9 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 a Generalized 𝛃𝛃𝐑𝐑 βˆ’ Birecurrent Finsler Space Fahmi Yaseen Abdo Qasema*, Wafa'a Hadi Ali Hadib aDepartment of Mathematics , Faculty of Education-Aden, University of Aden, Khormaksar , Aden, Yemen bDepartment of Mathematics , Community College, Dar Saad , Aden, Yemen aEmail: Fahmiyassen1@gmail.com bEmail: wf_hadi@yahoo.com Abstract In the present paper, we introduced a Finsler space whose Cartan's third curvature tensor π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π›Ώπ›Ώπ‘—π‘—π‘–π‘– π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗� βˆ’ 2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 πΆπΆπ‘—π‘—β„Žπ‘šπ‘š βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š). where π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› and π‘π‘π‘šπ‘šπ‘›π‘› are non- zero covariant tensor fields of second order called recurrence tensor fields, such space is called as a generalized 𝛽𝛽𝑅𝑅 βˆ’birecurrent Finsler space. The curvature tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– , the torsion tensor π»π»π‘—π‘—β„Žπ‘–π‘– ,the deviation tensor π»π»β„Žπ‘–π‘– , the Ricci tensors ( 𝐻𝐻𝑗𝑗𝑗𝑗 ,𝑅𝑅𝑗𝑗𝑗𝑗), the vector 𝐻𝐻𝑗𝑗 and the scalar curvature tensor 𝐻𝐻of such space are non-vanishing. Under certain conditions, a generalized 𝛽𝛽𝑅𝑅 βˆ’birecurrent Finsler space becomes Landsberg space . Some conditions have been pointed out which reduce a generalized 𝛽𝛽𝑅𝑅 βˆ’birecurrent Finsler space 𝐹𝐹𝑛𝑛(𝑛𝑛 > 2) into Finsler space of scalar curvature. Keywords: Finsler space; Generalized 𝛽𝛽𝑅𝑅 βˆ’birecurrent Finsler space; Ricci tensor; Landsberg space; Finsler space of scalar curvature. 1. Introduction H.S. Ruse[4] considered a three dimensional Riemannian space having the recurrent of curvature tensor and he called such space as Riemannian space of recurrent curvature. This idea was extended to n-dimensional Riemannian and non- Riemannian space by A.G. Walker [1],Y.C. Worg [14] ,Y.C. Worg and K. Yano [15] and others. ------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 10 S. Dikshit [13] introduced a Finsler space whose Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies recurrence property in the sense of Berwald, F.Y.A.Qasem and A.A.M.Saleem [3] discussed general Finsler space for the β„Žπ‘£π‘£ βˆ’curvature tensor π‘ˆπ‘ˆπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies the birecurrence property with respect to Berwald's coefficient 𝐺𝐺𝑗𝑗𝑗𝑗𝑖𝑖 and they called it UBR- Finsler space. P.N.pandey, S.Saxena and A.Goswami [8] introduced a Finsler space whose Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies generalized recurrence property in the sense of Berwald they called such space generalized 𝐻𝐻-recurrent Finsler space . (1.1) a) 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 = 𝐹𝐹2 b) g𝑖𝑖𝑗𝑗 = οΏ½Μ‡οΏ½πœ•π‘–π‘–π‘¦π‘¦π‘—π‘— = οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘¦π‘¦π‘–π‘– c) ℬ𝑗𝑗 𝑦𝑦𝑖𝑖= 0 d) 𝐢𝐢 𝑖𝑖𝑗𝑗𝑗𝑗 𝑦𝑦𝑖𝑖 = 𝐢𝐢 𝑗𝑗𝑖𝑖𝑗𝑗 𝑦𝑦𝑖𝑖 = 𝐢𝐢 𝑗𝑗𝑗𝑗𝑖𝑖 𝑦𝑦𝑖𝑖 = 0 e) ℬ𝑗𝑗 𝑔𝑔𝑖𝑖𝑗𝑗 = βˆ’2πΆπΆπ‘–π‘–π‘—π‘—π‘—π‘—β„Žπ‘¦π‘¦β„Ž = - 2π‘¦π‘¦β„Žβ„¬ β„ŽπΆπΆπ‘–π‘–π‘—π‘—π‘—π‘— f) 𝐺𝐺 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 𝑦𝑦 𝑗𝑗 = 𝐺𝐺 β„Žπ‘—π‘—π‘—π‘— 𝑖𝑖 𝑦𝑦 𝑗𝑗 = 𝐺𝐺 π‘—π‘—β„Žπ‘—π‘— 𝑖𝑖 𝑦𝑦 𝑗𝑗 = 0 . The unit vector Ι­ 𝑖𝑖 and the associate vector Ι­ i is defined by (1.2) π‘Žπ‘Ž) Ι­ 𝑖𝑖 = 𝑦𝑦𝑖𝑖 𝐹𝐹 𝑏𝑏) Ι­ 𝑖𝑖 = g 𝑖𝑖𝑗𝑗ɭ𝑗𝑗 = οΏ½Μ‡οΏ½πœ•π‘–π‘– 𝐹𝐹 = 𝑦𝑦𝑖𝑖 𝐹𝐹 . The processes of Berwald's covariant differentiation and the partial differentiation commute according to (1.3) ( οΏ½Μ‡οΏ½πœ• π‘—π‘—β„¬β„Ž βˆ’ β„¬π‘—π‘—οΏ½Μ‡οΏ½πœ• β„Ž) 𝑇𝑇𝑗𝑗𝑖𝑖 = π‘‡π‘‡π‘—π‘—π‘Ÿπ‘ŸπΊπΊπ‘—π‘—β„Žπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’ π‘‡π‘‡π‘Ÿπ‘Ÿπ‘–π‘–πΊπΊπ‘—π‘—β„Žπ‘—π‘—π‘Ÿπ‘Ÿ . The tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies the relation (1.4) 𝐻𝐻 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 𝑦𝑦 𝑗𝑗 = 𝐻𝐻 π‘—π‘—β„Ž 𝑖𝑖 . (1.5) 𝐻𝐻 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 = οΏ½Μ‡οΏ½πœ•π‘—π‘—π»π» π‘—π‘—β„Ž 𝑖𝑖 . The torsion tensor 𝐻𝐻 π‘—π‘—β„Ž 𝑖𝑖 satisfies (1.6) 𝐻𝐻 π‘—π‘—β„Ž 𝑖𝑖 𝑦𝑦 𝑗𝑗 = 𝐻𝐻 β„Ž 𝑖𝑖 , (1.7) 𝑅𝑅 π‘—π‘—π‘—π‘—β„Ž 𝑖𝑖 𝑦𝑦 𝑗𝑗 = 𝐻𝐻 π‘—π‘—β„Ž 𝑖𝑖 , (1.8) 𝐻𝐻 𝑗𝑗𝑗𝑗 = 𝐻𝐻 𝑗𝑗𝑗𝑗𝑖𝑖 𝑖𝑖 , (1.9) 𝐻𝐻 𝑗𝑗 = 𝐻𝐻 𝑗𝑗𝑖𝑖 𝑖𝑖 , and (1.10) 𝐻𝐻 = 1 π‘›π‘›βˆ’1 𝐻𝐻 𝑖𝑖 𝑖𝑖 . where 𝐻𝐻 𝑗𝑗𝑗𝑗 and 𝐻𝐻 are called β„Ž -Ricci tensor [7] and curvature scalar respectively. Since contraction of the American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 11 indices does not affect the homogeneity in 𝑦𝑦𝑖𝑖 , hence the tensors 𝐻𝐻 π‘Ÿπ‘Ÿπ‘—π‘— , 𝐻𝐻 π‘Ÿπ‘Ÿ and the scalar 𝐻𝐻 are also homogeneous of degree zero, one and two in 𝑦𝑦𝑖𝑖 respectively . The above tensors are also connected by (1.11) 𝐻𝐻 𝑗𝑗𝑗𝑗 𝑦𝑦𝑗𝑗 = 𝐻𝐻 𝑗𝑗 , (1.12) 𝐻𝐻 𝑗𝑗𝑗𝑗 = οΏ½Μ‡οΏ½πœ•π‘—π‘— 𝐻𝐻 𝑗𝑗 , (1.13) 𝐻𝐻 𝑗𝑗 𝑦𝑦 𝑗𝑗 = (𝑛𝑛 βˆ’ 1)𝐻𝐻 . (1.14) 𝐻𝐻 π‘—π‘—β„Ž 𝑖𝑖 = οΏ½Μ‡οΏ½πœ•π‘—π‘— 𝐻𝐻 β„Ž 𝑖𝑖 . The necessary and sufficient condition for a Finsler space 𝐹𝐹𝑛𝑛(𝑛𝑛 > 2) to be a Finsler space of scalar curvature is given by (1.15) π»π»β„Žπ‘–π‘– = 𝐹𝐹2𝑅𝑅( π›Ώπ›Ώβ„Žπ‘–π‘– βˆ’ Ι­ 𝑖𝑖 Ι­ β„Ž ) . A Finsler space 𝐹𝐹𝑛𝑛 is said to be Landsberg space if satisfies (1.16) π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘–π‘–π‘—π‘—π‘—π‘—π‘Ÿπ‘Ÿ = 0 . The Ricci tensor 𝑅𝑅𝑗𝑗𝑗𝑗 is given by (1.17) 𝑅𝑅 𝑗𝑗𝑗𝑗𝑖𝑖 𝑖𝑖 = 𝑅𝑅 𝑗𝑗𝑗𝑗 . 2. Generalized πœ·πœ·π‘Ήπ‘Ή βˆ’Birecurrent Finsler Space A Finsler space whose Cartan's third curvature tensor π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies (2.1) β„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– = πœ†πœ†π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + πœ‡πœ‡π‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗) , π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β‰  0, where πœ†πœ†π‘›π‘› and πœ‡πœ‡π‘›π‘› are non-zero covariant vector fields and called the recurrence vector fields, we shall call such Finsler space as a generalized 𝑅𝑅- recurrent Finsler space. Differentiating (2.1) covariantly with respect to π‘₯π‘₯π‘šπ‘š in the sense of Berwald and using (1.1e) ,we get (2.2) β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– = ( β„¬π‘šπ‘šπœ†πœ†π‘›π‘› + πœ†πœ†π‘›π‘› πœ†πœ†π‘šπ‘š)π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + ( πœ†πœ†π‘›π‘›πœ‡πœ‡π‘šπ‘š + β„¬π‘šπ‘šπœ‡πœ‡π‘›π‘›)�𝛿𝛿𝑗𝑗𝑖𝑖 π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗� βˆ’2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 πΆπΆπ‘—π‘—β„Žπ‘šπ‘š βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š). Which can be written as (2.3) β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π›Ώπ›Ώπ‘—π‘—π‘–π‘– π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗� βˆ’ 2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 πΆπΆπ‘—π‘—β„Žπ‘šπ‘š βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š). American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 12 where π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› = β„¬π‘šπ‘šπœ†πœ†π‘›π‘› + πœ†πœ†π‘›π‘› πœ†πœ†π‘šπ‘š and π‘π‘π‘šπ‘šπ‘›π‘› = πœ†πœ†π‘›π‘›πœ‡πœ‡π‘šπ‘š + β„¬π‘šπ‘šπœ‡πœ‡π‘›π‘› are non-zero covariant tensor fields of second order . Definition 2.1. A Finsler space 𝐹𝐹𝑛𝑛 whose Cartan's third curvature tensor π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– satisfies the condition (2.3) will be called generalized 𝛽𝛽𝑅𝑅-birecurrent Finsler space ,we shall denote it 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛. Transvecting (2.3) by 𝑦𝑦𝑗𝑗 , using (1.1c) , (1.7) and (1.1d), we get (2.4) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– + π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π›Ώπ›Ώπ‘—π‘—π‘–π‘– π‘¦π‘¦β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑦𝑦𝑗𝑗�. Further transvecting (2.4) by 𝑦𝑦𝑗𝑗 , using (1.1c) , (1.6) and (1.1a) , we get (2.5) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»β„Žπ‘–π‘– = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»β„Žπ‘–π‘– + π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π‘¦π‘¦π‘–π‘–π‘¦π‘¦β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝐹𝐹2οΏ½. Thus, we have Theorem 2.1. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , Berwald second covariant derivative of the β„Ž(𝑣𝑣) βˆ’torsion tensor π»π»π‘—π‘—β„Žπ‘–π‘– and the deviation tensor π»π»β„Žπ‘–π‘– is given by the equations (2.4) and (2.5) ,respectively. Contracting the indices 𝑖𝑖 and β„Ž in (2.3), (2.4) and (2.5) , respectively and using (1.17) ,(1.9) and (1.10) , we get (2.6) β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘— = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘— + (1 βˆ’ 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— βˆ’ 2(1 βˆ’ 𝑛𝑛)π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘ŸπΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š. (2.7) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘— + (1 βˆ’ 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘—π‘— . and (2.8) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π» = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π» βˆ’ π‘π‘π‘šπ‘šπ‘›π‘›πΉπΉ2. Thus, we have Theorem 2.2. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the 𝑅𝑅 βˆ’ Ricci tensor 𝑅𝑅𝑗𝑗𝑗𝑗 , the curvature vector 𝐻𝐻𝑗𝑗 and the scalar curvature 𝐻𝐻 are non-vanishing . Differentiating (2.7) partially with respect to 𝑦𝑦𝑗𝑗 and using (1.1b), we get (2.9) οΏ½Μ‡οΏ½πœ•π‘—π‘—(β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—) = (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)𝐻𝐻𝑗𝑗 + π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π»π»π‘—π‘—οΏ½ + (1 βˆ’ 𝑛𝑛)οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π‘¦π‘¦π‘—π‘— + (1 βˆ’ 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘—. Using commutation formula exhibited by (1.3) for (ℬ𝑛𝑛𝐻𝐻𝑗𝑗) in (2.9) and using (1.12), we get (2.10) β„¬π‘šπ‘šοΏ½Μ‡οΏ½πœ•π‘—π‘—(ℬ𝑛𝑛𝐻𝐻𝑗𝑗) βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π»π‘—π‘— +π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— + (1 βˆ’ 𝑛𝑛)οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π‘¦π‘¦π‘—π‘— + (1 βˆ’ 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 13 Again applying the commutation formula exhibited by (1.3) for (𝐻𝐻𝑗𝑗), we get (2.11) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘— βˆ’ β„¬π‘šπ‘šοΏ½π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½ βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π»π‘—π‘— + π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— + (1 βˆ’ 𝑛𝑛)οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π‘¦π‘¦π‘—π‘— + (1 βˆ’ 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— . This shows that (2.12) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘— = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— + (1 βˆ’ 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘—. if and only if (2.13) βˆ’β„¬π‘šπ‘šοΏ½π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½ βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π»π‘—π‘— + (1 βˆ’ 𝑛𝑛)οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π‘¦π‘¦π‘—π‘—. Thus, we have Theorem 2.3. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the 𝐻𝐻 βˆ’Ricci tensor 𝐻𝐻𝑗𝑗𝑗𝑗 is given by equation (2.12) if and only if (2.13) holds good. Transvecting (2.11) by 𝑦𝑦𝑗𝑗 , using (1.1c) , (1.1f), (1.11) (1.13) and (1.1a), we get (2.14) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— βˆ’ (𝑛𝑛 βˆ’ 1)(β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ = (𝑛𝑛 βˆ’ 1)οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π» + π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘— + (1 βˆ’ 𝑛𝑛)οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½πΉπΉ2 +(1 βˆ’ 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘—π‘—. Using (2.7) in (2.14), we get (2.15) (β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ = βˆ’οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π» + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½πΉπΉ2. Suppose (β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ = 0 , in view of (2.15) , we get (2.16) βˆ’οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π» + οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½πΉπΉ2 = 0. Which can be written as (2.17) οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›οΏ½ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘šπ‘šοΏ½π»π» 𝐹𝐹2 . If the covariant tensor field π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is independent of 𝑦𝑦𝑖𝑖 , equation (2.17) shows that the covariant tensor field π‘π‘π‘šπ‘šπ‘›π‘› is independent of 𝑦𝑦𝑖𝑖 . conversely , if the covariant tensor π‘π‘π‘šπ‘šπ‘›π‘› is independent of 𝑦𝑦𝑖𝑖 , we get π»π»οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½ = 0. In view theorem2.2 , the condition π»π»οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½ = 0 implies οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› = 0 ,i.e. the covariant tensor field π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is also independent of 𝑦𝑦𝑖𝑖 . this leads to American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 14 Theorem 2.4. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the covariant tensor field π‘π‘π‘šπ‘šπ‘›π‘› is independent of the directional arguments if and only if the covariant tensor field π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is independent of the directional arguments provided (β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ = 0. Suppose the tensor π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is not independent of 𝑦𝑦𝑖𝑖 and in view of (2.11) , (2.12) and (2.17), we get (2.18) βˆ’β„¬π‘šπ‘šοΏ½π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½ βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›(𝐻𝐻𝑗𝑗 βˆ’ (π‘›π‘›βˆ’1) 𝐹𝐹2 𝐻𝐻𝑦𝑦𝑗𝑗). Transvecting (2.18) by π‘¦π‘¦π‘šπ‘š , using (1.1c) and (1.1f) , we get (2.19) βˆ’β„¬π‘šπ‘šοΏ½π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘šπ‘š = ( οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π‘¦π‘¦π‘šπ‘š(𝐻𝐻𝑗𝑗 βˆ’ (π‘›π‘›βˆ’1) 𝐹𝐹2 𝐻𝐻𝑦𝑦𝑗𝑗). Which implies (2.20) βˆ’β„¬π‘šπ‘šοΏ½π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘šπ‘š = (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› βˆ’ π‘Žπ‘Žπ‘—π‘—π‘›π‘›)(𝐻𝐻𝑗𝑗 βˆ’ (π‘›π‘›βˆ’1) 𝐹𝐹2 𝐻𝐻𝑦𝑦𝑗𝑗) . where π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘šπ‘š = π‘Žπ‘Žπ‘›π‘› Suppose β„¬π‘šπ‘šοΏ½π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘šπ‘š = 0 , equation (2.20) has at least one of the following conditions (2.21) a) π‘Žπ‘Žπ‘—π‘—π‘›π‘› = οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› , b) 𝐻𝐻𝑗𝑗 = (π‘›π‘›βˆ’1) 𝐹𝐹2 𝐻𝐻𝑦𝑦𝑗𝑗 . Thus, we have Theorem 2.5. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , which the covariant tensor field π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is not independent of the directional argument at least one of the conditions(2.21a) and (2.21b) hold. Suppose (2.21b) holds, then (2.18) implies (2.22) βˆ’β„¬π‘šπ‘š οΏ½(π‘›π‘›βˆ’1)𝐻𝐻 𝐹𝐹2 π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½ βˆ’ οΏ½β„¬π‘Ÿπ‘Ÿ (π‘›π‘›βˆ’1)𝐻𝐻 𝐹𝐹2 𝑦𝑦𝑗𝑗� πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ βˆ’ �ℬ𝑛𝑛 (π‘›π‘›βˆ’1)𝐻𝐻 𝐹𝐹2 π‘¦π‘¦π‘Ÿπ‘ŸοΏ½ πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = 0. Transvecting (2.22) by π‘¦π‘¦π‘šπ‘š and using (1.1c) and (1.1f), we get (2.23) β„¬π‘šπ‘š(𝐻𝐻)π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π‘¦π‘¦π‘šπ‘š + π»π»οΏ½β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š = 0. If π»π»οΏ½β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š = 0 , the equation (2.23) implies (2.24) π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ = 0 , since β„¬π‘šπ‘š(𝐻𝐻)π‘¦π‘¦π‘šπ‘š β‰  0 Therefore the space is Landsberg space Thus , we have American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 15 Theorem 2.6. An 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 is Landsberg space if condition (2.21b) holds and provided π»π»οΏ½β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š = 0 . If the covariant tensor field π‘Žπ‘Žπ‘—π‘—π‘›π‘› β‰  οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› , in view of theorem2.5 , (2.21b) holds good. In view of this fact, we may rewrite theorem2.6 in the following Theorem 2.7. An 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 is necessarily Landsberg space provided π‘Žπ‘Žπ‘—π‘—π‘›π‘› β‰  οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› and π»π»οΏ½β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š = 0 . Differentiating (2.4) partially with respect to 𝑦𝑦𝑗𝑗 , using (1.5) and (1.1b), we get (2.25) οΏ½Μ‡οΏ½πœ•π‘—π‘—οΏ½β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– οΏ½ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π»π‘—π‘—β„Žπ‘–π‘– + π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›)�𝛿𝛿𝑗𝑗𝑖𝑖 π‘¦π‘¦β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑦𝑦𝑗𝑗� +π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗) . Using commutation formula exhibited by (1.3) for (β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– ) in (2.25) , we get (2.26) β„¬π‘šπ‘šοΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– οΏ½ βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ + (β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ )πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ βˆ’(β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– + π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›)�𝛿𝛿𝑗𝑗𝑖𝑖 π‘¦π‘¦β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑦𝑦𝑗𝑗� +π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗). Again applying the commutation formula exhibited by (1.3) for (π»π»π‘—π‘—β„Žπ‘–π‘– ) in (2.26) and using (1.5), we get β„¬π‘šπ‘šοΏ½β„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’ π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½ βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ +(β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ )πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ βˆ’ (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– +π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›)�𝛿𝛿𝑗𝑗𝑖𝑖 π‘¦π‘¦β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑦𝑦𝑗𝑗� + π‘π‘π‘šπ‘šπ‘›π‘›οΏ½π›Ώπ›Ώπ‘—π‘—π‘–π‘– π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗�. Above equation can be written as (2.27) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + (β„¬π‘šπ‘šπ»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ )πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– + π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ (β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ) βˆ’ (β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– )πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– (β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ) βˆ’ οΏ½β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– οΏ½πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ βˆ’ π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– (β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ ) βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ +(β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ )πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ βˆ’ (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– +π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›)�𝛿𝛿𝑗𝑗𝑖𝑖 π‘¦π‘¦β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑦𝑦𝑗𝑗� + π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗) . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 16 This shows that (2.28) β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– = π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– + π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘”π‘”π‘—π‘—β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑔𝑔𝑗𝑗𝑗𝑗) . if and only if (2.29) (β„¬π‘šπ‘šπ»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ )πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– + π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ (β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ) βˆ’ (β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– )πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– (β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ) βˆ’οΏ½β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– οΏ½πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ βˆ’ π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– (β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ ) βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ + (β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ )πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ βˆ’ (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ = (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– + (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›)�𝛿𝛿𝑗𝑗𝑖𝑖 π‘¦π‘¦β„Ž βˆ’ π›Ώπ›Ώβ„Žπ‘–π‘– 𝑦𝑦𝑗𝑗�. Thus, we have Theorem 2.8. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– is non-vanishing if and only if (2. 29) holds good. Transvecting (2.29) by 𝑦𝑦𝑗𝑗 , using (1.1c) , (1.1f), (1.1a) and (1.6), we get (2.30) (β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– + π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ) βˆ’ (β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ) βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ + (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›οΏ½π»π»β„Žπ‘–π‘– βˆ’ (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›)οΏ½π›Ώπ›Ώβ„Žπ‘–π‘– 𝐹𝐹2 βˆ’ π‘¦π‘¦π‘–π‘–π‘¦π‘¦β„ŽοΏ½ . In view of (2.17) and (2.30), we get (2.31) (β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– + π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ) βˆ’ (β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ) βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ + (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ = οΏ½οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žβ„“π‘šπ‘šοΏ½[π»π»β„Žπ‘–π‘– βˆ’ π»π»οΏ½π›Ώπ›Ώβ„Žπ‘–π‘– βˆ’ Ι­ 𝑖𝑖ɭ β„ŽοΏ½]. If (2.32) (β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– + π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ) βˆ’ (β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ) βˆ’(β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ + (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ = 0 . We have at least one of the following conditions (2.33) a) (οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›) = 0 , b) π»π»β„Žπ‘–π‘– = 𝐻𝐻(π›Ώπ›Ώβ„Žπ‘–π‘– βˆ’ Ι­ 𝑖𝑖ɭ β„Ž). American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 19, No 1, pp 9-18 17 Putting 𝐻𝐻 = 𝐹𝐹2𝑅𝑅 ,𝑅𝑅 β‰  0 , (2.33) may be written as (2.34) π»π»β„Žπ‘–π‘– = 𝐹𝐹2𝑅𝑅(π›Ώπ›Ώβ„Žπ‘–π‘– βˆ’ Ι­ 𝑖𝑖ɭ β„Ž). Therefore the space is a Finsler space of scalar curvature Thus , we have Theorem 2.9. An 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , for ( 𝑛𝑛 > 2) admitting (β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– + π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ) βˆ’ (β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ) βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ + (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ = 0 is a Finsler space of scalar curvature provided 𝑅𝑅 β‰ 0 and the covariant tensor filed π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is not independent of the directional arguments. 3. Conclusion 1. The space whose defined by condition (2.3) is called generalized 𝛽𝛽𝑅𝑅 βˆ’ birecurrent Finsler space. 2. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , Berwald second covariant derivative of the β„Ž(𝑣𝑣) βˆ’torsion tensor π»π»π‘—π‘—β„Žπ‘–π‘– and the deviation tensor π»π»β„Žπ‘–π‘– is given by the equations (2.4) and (2.5) ,respectively. 3. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the 𝑅𝑅 βˆ’ Ricci tensor 𝑅𝑅𝑗𝑗𝑗𝑗 , the curvature vector 𝐻𝐻𝑗𝑗 and the scalar curvature 𝐻𝐻 are non-vanishing . 4. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the 𝐻𝐻 βˆ’Ricci tensor 𝐻𝐻𝑗𝑗𝑗𝑗 is given by equation (2.12) if and only if (2.13) holds good. 5. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the covariant tensor field π‘π‘π‘šπ‘šπ‘›π‘› is independent of the directional arguments if and only if the covariant tensor field π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is independent of the directional arguments provided (β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ = 0. 6. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , which the covariant tensor field π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is not independent of the directional argument at least one of the conditions (2.21a) and (2.21b) hold. 7. An 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 is Landsberg space if condition (2.21b) holds and provided π»π»οΏ½β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š = 0 8. An 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 is necessarily Landsberg space provided π‘Žπ‘Žπ‘—π‘—π‘›π‘› β‰  οΏ½Μ‡οΏ½πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› and π»π»οΏ½β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ οΏ½π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š = 0 9. In 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , the Berwald curvature tensor π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– is non-vanishing if and only if (2. 29) holds good. 10. An 𝐺𝐺𝛽𝛽𝑅𝑅 βˆ’ 𝐡𝐡𝑅𝑅 βˆ’ 𝐹𝐹𝑛𝑛 , for ( 𝑛𝑛 > 2) admitting (β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– + π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ) βˆ’ (β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’ π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ) βˆ’ (β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– )πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ + (β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’ (β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ = 0 is a Finsler space of scalar curvature provided 𝑅𝑅 β‰ 0 and the covariant tensor filed π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› is not independent of the directional arguments. 4. Recommendations The authors recommend the research should be continued in the Finsler spaces because it has many applications in Biology , relativity physics and other fields . 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