A physical classification of Killing magnetic fields in Thurston geometries

In recent years, numerous studies have appeared that considered Killing vectors of three‐dimensional Riemannian manifolds as magnetic fields, since these vector fields are divergenceless by definition. The existence of adivergenceless vector field modeled as a magnetic field does not imply that it is physically realizable. In this study, we propose a physical classification scheme based on the divergences of the integral that defines the energy of a Killing magnetic field. We consider all the Killing magnetic fields of Thurston geometries studied in the literature and classify them as either physical or nonphysical.