Finiteness of totally magnetic hypersurfaces

By introducing a dynamical version of the second fundamental form, we generalize a recent result of Filip–Fisher–Lowe to the setting of magnetic systems. Namely, we show that a real-analytic negatively s-curved magnetic system on a closed real-analytic manifold has only finitely many closed totally s-magnetic hypersurfaces, unless the magnetic 2-form is trivial and the underlying metric is hyperbolic.

💡 Research Summary

The paper extends the recent finiteness result of Filip–Fisher–Lowe (2024) from the purely Riemannian setting to magnetic systems, i.e., pairs ((M,g,\sigma)) consisting of a closed real‑analytic manifold, a Riemannian metric, and a closed 2‑form (magnetic field). For a fixed speed parameter (s>0) the magnetic flow is the second‑order ODE \