Closed Magnetic geodesics on Heisenberg nilmanifolds

TL;DR

This paper investigates the existence and properties of closed magnetic geodesics on three-dimensional Heisenberg nilmanifolds, establishing conditions for their existence across different homotopy classes and energy levels.

Contribution

It provides new results on the existence of closed magnetic geodesics on Heisenberg nilmanifolds, including conditions for contractible and non-contractible cases, and examples illustrating these phenomena.

Findings

Existence of closed contractible magnetic geodesics below Mañé critical value.

Not all homotopy classes necessarily contain closed magnetic geodesics.

Examples of manifolds with infinitely many or no non-contractible closed magnetic trajectories.

Abstract

In this work we study the existence of closed magnetic geodesics on three-dimensional Heisenberg nilmanifolds for every left-invariant Lorentz force. Our first objective is to establish the existence of closed contractible magnetic geodesics on $H_3$. Once the invariant magnetic field is induced to a compact quotient $M=\Lambda \backslash H_3$, we study magnetic geodesics on $M$. Firstly, we determine conditions on a lattice $\Lambda \subset H_3$ to ensure that a given magnetic geodesic projects to a closed curve on $M$. In particular, we prove that for {\it any} energy level below the Ma\~n\'e critical value there always exists a contractible closed magnetic geodesic on the compact manifold $M$. On the other hand, we show that closed magnetic geodesics do not necessarily exist in every homotopy class. Finally, we present examples of compact quotients $\Gamma_k\backslash H_3$ that admit infinitely many closed magnetic trajectories, as well as examples for which no closed non-contractible magnetic trajectories exist for a given left-invariant Lorentz force.