Surfaces of coordinate finite $ II $-type

Mutaz Al-Sabbagh; Mutaz Al-Sabbagh

doi:10.3934/math.2025285

AIMS Mathematics

2025, Volume 10, Issue 3: 6258-6269. doi: 10.3934/math.2025285

Previous Article Next Article

Research article

Surfaces of coordinate finite $ II $-type

Mutaz Al-Sabbagh ^,

Department of Basic Engineering Sciences, Imam Abdulrahman bin Faisal University, Dammam 31441, Saudi Arabia

Received: 23 January 2025 Revised: 05 March 2025 Accepted: 12 March 2025 Published: 20 March 2025
MSC : 53A05, 53A45

We study the class of surfaces of revolution in the 3-dimensional Euclidean space $ E^{3} $ with nonvanishing Gauss curvature whose position vector $ \boldsymbol{x} $ satisfies the condition $ \Delta^{II}\boldsymbol{x} = A\boldsymbol{x} $, where $ A $ is a square matrix of order 3 and $ \Delta^{II} $ denotes the Laplace operator of the second fundamental form $ II $ of the surface. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.
- surfaces in $ E^{3} $,
- surfaces of revolution,
- surfaces of coordinate finite type,
- Beltrami operator
Citation: Mutaz Al-Sabbagh. Surfaces of coordinate finite $ II $-type[J]. AIMS Mathematics, 2025, 10(3): 6258-6269. doi: 10.3934/math.2025285

Related Papers:

Abstract

We study the class of surfaces of revolution in the 3-dimensional Euclidean space $ E^{3} $ with nonvanishing Gauss curvature whose position vector $ \boldsymbol{x} $ satisfies the condition $ \Delta^{II}\boldsymbol{x} = A\boldsymbol{x} $, where $ A $ is a square matrix of order 3 and $ \Delta^{II} $ denotes the Laplace operator of the second fundamental form $ II $ of the surface. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.

References

[1]	B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22 (1996), 117–337.
[2]	S. I. Abdelsalam, A. Magesh, P. Tamizharasi, Optimizing fluid dynamics: An in-depth study for nano-biomedical applications with a heat source, J. Therm. Anal. Calorim., 2024. https://doi.org/10.1007/s10973-024-13472-2
[3]	B. Y. Chen, Total mean curvature and submanifolds of finite type, 2 Eds., World Scientific Publisher, 2014. https://doi.org/10.1142/9237
[4]	T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385. https://doi.org/10.2969/jmsj/01840380 doi: 10.2969/jmsj/01840380
[5]	O. J. Garay, An extension of Takahashi's theorem, Geom. Dedicata., 34 (1990), 105–112. https://doi.org/10.1007/BF00147319 doi: 10.1007/BF00147319
[6]	F. Dilen, J. Pas, L. Verstraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J., 13 (1990), 10–21. https://doi.org/10.2996/kmj/1138039155 doi: 10.2996/kmj/1138039155
[7]	B. Y. Chen, P. Piccini, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc., 35 (1987), 161–186.
[8]	H. Al-Zoubi, H. Alzaareer, T. Hamadneh, M. Al Rawajbeh, Tubes of coordinate finite type Gauss map in the Euclidean 3-space, Indian J. Math., 62 (2020), 171–182.
[9]	H. Alzaareer, H. Al-Zoubi, F. Abdel-Fattah, Quadrics with finite Chen-type Gauss map, J. Prime Res. Math., 18 (2022), 96–107.
[10]	C. Baikoussis, L. Verstraelen, The Chen-type of the spiral surfaces, Results Math., 28 (1995), 214–223. https://doi.org/10.1007/BF03322254 doi: 10.1007/BF03322254
[11]	C. Baikoussis, L. Verstraelen, On the Gauss map of translation surfaces, Rend. Semi. Mat. Messina Ser II, in press.
[12]	C. Baikoussis, L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II, 16 (1993), 31–42.
[13]	F. Dillen, J. Pass, L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sin., 18 (1990), 239–246.
[14]	H. Al-Zoubi, M. Al-Sabbagh, Anchor rings of finite type Gauss map in the Euclidean 3-space, Int. J. Math. Comput. Methods, 5 (2020), 9–13.
[15]	H. Al-Zoubi, F. Abdel-Fattah, M. Al-Sabbagh, Surfaces of finite III-type in the Euclidean 3-space, WSEAS Trans. Math., 20 (2021), 729–735. https://doi.org/10.37394/23206.2021.20.77 doi: 10.37394/23206.2021.20.77
[16]	S. Stamatakis, H. Al-Zoubi, On surfaces of finite Chen-type, Results Math., 43 (2003), 181–190. https://doi.org/10.1007/BF03322734 doi: 10.1007/BF03322734
[17]	H. Al-Zoubi, H. Alzaareer, A. Zraiqat, T. Hamadneh, W. Al Mashaleh, On ruled surfaces of coordinate finite type, WSEAS Trans. Math., 21 (2022), 765–769. https://doi.org/10.37394/23206.2022.21.87 doi: 10.37394/23206.2022.21.87
[18]	H. Al-Zoubi, H. Alzaareer, M. Al-Rawajbeh, M. Al-Kafaween, Characterization of tubular surfaces in terms of finite $III$-type, WSEAS Trans. Syst. Control, 19 (2024), 22–26. https://doi.org/ 10.37394/23203.2024.19.3 doi: 10.37394/23203.2024.19.3
[19]	H. Al-Zoubi, T. Hamadneh, M. A. Hammad, M. Al-Sabbagh, M. Ozdemir, Ruled and quadric surfaces satisfying $\triangle ^II\mathbf{N} = \Lambda\mathbf{N}$, Symmetry, 15 (2023), 300. https://doi.org/10.3390/sym15020300 doi: 10.3390/sym15020300
[20]	J. Arroyo, O. J. Garay, J. J. Mencía, On a family of surfaces of revolution of finite Chen-type, Kodai Math. J., 21 (1998) 73–80.
[21]	H. Al-Zoubi, T. Hamadneh, H. Alzaareer, M. Al-Sabbagh, Tubes in the Euclidean 3-space with coordinate finite type Gauss map, In: 2021 International conference on information technology (ICIT), Jordan: IEEE, 2021. https://doi.org/10.1109/ICIT52682.2021.9491118
[22]	H. Al-Zoubi, A. Dababneh, M. Al-Sabbagh, Ruled surfaces of finite $II$-type, WSEAS Trans. Math., 18 (2019), 1–5.
[23]	S. Stamatakis, H. Al-Zoubi, Surfaces of revolution satisfying $\triangle^III\mathbf{x} = A\mathbf{x}$, arXiv: 1510.08479, 2010. https://doi.org/10.48550/arXiv.1510.08479
[24]	H. Al-Zoubi, T. Hamadneh, A. Alkhatib, Quadric surfaces of coordinate finite type Gauss map in the Euclidean 3-space, Indian J. Math., 64 (2022), 385–399.
[25]	H. Al-Zoubi, H. Alzaareer, Non-degenerate translation surfaces of finite $III$-type, Indian J. Math., 65 (2023), 395–407.
[26]	H. Huck, R. Roitzsch, U. Simon, W. Vortisch, R. Walden, B. Wegner, et al., Beweismethoden der Differentialgeometrie im Grossen, In: Lecture notes in mathematics, Heidelberg: Springer Berlin, 335 (1973). https://doi.org/10.1007/BFb0061770
[27]	H. Al-Zoubi, W. Al Mashaleh, Surfaces of finite type with respect to the third fundamental form, In: 2019 IEEE Jordan international joint conference on electrical engineering and information technology (JEEIT), Jordan: IEEE, 2019,174–178. https://doi.org/10.1109/JEEIT.2019.8717507
[28]	Y. H. Kim, C. W. Lee, D. W. Yoon, On the Gauss map of surfaces of revolution without parabolic points, Bull. Korean Math. Soc., 46 (2009), 1141–1149. https://doi.org/10.4134/BKMS.2009.46.6.1141 doi: 10.4134/BKMS.2009.46.6.1141

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)