Contact topology and electromagnetism: The Weinstein conjecture and Beltrami-Maxwell fields.

This article establishes connections between contact topology and Maxwell fields in vacuum on closed three-dimensional Riemannian manifolds embedded in four-dimensional Lorentzian manifolds. Focusing on a class of Maxwell fields where electric and magnetic fields are parallel and composed of Beltrami fields—1-forms satisfying a specific eigenvalue equation—the study employs theorems resolving the Weinstein conjecture to prove the existence of closed electromagnetic field lines (periodic orbits of Reeb vector fields) on these manifolds. It shows that such Maxwell fields induce stable Hamiltonian structures and contact forms, with associated Reeb vector fields preserving electromagnetic energies. The main results demonstrate that under these geometric and topological conditions, closed field lines of the electric and magnetic fields necessarily exist, providing a novel topological perspective on Maxwell's equations and suggesting potential applications in plasma physics and electromagnetic knot theory.