Magnetic unit vector fields

TL;DR

This paper characterizes magnetic unit vector fields on Riemannian manifolds, linking critical points of the Landau Hall and Dirichlet functionals, and classifies magnetic left invariant vector fields on 3D Lie groups.

Contribution

It establishes a new equivalence between critical points of two functionals and classifies magnetic vector fields on specific Lie groups.

Findings

Unit vector fields are critical points of both functionals under certain conditions.

Classification of magnetic left invariant vector fields on 3D Lie groups.

Characterization of magnetic maps into unit tangent sphere bundles.

Abstract

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups.