The Chen type of Hasimoto surfaces in the Euclidean 3-space

Hassan Al-Zoubi; Bendehiba Senoussi; Mutaz Al-Sabbagh; Mehmet Ozdemir; Hassan Al-Zoubi; Bendehiba Senoussi; Mutaz Al-Sabbagh; Mehmet Ozdemir

doi:10.3934/math.2023819

AIMS Mathematics

2023, Volume 8, Issue 7: 16062-16072. doi: 10.3934/math.2023819

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The Chen type of Hasimoto surfaces in the Euclidean 3-space

1.
Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman, Jordan 11733
2.
Department of Mathematics, Ecole Normale Superieure, 45 rue d'Ulm, 75230, Mostaganem, Algeria
3.
Department of Basic Engineering Sciences, Imam Abdulrahman bin Faisal University, Dammam 31441, Saudi Arabia

Received: 25 February 2023 Revised: 23 April 2023 Accepted: 24 April 2023 Published: 05 May 2023
MSC : 14J25, 53Z05

A surface $ \mathcal{M}^{2} $ with position vector $ r = r(s, t) $ is called a Hasimoto surface if the relation $ r_{t} = r_{s} \wedge r_{ss} $ holds. In this paper, we first define the Beltrami-Laplace operator according to the three fundamental forms of the surface, then we classify the $ J $-harmonic Hasimoto surfaces and their Gauss map in $ \mathbb{E}^{3} $, for $ J = II $ and $ III $.
- Hasimoto surface,
- Gauss map,
- Euclidean space,
- Beltrami-Laplace operator,
- surfaces of finite type
Citation: Hassan Al-Zoubi, Bendehiba Senoussi, Mutaz Al-Sabbagh, Mehmet Ozdemir. The Chen type of Hasimoto surfaces in the Euclidean 3-space[J]. AIMS Mathematics, 2023, 8(7): 16062-16072. doi: 10.3934/math.2023819

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Abstract

A surface $ \mathcal{M}^{2} $ with position vector $ r = r(s, t) $ is called a Hasimoto surface if the relation $ r_{t} = r_{s} \wedge r_{ss} $ holds. In this paper, we first define the Beltrami-Laplace operator according to the three fundamental forms of the surface, then we classify the $ J $-harmonic Hasimoto surfaces and their Gauss map in $ \mathbb{E}^{3} $, for $ J = II $ and $ III $.

References

[1]	A. Kelleci, M. Bektaş, M. Ergüt, The Hasimoto surface according to bishop frame, Adıyaman University Journal of Science, 9 (2019), 13–22.
[2]	N. Abdel-All, R. Hussien, T. Youssef, Hasimoto surfaces, Life Sci. J., 9 (2012) 556–560.
[3]	W. Schief, C. Rogers, Binormal motion of curves of constant curvature and torsion. generation of soliton surfaces, Proc. R. Soc. Lond., 455 (1999), 3163–3188. http://dx.doi.org/10.1098/rspa.1999.0445 doi: 10.1098/rspa.1999.0445
[4]	C. Rogers, W. Schief, Intrinsic geometry of the NLS equation and its backlund transformation, Stud. Appl. Math., 101 (1998), 267–287. http://dx.doi.org/10.1111/1467-9590.00093 doi: 10.1111/1467-9590.00093
[5]	M. Erdogdu, M. Özdemir, Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom., 17 (2014), 169–181. http://dx.doi.org/10.1007/s11040-014-9148-3 doi: 10.1007/s11040-014-9148-3
[6]	M. Elzawy, Hasimoto surfaces in Galilean space $G_{3}$, J. Egypt. Math. Soc., 29 (2021), 5. http://dx.doi.org/10.1186/s42787-021-00113-y doi: 10.1186/s42787-021-00113-y
[7]	M. Mhailan, M. Abu Hammad, M. Al Horani, R. Khalil, On fractional vector analysis, Journal of Mathematical and Computational Science, 10 (2020), 2320–2326.
[8]	Y. Li, Z. Chen, S. Nazra, R. Abdel-Baky, Singularities for timelike developable surfaces in Minkowski 3-space, Symmetry, 15 (2023), 277. http://dx.doi.org/10.3390/sym15020277 doi: 10.3390/sym15020277
[9]	Y. Li, M. Aldossary, R. Abdel-Baky, Spacelike circular surfaces in Minkowski 3-space, Symmetry, 15 (2023), 173. http://dx.doi.org/10.3390/sym15010173 doi: 10.3390/sym15010173
[10]	Y. Li, O. Tuncer, On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space, Math. Method. Appl. Sci., in press. http://dx.doi.org/10.1002/mma.9173
[11]	Y. Li, A. Alkhaldi, A. Ali, R. Abdel-Baky, M. Khalifa Saad, Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean 3-space, AIMS Mathematics, 8 (2023), 13875–13888. http://dx.doi.org/10.3934/math.2023709 doi: 10.3934/math.2023709
[12]	M. Erdo$\widetilde {\rm{g}}$du, A. Yavuz, Differential geometric aspects of nonlinear Schrödinger equation, Commun. Fac. Sci. Univ., 70 (2021), 510–521. http://dx.doi.org/10.31801/cfsuasmas.724634 doi: 10.31801/cfsuasmas.724634
[13]	B. Senoussi, M. Bekkar, Helicoidal surfaces with $\Delta^{J}r = Ar$ in $3$-dimensional Euclidean space, Stud. Univ. Babes-Bol. Math., 60 (2015), 437–448.
[14]	S. Stamatakis, H. Al-Zoubi, On surfaces of finite Chen-type, Results Math., 43 (2003), 181–190. http://dx.doi.org/10.1007/BF03322734 doi: 10.1007/BF03322734
[15]	S. Stamatakis, H. Al-Zoubi, Surfaces of revolution satisfying $\Delta^IIIx = Ax$, J. Geom. Graph., 14 (2010), 181–186.
[16]	B. Senoussi, A. Akbay, Characterizations of Hasimoto surfaces in Euclidean 3-spaces $\mathbb{E}^{3}$, Appl. Math. Inf. Sci., 16 (2022), 689–694. http://dx.doi.org/10.18576/amis/160504 doi: 10.18576/amis/160504

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