Classification of Surfaces of Finite Chen II-Type

228f2282-6823-4e7e-9b2f-b01672aeb01820250120043959340wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051120202512020252410.37394/23206.2025.24https://wseas.com/journals/mathematics/2025.phpClassification of Surfaces of Finite Chen II-TypeHamzaAlzaareerDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDANHassanAlzoubiDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDANWaseemAlmashalehDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDANIn this paper, we delve into the fascinating realm of quadric surfaces, with a specific focus on those of finite type. We first define relations regarding the first and the second Laplace operators corresponding to the second fundamental form II of a surface in the Euclidean space E 3 . We focus on quadric surfaces from two sides, on one side, we study quadric surfaces of the first kind whose Gauss map N satisfies a relation of the form ΔΙΙn = AN, where A is a square matrix of order 3 and ∆ is the second Laplace operator. On the other side, we study quadric surfaces of the second kind with the same property.12020251202025171https://wseas.com/journals/mathematics/2025/a025106-001(2025).pdf10.37394/23206.2025.24.1https://wseas.com/journals/mathematics/2025/a025106-001(2025).pdfB.-Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math, Vol. 17, 1991. 10.3390/sym15020300H. Al-Zoubi, T. Hamadneh, M. Abu Hammad, M. Al- Sabbagh, and Mehmet Ozdemir, Ruled and Quadric Surfaces Satisfying ΔIIN = ΛN, Symmetry, Vol. 15 No. 2, 2023, 1-15. https://doi.org/10.3390/sym15020300 B.-Y. Chen, F. Dillen, Quadrics, of finite type, J. Geom. Vol. 38, 1990, 16-22. J. S. Ro, D. W. Yoon. Tubes of Weingarten types in Euclidean 3-space, J. Cungcheong Math. Soc. Vol 22 (2009), 359-366. B.-Y. Chen, Surfaces of finite type in Euclidean 3-space, Bull. Soc. Math. Belg., Vol 39 (1987), 243-254. 10.14712/1213-7243.2020.018B. Senousi, H. Al-Zoubi, Translation surfaces of finite type in Sol3, Commentationes Mathematicae Universitatis Carolinae, Vol. 22, No. 2, 2020, pp 237–256. DOI: 10.14712/1213-7243.2020.018 10.1007/s00022-012-0136-0M. Bekkar and B. Senoussi, Translation surfaces in the 3-dimensional space satisfying Δ IIIri=miri, J. Geom., Vol 103 (2012), 367- 374. 10.37394/23206.2022.21.87H. Al-Zoubi, H. Alzaareer, A. Zraiqat, T. Hamadneh, and W. Al-Mashaleh, On Ruled Surfaces of Coordinate Finite Type, WSEAS Transactions on Mathematics, Vol. 21, 2022, 765-769. DOI: 10.37394/23206.2022.21.87 10.1017/s0004972700028616B.-Y. Chen, F. Dillen, L. Verstraelen, L. Vrancken, Ruled surfaces of finite type, Bull. Austral. Math. Soc. 42, 447-553 (1990). F. Dillen, J. Pass, L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica Vol. 18, 1990, 239- 246. 10.2996/kmj/1138038815O. Garay, On a certain class of finite type surfaces of revolution, Kodai Math. J. Vol. 11, 1988, 25-31. 10.4134/bkms.2009.46.6.1141Y. H. Kim, C. W. Lee, and D. W. Yoon, On the Gauss map of surfaces of revolution without parabolic points, Bull. Korean Math. Soc. Vol 46 (2009), 1141–1149. 10.1007/bf03322254Ch. Baikoussis, L. Verstraelen, The Chentype of the spiral surfaces, Results. Math. Vol. 28, 1995, 214-223. 10.1017/s001708950003055xF. Denever, R. Deszcz, L. Verstraelen, The Chen type of the noncompact cyclides of Dupin, Glasg. Math. J. 36, 1994, 71-75. 10.1007/bf01230997F. Denever, R. Deszcz, L. Verstraelen, The compact cyclides of Dupin and a conjecture by B.-Y Chen, J. Geom. Vol. 46, 1993, 33-38. Ch. Baikoussis, L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Semi. Mat. Messina Ser II Vol. 16, 1993, 31-42. B. Senoussi and M. Bekkar, Helicoidal surfaces with △ J r = Ar in 3-dimensional Euclidean space, Stud. Univ. Babes-Bolyai Math. 60 2015, pp 437–448. 10.1007/bf03322734S. Stamatakis, H. Al-Zoubi, On surfaces of finite Chen type. Result. Math. Vol. 43, 2003, 181-190. B.-Y. Chen, Total mean curvature and submanifolds of finite type, world Scientific, Singapore, 1984. M. Mhailan, M. Abu Hammad, M. Al Horani, R. Khalil, On fractional vector analysis, J. Math. Comput. Sci. \Vol 10 (2020), 2320- 2326. https://doi.org/10.28919/jmcs/4863 I. Batiha, J. Oudetallah, A. Ouannas, A. AlNana, I. Jebril. Tuningthe fractional-order PID-controller for blood Glucose level of diabetic patients, Int. J. Adv. Soft Comp. App. 13 No. 2 1-10 (2021). 10.15849/ijasca.220720.07I. Batiha, S. Njadat R. Batyha. A. Zraiqat, A. Dabbabneh, Sh. Momani. Design fractionalorder PID controller for single-joint robot arm model. Int. J. Adv. Soft Comp. App. 14 No.2 96-114 (2022).