Differential Geometry of Magnetic and Killing Magnetic Trajectories in de Sitter 3-Space

This paper is concerned to study normal magnetic trajectories of charged particles which move under the influence of Lorentz force generated by static magnetic fields on de Sitter 3-space S4 so that our study coincides with the theory of magnetostatics in the ambience of Physics. First, we observe how the associated Lorentz force acts on the vector fields in the dynamic Frenet frame or pseudo-orthonormal frame along the trajectory depending on its causality. Based on this, we characterize magnetic trajectories in respect of their Frenet apparatus. Afterwards, we present a geometrical model of hyperbolic quaternions for S-1(3) where we identified S-1(3) with a subspace of the Lie group H of hyperbolic quaternions and obtain Killing vector fields on S-1(3) forming a basis for the concerned six-dimensional Lie algebra. We also characterize Killing vector fields along regular curves in S-1(3) and observe that they can be extended to Killing vector fields on S-1(3) (in respect of their causality) and vice-versa. Finally, we characterize and classify normal Killing magnetic trajectories in S4 in terms of their quasi-slope, curvature and torsion or pseudo-torsion based on their causality.