Finiteness of totally magnetic hypersurfaces

TL;DR

The paper investigates finiteness of closed totally $s$-magnetic hypersurfaces for real-analytic magnetic systems with negative $s$-magnetic curvature, showing rigidity unless the magnetic form is trivial and the metric is hyperbolic arithmetic. It develops a dynamical framework based on the dynamical second fundamental form $\mathrm{II}^{\varphi}$ and the dynamical exponential map $\exp^{\varphi}$, and analyzes principal isometric extensions via the Brin group to propagate invariance along the flow. The main result extends Filip–Fisher–Lowe-type finiteness theorems to magnetic flows, concluding that if infinitely many closed totally $s$-magnetic hypersurfaces exist, then $\sigma=0$ and $(M,g)$ must be a hyperbolic arithmetic manifold; otherwise the finiteness holds. This work enhances rigidity phenomena in negatively curved dynamical systems by linking geometric realizability of invariant submanifolds to global curvature and arithmetic structure, using a dynamical-Cartan approach that may inform broader rigidity questions for odd semi-spray flows.

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.

Theorems & Definitions (27)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Example 2.1
• Lemma 2.2: filip2024finiteness
• Lemma 3.1
• proof
• Example 3.2
• Definition 3.3