Magnetic flows on 3D contact sub-Riemannian manifolds via the Rumin complex

Abstract

We show that the appropriate notion of magnetic field on three-dimensional contact sub-Riemannian manifolds is given by a closed Rumin differential two-form. We introduce horizontal magnetic flows starting from magnetic potential one-forms, proving that the flow depends only on the Rumin differential of the potential. Notably, in dimension three the Rumin differential acts on one-forms as a second-order differential operator. We further prove that such magnetic flows can be interpreted as a geodesic flow on a suitably lifted sub-Riemannian structure, which is of Engel type when the magnetic field is non-vanishing. In the general case, when the magnetic field may vanish, we analyze the geometry of the lifted structure, characterizing its step and abnormal trajectories in terms of the analytical properties of the magnetic field. Our work is inspired by the classical correspondence, first observed by Montgomery, between Riemannian magnetic flows and sub-Riemannian geometry.