Finiteness of Totally Magnetic Hypersurfaces

TL;DR

This paper extends a recent mathematical result to magnetic systems, proving finiteness of certain hypersurfaces under specific conditions on the magnetic form and metric.

Contribution

It introduces a dynamical second fundamental form and generalizes finiteness results to negatively curved magnetic systems on real-analytic manifolds.

Findings

Negatively s-curved magnetic systems have finitely many closed totally s-magnetic hypersurfaces.

Finiteness holds unless the magnetic 2-form is trivial and the metric is hyperbolic.

The approach uses a generalized dynamical second fundamental form.

Abstract

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.