Classification of Killing Magnetic Curves In H^3

TL;DR

This paper classifies magnetic curves in hyperbolic 3-space generated by Killing vector fields, solving geodesic equations analytically and comparing with numerical solutions, while also exploring properties of magnetic fields in certain manifolds.

Contribution

It provides an analytical classification of magnetic curves in H^3 associated with all Killing vector fields and examines magnetic fields in specific geometric manifolds.

Findings

Analytical solutions for magnetic trajectories in H^3

Comparison between analytical and numerical solutions

Magnetic vector fields cannot align with Reeb vector in certain manifolds

Abstract

In this paper, we study classification of magnetic curves corresponding to Killing vector fields of H^3 (hyperbolic 3-space). First, we solve the geodesic equation analytically. Then we calculate the trajectories generated by all the six Killing vector fields, which are considered as magnetic field vectors, by using perturbation method up to first order with respect to the strength of the magnetic field. We present a comparison of our solution with the numerical solution for one case. We also prove that 3-dimensional ({\alpha})-Kenmotsu manifolds cannot have any magnetic vector field in the direction of their Reeb vector fields.