Magnetic flows on 3D contact sub-Riemannian manifolds via the Rumin complex
Davide Barilari, Tania Bossio, Valentina Franceschi
TL;DR
This work proposes a consistent magnetic theory on three-dimensional contact sub-Riemannian manifolds by representing magnetic fields as closed horizontal Rumin 2-forms $\beta\in\Omega^2_H(M)$. Magnetic geodesics are shown to depend only on $\beta$ via a magnetic Hamiltonian $H_A$ with $\beta=d_H A$, and the Rumin complex provides the correct second-order differential structure for fields living in dimension three. The dynamics are interpreted geometrically by lifting to $\overline{M}=M\times\mathbb{R}$, where the lifted distribution is Engellike when $\beta\neq0$, and the zero locus $\mathcal{Z}$ of $\beta$ governs the step, abnormal trajectories, and the Maxwell-type closure condition $d\beta=0$ expressed as a divergence constraint. The paper combines precise algebraic criteria for the lift's step with explicit Heisenberg-group examples to connect magnetic flow, lifted sub-Riemannian geometry, and Montgomery's classical correspondence in a sub-Riemannian setting, yielding a framework for future analysis of zero loci and abnormal dynamics in low dimensions.
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.