Real-analytic negatively s-curved magnetic systems on closed manifolds have only finitely many closed totally s-magnetic hypersurfaces unless the magnetic form is trivial and the metric is hyperbolic.
On the creation of conjugate points for thermostats
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abstract
Let $(M, g)$ be a closed oriented Riemannian surface, and let $SM$ be its unit tangent bundle. We show that the interior in the $\mathcal{C}^2$ topology of the set of smooth functions $\lambda:SM\to \mathbb{R}$ for which the thermostat $(M, g, \lambda)$ has no conjugate points is a subset of those functions for which the thermostat is projectively Anosov. Moreover, we prove that if a reversible thermostat is projectively Anosov, then its non-wandering set contains no conjugate points.
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math.DG 1years
2026 1verdicts
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Finiteness of Totally Magnetic Hypersurfaces
Real-analytic negatively s-curved magnetic systems on closed manifolds have only finitely many closed totally s-magnetic hypersurfaces unless the magnetic form is trivial and the metric is hyperbolic.