Null Cartan curve variations in 3D semi-Riemannian manifold

Year 2021, Volume: 50 Issue: 2, 351 - 360, 11.04.2021

Zehra Özdemir

https://doi.org/10.15672/hujms.569423

Cited By: 1

Abstract

The aim of this study is to investigate the variations of the Bishop frame curvatures for null Cartan curves in semi-Riemannian manifolds. The Killing equations in terms of the variations of the Bishop curvatures along the null Cartan curve is especially derived. Killing equations are used to interpret the movement of the charged particles in a magnetic field. The charged particle motion along a null Cartan curve through the Killing equations is examined as an application in the study. It is found that the charged particle traces a trajectory in the form of the null cubic during its movement in the Killing magnetic vector field. According to the results obtained, an example of the null Cartan magnetic trajectories is presented.

Keywords

variation, killing vector field, Bishop frame, Cartan frame, null Cartan curve

References

• [1] M. Barros, J.L. Cabrerizo, M. Fernández, and A. Romero, Magnetic vortex filament flows, J. Math. Phys. 48 (8), 27 pp., 2007.
• [2] M. Barros, A. Ferrández, P. Lucas, and M.A. Merono, General helices in the 3- dimensional Lorentzian space forms, Rocky Mountain J. Math. 31, 373-388, 2001.
• [3] L.R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly. 82 (3), 246-251, 1975.
• [4] Z. Bozkurt, İ. Gök, Y. Yaylı, and F.N. Ekmekci, A new approach for magnetic curves in 3D Riemannian manifolds, J. Math. Physics. 55, 053501, 2014.
• [5] 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.
• [6] K.L. Duggal and D.H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, Singapore, 2007.
• [7] A. Ferrández, A. Giménez, and P. Lucas, Geometrical particles models on 3D null curves, Physics Letters B 543 (3-4), 311-317, 2002.
• [8] A. Ferrández, A. Giménez, and P. Lucas, Relativistic particles and the geometry of 4D null curves, J. Geom. Phys. 57 (10), 2124-2135, 2007.
• [9] A. Giménez, Relativistic particles along null curves in 3D Lorenzian space forms, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (9), 2851-2859, 2010.
• [10] M. Grbović and E. Nešović, On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space, J. Math. Anal. Appl. 461, 219-233, 2018.
• [11] L.P. Hughston and W.T. Shaw, Real classical strings, Proc. Roy. Soc. London Ser. A. 414, 415-422, 1987.
• [12] L.P. Hughston and W.T. Shaw, Classical strings in ten dimensions, Proc. Roy. Soc. London Ser. A. 414, 423-431, 1987.
• [13] L.P. Hughston and W.T. Shaw, Constraint-free analysis of relativistic strings, Classical Quantum Gravity 5, 69-72, 1988.
• [14] J. Inoguchi and S. Lee, Null curves in Minkowski 3-space, Int. Electronic J. Geom. 1 (2), 40-83, 2008.
• [15] A. Kazan and H.B. Karadag, Magnetic pseudo null and magnetic null curves in Minkowski 3-space, Int. Math. Forum 12 (3), 119-132, 2017.
• [16] A. Kazan and H.B. Karadag, Magnetic Curves According to Bishop Frame and Type-2 Bishop Frame in Euclidean 3-Space, British J. Math. Comp. Sci. 22 (4), 1-18, 2017.
• [17] Z. Özdemir, Pseudo Null Curve Variations for Bishop Frame in 3D semi-Riemannian Manifold, Int. J. Geom. Methods Modern Phys. 16 (3), 1950043, 2019.
• [18] M. Özdemir and A.A. Ergin, Parallel frame of non-lightlike curves, Missouri J. Math. Sci. 20 (2), 127-137, 2008.
• [19] Z. Özdemir, İ. Gök, Y. Yaylı and F.N. Ekmekci, Notes on Magnetic Curves in 3D semi-Riemannian Manifolds, Turk. J. Math. 39, 412-426, 2015.
• [20] W. T. Shaw, Twistors and strings (Santa Cruz, CA). 337-363 (1986). Amer. Math. Soc., RI, 1988.
• [21] H. Urbantke, On Pinl’s representation of null curves in n dimensions, In Relativity Today, Budapest, 34-36, 1987, World Sci. Publ., Teaneck, New York, 1988.