sense of Berwald coincide with Ricci 65-) Volume 20, No 1, pp 522016( )Research Journal for Engineering, Technology, and Sciences (ASRJETS ScientificAmerican 65 tensor Kjk of Cartan's fourth curvature. (3.7) In Kh– GBR–affinely connected space, the associate curvature tensor Hjpkh of Berwald curvature tensor coincide with the associate curvature tensor Kjpkh of Cartan's fourth curvature tensor. 5. Recommendations Authors recommend the need for the continuing research and development in affinely connected space and its relation with other spaces. References [1] A. A. A. Muhib. "On independent components of tensor, I- relative tensor and R h– generalized trirecurrent Finsler space", M. Sc. Thesis, University of Aden, Aden, Yemen, 2009. [2] A. A. M. Saleem. "On Generalized Birecurrent and Trirecurrent Finsler Spaces", M. Sc. Thesis, University of Aden, Aden, Yemen, 2011. [3] H. Rund: The differential geometry of Finsler space, Springer - Verlag, Berlin Gottingen–Heidelberg, 1959; 2nd Edit. (in Russian), Nauka, (Moscow), 1981. [4] H.D. Pande and B. Single. ''On existence of the affinely connected Finsler spaces with recurrent tensor field, Reprinted from India Journal Pure and Applied of Mathematics, Vol. 8, No. 3,(March 1977), pp. 295-301. [5] M. A. A. Ali ''On Kh - birecurrent Finsler space.'' M.Sc. Thesis, University of Aden, Aden, Yemen, 2014. [6] N.S.H. Hussien ''On Kh- recurrent Finsler space.'' M.sc. Thesis, University of Aden, Aden, Yemen,2014. [7] P. K. Dwivedi: Pβˆ—- Reducible Finsler spaces and application, Int. Journal of Math. Analysis, Vol. 5, No. 5, 2011, pp. 223-229.