NONE 20260604121207403 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Interrelations Among Concircular, Conformal and Conharmonic Curvatures in Fifth-Order Recurrent Finsler Spaces Adel M. Al-Qashbari Department of Mathematics, Education Faculty, University of Aden, YEMEN Alaa A. Abdallah Department of Mathematics, Education Faculty, Abyan University, Zinjibar, Abyan, YEMEN Saeedah M. Baleedi Department of Mathematics, Education Faculty, Abyan University, Zinjibar, Abyan, YEMEN The significance of this research lies in the generalization of curvature tensors, particularly the concircular curvature tensor, which is studied as a fifth recurrent tensor within the framework of Cartan’s fourth curvature tensor. It contributes to the fundamental understanding of geometric properties in Finsler spaces. This paper builds upon new the concircular curvature tensor π‘€π‘–π‘—π‘˜β„Ž in generaralized fifth recurrent Finsler space that Cartan's fourth curvature tensor πΎπ‘—π‘˜β„Ž 𝑖 in sense of Berwald (𝐺ℬ𝐾 βˆ’ 5𝑅𝐹𝑛) via Lie derivative. We obtain the relation between the concircular curvature tensor π‘€π‘–π‘—π‘˜β„Ž , conformal curvature tensor πΆπ‘–π‘—π‘˜β„Ž , conharmonic curvature tensor πΏπ‘–π‘˜β„Ž π‘Ÿ and associate curvature tensor π‘…π‘–π‘—π‘˜β„Ž if the metric tensor π‘”π‘Ÿπ‘— = 1 by Lie - derivative. Also, we show these curvature tensors have the same extension and direction. In addition, the concircular curvature tensor and conharmonic curvature tensor πΏπ‘—π‘˜β„Ž 𝑖 are co-directional. We prove that the concircular curvature tensor π‘€π‘–π‘—π‘˜β„Ž behaves as fifth recurrent by Lie derivative in the main space. 12 31 2025 12 31 2025 802 80 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2025.24.80 https://wseas.com/journals/mathematics/2025/b625106-040(2025).pdf M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Otsu, Japan, 1994. Z. Ahsan and M. Ali, On some properties of π‘Š βˆ’curvature tensor, Palestine Journal of Mathematics, Vol.3, No.1, 2014, 61-69. M. Ali, M. Salman and M. Bilal, Conharmonic curvature inheritance in spacetime of general relativity, Universe, Vol.7, No.12, 2021, 1-21. T. J. Willmore, The definition of Lie derivative, Proceedings of the Edinburgh Mathematical Society, Vol.12, No.1, 1960, 27-29. K. Yano, The theory of Lie derivatives and its applications, North-Holland Publishing Co., Groningen, 1957. M. A. Opondo, Study of projective curvature tensor π‘Šπ‘—π‘˜β„Ž 𝑖 in bi-recurrent Finsler space, Ph.D. Dissertation, Kenyatta University, (Nairobi), (Kenya), 2021. H. Gouin, Remarks on the Lie derivative in fluid mechanics, International Journal of Non - Linear Mechanics, Vol.150, 2023, 1-16. E. Pak and G. Kim, Reeb Lie derivatives on real hyper surfaces in complex hyperbolic twoplane Grass mannians, Faculty of Sciences and Mathematics, University of Nis, (Serbia), Filomat, Vol.37, No.3, 2023, 915-924. A. M. AL-Qashbari, A. A. Abdallah and S. M. Baleedi, Berwald covariant derivative and Lie derivative of conharmonic curvature tensors in generalized fifth recurrent Finsler space, GPHInternational Journal of Mathematics, Vol.8, No.1, 2025, 24-32. Q. Xia, Geometry and analysis on Finsler spaces, World Scientific, 2025. H. Rund, The differential geometry of Finsler spaces, Springer-Verlag, Berlin GΓΆttingen, 1959; 2nd Edit. (in Russian), Nauka, (Moscow), 1981. D. C. Agarwal, Tensor Calculus and Riemannian Geometry, Krishna Prakashan Media, 2013. R. Sidaoui, A. H. A. Alfedeel, A. A. Abdallah, K. Aldwoah, M. H. Alqahtani, A. H. Tedjani and B. Muflh, Geometric analysis of Lie and Berwald derivatives of inheritance tensors in Finsler spaces, Mathematics, Vol.13, No.3900, 2025. S. Saxena and P. N. Pandey, On Lie-recurrence in Finsler spaces, Differential GeometryDynamics Systems, Vol.13, 2011, 201-207. S. I. Ohta, Comparison Finsler geometry, Springer International Publishing, license to Springer Nature Switzerland, 2021.