Null Cartan curve variations in 3D semi-Riemannian manifold

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

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