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arXiv · 2407.05515

Closed Magnetic geodesics on Heisenberg nilmanifolds

Gabriela P. Ovando·Mauro Subils
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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 $M=Λ\backslash H_3$, we study magnetic geodesics on $M$. Firstly, we determine conditions on a lattice $Λ\subset H_3$ 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 Mañé 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 $Γ_k\backslash H_3$ 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.

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Gabriela P. Ovando, Mauro Subils. 2026-05-20. Closed Magnetic geodesics on Heisenberg nilmanifolds. https://arxiv.org/abs/2407.05515

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