III-harmonic Curves in 


SL2ℝ˜
\widetilde {{\rm{S}}{{\rm{L}}_2}\mathbb{R}}

 Space

Bendehiba Senoussi

doi:10.2478/awutm-2024-0010

$III-harmonic Curves in SL2ℝ˜ \widetilde {{\rm{S}}{{\rm{L}}_2}\mathbb{R}} Space Cover$

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III-harmonic Curves in SL2ℝ˜ \widetilde {{\rm{S}}{{\rm{L}}_2}\mathbb{R}} Space

Annals of West University of Timisoara - Mathematics and Computer Science

Volume 60 (2024): Issue 1 (January 2024)

By: Bendehiba Senoussi

Open Access

|Dec 2024

Abstract

Some work has been done in the study of non-geodesic III-harmonic curves in some model spaces. In this paper, we study III-harmonic curves in $\tilde{{SL}_{2} ℝ}$ \widetilde {{\rm{S}}{{\rm{L}}_2}\mathbb{R}} space. We give necessary and su cient conditions for helices to be III-harmonic. Also, we characterize III-harmonic curves in terms of their curvature and torsion.

References

A. T. Ali, Position vectors of general helices in Euclidean 3-space, Bull. Math. Anal. Appl. 3 (2011), 1-8.
Search in Google Scholar Back to article
M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125 (1997), 1503-1509.
Search in Google Scholar Back to article
M. Barros, A. Ferrández, P. Lucas, M. A. Meroño, General helices in the 3-dimensional Lorentzian space forms, Rocky Mountain J. Math. 31 (2001), 373-388.
Search in Google Scholar Back to article
V. Branding, A structure theorem for polyharmonic maps between Riemannian manifolds, arXiv preprint arXiv:1901.08445, 2019.
Search in Google Scholar Back to article
R. Caddeo, C. Oniciuc, P. Piu, Explicit formulas for non-geodesic biharmonic curves of the Heisenberg group, Rend. Sem. Mat. Univ. Pol. Torino 62 (3) (2004), 265-278.
Search in Google Scholar Back to article
R. Caddeo, S. Montaldo, C. Oniciuc, P. Piu, The classification of biharmonic curves of Cartan-Vranceanu 3-dimensional spaces, Modern trends in geometry and topology, Cluj Univ. Press, Cluj-Napoca (2006), 121-131.
Search in Google Scholar Back to article
[7] B. Divjak, Z. Erjavec, B. Szabolcs, B. Szilágyi, Geodesics and geodesic spheres in SL(2,ℝ)˜ \widetilde {SL\left( {2,\mathbb{R}} \right)} geometry, Math. Commun. 14 (2009), 413-424.
Back to article
J. Eells, L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics 50, 1983.
Search in Google Scholar Back to article
[9] Z. Erjavec, D. Horvat, Biharmonic curves in SL(2,ℝ)˜ \widetilde {SL\left( {2,\mathbb{R}} \right)} space, Math. Commun. 19 (2014), 291-299.
Back to article
[10] Z. Erjavec, Minimal surfaces in SL(2,ℝ)˜ \widetilde {SL\left( {2,\mathbb{R}} \right)} geometry, Glas. Mat. 50 (2015), 207-221.
Back to article
D. Fetcu, Biharmonic curves in Cartan-Vranceanu (2n+1)-dimensional spaces, Beitr. Algebra Geom. 46 (2) (2007), 513-521.
Search in Google Scholar Back to article
S. Maeta, k-Harmonic maps into a Riemannian manifold with constant sectional curvature, Proc. Amer. Math. Soc. 140 (2012), 1835-1847.
Search in Google Scholar Back to article
S. Maeta, The second variational formula of the k-energy and k-harmonic curves, Osaka J. Math. 49 (2012), 1035-1063.
Search in Google Scholar Back to article
E. Molnár, The projective interpretation of the eight 3-dimensional homogeneous geometries, Beitr. Algebra Geom. 38 (1997), 261-288.
Search in Google Scholar Back to article
E. Molnár, J. Szirmai, Symmetries in the 8 homogeneous 3− geometries, Symmetry Cult. Sci. 21 (2010), 87-117.
Search in Google Scholar Back to article
E. Molnár, J. Szirmai, A. Vesnin, The optimal packings by translation balls in SL(2, ℝ), J. Geom. 105 (2014), 287-306.
Search in Google Scholar Back to article
S. Montaldo, A. Pampano, Triharmonic curves in 3-dimensional homogeneous spaces, Mediterr. J. Math. 18 (2021), Article number 198.
Search in Google Scholar Back to article
S. Montaldo, C. Oniciuc, A. Ratto, Polyharmonic surfaces in 3-dimensional homogeneous spaces, Manuscripta Math. 174 (2024), 31-58.
Search in Google Scholar Back to article
S. Montaldo A. Ratto, New examples of r- harmonic immersions into the sphere, J. Math. Anal. Appl. 458 (1) (2018), 849-859.
Search in Google Scholar Back to article
J. Monterde, Curves with constant curvature ratios, Bol. Soc. Mat. Mexicana 13 (2007), 177-186.
Search in Google Scholar Back to article
B. Senoussi, Triharmonic curves in Heisenberg group H₃, J. Dyn. Syst. Geom. Theor. 20 (2022), 55-65.
Search in Google Scholar Back to article
B. Senoussi, Triharmonic curves in SOL₃space, Palestine J. Math. 11, (2022), 264-273.
Search in Google Scholar Back to article
B. Senoussi, Triharmonic curves in Bianchi-Cartan-Vranceanu spaces, Rendiconti Sem. Mat. Univ. Pol. Torino 80 (2) (2022), 5-13.
Search in Google Scholar Back to article
B. Senoussi, M. Bekkar, Characterization of general helix in the 3- dimensional Lorentz-Heisenberg space, Int. Electron. J. Geom. 6 (1) (2013), 46-55.
Search in Google Scholar Back to article
[25] B. Senoussi, M. Bekkar, Some characterization of curves in SL2ℝ˜ \widetilde {{\rm{S}}{{\rm{L}}_2}\mathbb{R}} space, Sib. Elektron. Mat. Izv. 16 (2019), 902-915.
Back to article
D. J. Struik, Lectures on classical di erential geometry, New York Dover Publications. Inc., 1988.
Search in Google Scholar Back to article
S. B. Wang, The first variation formula for k-harmonic mapping, J. Nanchang Univ. 13 (1989).
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/awutm-2024-0010 | Journal eISSN: 1841-3307 | Journal ISSN: 1841-3293

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Language: English

Submitted on: Jan 4, 2024

Accepted on: Dec 9, 2024

Published on: Dec 24, 2024

Published by: West University of Timisoara

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

SL2ℝ˜

\widetilde {{\rm{S}}{{\rm{L}}_2}\mathbb{R}}

space,

Related subjects:

© 2024 Bendehiba Senoussi, published by West University of Timisoara
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 60 (2024): Issue 1 (January 2024)