Variational equations and Killing magnetic trajectories on timelike surfaces in semi-Riemannian manifolds

Yıl 2022, Cilt: 51 Sayı: 4, 1058 - 1071, 01.08.2022

Kübra Şahin Zehra Ozdemir

https://doi.org/10.15672/hujms.967923

Öz

In this article, Darboux frame variations for timelike surfaces in semi-Riemannian manifolds are discussed. In addition, the Killing equations are examined by using the Darboux frame curvature variations. Then, magnetic trajectories are generated by means of the variational vector fields. Furthermore, parametric representations of all magnetic trajectories on the de Sitter space $\mathbb{S}_{1}^{2}$ are presented. Moreover, various examples of magnetic trajectories are given in order to illustrate the theoretical results.

Anahtar Kelimeler

vector fields, magnetic flows, ordinary differential equations, special curves, motion of a rigid body with a fixed point, variational methods

Kaynakça

• [1] M. Barros, J.L. Cabrerizo, M. Fernández and A. Romero, Magnetic vortex flament flows, J. Math. Phys. 48, 1-27, 2007.
• [2] M. Barros, A. Ferrandez, P. Lucas and M.A. Merono, General helices in the 3- dimensional Lorentzian space forms, Rocky. Mt. J. Math. 32, 373-388, 2001.
• [3] M. Barros, A. Romero, J.L. Cabrerizo and M. Fernández, The Gauss Landau Hall problem on Riemannian surfaces. J. Math. Phys.46, 112905, 2005.
• [4] Z. Bozkurt, I. Gök, Y. Yaylı and F.N. Ekmekci, A new approach for magnetic curves in 3D Riemannian manifolds, J. Math. Phys. 55, 053501, 2014.
• [5] J.L. Cabrerizo, Magnetic fields in 2D and 3D sphere, J. Nonlinear. Math. Phys. 20, 440-450, 2013.
• [6] H.S.M. Coxeter, A geometrical background for de Sitter’s world, Am. Math. Mon. (Math. Assoc. Am.) 50 (4), 217-228 (JSTOR 2303924), 1943.
• [7] W. de Sitter, On the relativity of inertia remarks concerning Einstein’s latesthypothesis, Proc. Kon. Ned. Acad. Wet. 19, 1217-1225, 1917.
• [8] S.L. Druta-Romaniuc and M.I. Munteanu, Magnetic Curves corresponding to Killing magnetic fields in E3, J. Math. Phys. 52, 113506, 2011.
• [9] S.L. Druta-Romaniuc and M.I. Munteanu, Killing magnetic curves in a Minkowski 3-space, Nonlinear. Anal. Real. World. Appl. 14, 383-396, 2013.
• [10] N. Gurbuz, p-Elastica in the 3-Dimensional Lorentzian Space Forms, Turkish J. Math. 30, 33-41, 2006.
• [11] R. Lopez, Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. Int. Electronic. J. Geom. 7 (1), 44-107, 2014.
• [12] B. O’Neill, Semi-Riemannian geometry with applications to relativity, Academic press, New York, 1983.
• [13] Z. Özdemir, Pseudo Null Curve Variations for Bishop Frame in 3D semi-Riemannian Manifold, Int. J. Geom. Methods and Modern Phys. 16 (3), 1950043, 2019.
• [14] Z. Özdemir, I. Gok, Y. Yaylı and F.N. Ekmekci, Notes on Magnetic Curves in 3D semi-Riemannian Manifolds. Turkish J. Math. 39, 412-426, 2015.
• [15] Z. Özdemir, Null Cartan curve variations in 3D semi-Riemannian manifold, Hacettepe J. Math. Stat. 50 (2), 351-360, 2021.
• [16] T. Sunada, Magnetic flows on a Riemann surface, In Proceedings of the KAIST Mathematics Workshop:Analysis and Geometry, Taejeon, Korea, 3-6, 1993.
• [17] H.H. Ugurlu, H. Kocayigit, The Frenet and Darboux Instantaneous Rotain Vectors of Curves on Time-like Surfaces, Math. Comput. Appl. 1 (2), 133-141, 1996.