Magnetic trajectories corresponding to Killing magnetic fields in a three-dimensional warped product

Abstract

The aim of this paper is to investigate Killing magnetic trajectories of varying electrically charged particles in a three-dimensional warped product [Formula: see text] with positive warping function [Formula: see text], where [Formula: see text] is an open interval in [Formula: see text] equipped with an induced semi-Euclidean metric on [Formula: see text]. First, Killing vector fields on [Formula: see text] are characterized and it is observed that lifts to [Formula: see text] of Killing vector fields tangent to [Formula: see text] are also Killing on [Formula: see text]. Now, any Killing vector field on [Formula: see text] corresponds to a Killing magnetic field on [Formula: see text]. Magnetic trajectories (also known as magnetic curves) of charged particles which move under the influence of Lorentz force generated by Killing magnetic fields on [Formula: see text] are obtained in both Riemannian and Lorentzian cases. Moreover, some examples are exhibited with pictures determining Killing magnetic trajectories in hyperbolic [Formula: see text]-space [Formula: see text] modeled by the Riemannian warped product [Formula: see text]. Furthermore, some examples of spacelike, timelike and lightlike Killing magnetic trajectories are given with their possible graphs in the Lorentzian warped product [Formula: see text].

Explicit formulas for Killing magnetic curves in Heisenberg group

“…Interesting results on Killing magnetic curves are given in Euclidian 3space ( [9]), Minkowski 3-space ( [8]), S 2 × R ( [17]), SL(2, R) ( [10]), Sol Space ( [12]), warped product manifold ( [15]), almost paracontact manifold ( [5]), Almost Cosymplectic Sol Space ( [11]) and in Walker manifolds ( [2]).…”