Research ArticleNo Access

Explicit formulas for Killing magnetic curves in three-dimensional Heisenberg group

Khadidja Derkaoui

Department of Mathematics, University of Chlef, Algeria

E-mail Address: derkaouikhdidja248@hotmail.com

E-mail Address: k.derkaoui@univ-chlef.dz

Corresponding author.

Search for more papers by this author

and

Fouzi Hathout

Department of Mathematics, University of Saïda, Algeria

E-mail Address: f.hathout@gmail.com

E-mail Address: fouzi.hathout@univ-saida.dz

Search for more papers by this author

https://doi.org/10.1142/S0219887821501358Cited by:2 (Source: Crossref)

Abstract

In this paper, we present the geometry of three-dimensional Heisenberg group $(ℍ_{3}, g)$ and its geodesics curves. After studying the Killing magnetic curves and some geodesic Killing magnetic curves, we give the explicit formulas for such curves.

Keywords:

AMSC: 53A04, 53C25

References

1. C. Bejan and S. L. Drută-Romaniuc , Walker manifolds and Killing magnetic curves, Differ. Geom. Appl. 35 (2014) 106–116, https://doi.org/10.1016/j.difgeo.2014.03.001. Crossref, Web of Science, Google Scholar
2. M. Barros and A. Romero , Magnetic vortices, Europhys. Lett. 77 (2007) 34002, https://doi.org/10.1209/0295-5075/77/34002EPL. Crossref, Google Scholar
3. W. Batat and A. Zaeim, On symmetries of the Heisenberg group, preprint (2017), arXiv:1710.04539v1. Google Scholar
4. G. Calvaruso, M. I. Munteanu and A. Perrone , Killing magnetic curves in three-dimensional almost paracontact manifolds, J. Math. Anal. Appl. 426(1) (2015) 423–439, https://doi.org/10.1016/j.jmaa.2015.01.057. Crossref, Web of Science, Google Scholar
5. L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson , An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics (Birkhäuser Verlag, Basel, 2007). Google Scholar
6. S. L. Drută-Romaniuc, J. Inoguchi, M. I. Munteanu and A. I. Nistor , Magnetic curves in cosymplectic manifolds, Rep. Math. Phys. 78 (2016) 33. Crossref, Web of Science, Google Scholar
7. S. L. Drută-Romaniuc, J. Inoguchi, M. I. Munteanu and A. I. Nistor , Magnetic curves in Sasakian manifolds, J. Nonlinear Math. Phys. 22 (2015) 428. Crossref, Web of Science, Google Scholar
8. S. L. Drută-Romaniuc and M. I. Munteanu , Killing magnetic curves in a Minkowski 3-space, Nonlinear Anal. Real World Appl. 14 (2013) 383. Crossref, Web of Science, Google Scholar
9. S. L. Drută-Romaniuc and M. I. Munteanu , Magnetic curves corresponding to killing magnetic fields in $𝔼^{3}$ , J. Math. Phys. 52 (2011) 113506. Crossref, Web of Science, Google Scholar
10. Z. Erjavec and J. Inoguchi , On magnetic curves in almost cosymplectic sol space, Results Math. 75 (2020) 113, https://doi.org/10.1007/s00025-020-01235-y. Crossref, Web of Science, Google Scholar
11. Z. Erjavec , On killing magnetic curves in $S L (2, ℝ)$ geometry, Rep. Math. Phys. 84(3) (2019) 333–350. Crossref, Web of Science, Google Scholar
12. Z. Erjavec and J. Inoguchi , Killing magnetic curves in sol space, Math. Phys. Anal. Geom. 21 (2018) 15, https://doi.org/10.1007/s11040-018-9272-6. Crossref, Web of Science, Google Scholar
13. C. Figueroa , The Gauss map of minimal graphs in the Hzisenberg group, J. Geom. Symmetry Phys. 25 (2012) 1–21. Google Scholar
14. T. Hangan , Au sujet des flots riemanniens sur le groupe nilpotent de Heisenberg, Rend. Circ. Mat. Palermo 35 (1986) 291–305, https://doi.org/10.1007/BF02844738. Crossref, Google Scholar
15. Z. Iqbal, J. Sengupta and S. Chakraborty , Magnetic trajectories corresponding to killing magnetic fields in a three-dimensional warped product 2020, Int. J. Geom. Methods Mod. Phys. 17(14) (2020) 2050212, https://doi.org/10.1142/S0219887820502126. Link, Web of Science, Google Scholar
16. J. Milnor , Curvature of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293–329. Crossref, Web of Science, Google Scholar
17. M. I. Munteanua and A. Nistor , The classification of killing magnetic curves in $𝕊^{2} \times ℝ$ , J. Geom. Phys. 62 (2012) 170–182, https://doi.org/10.1016/j.geomphys.2011.10.002. Crossref, Web of Science, Google Scholar
18. C. Özgür , On magnetic curves in $3$ -dimensional Heisenberg group, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerbaijan 43(2) (2017) 278–286. Web of Science, Google Scholar