SIGMA 22 (2026), 043, 15 pages      arXiv:2602.00439      https://doi.org/10.3842/SIGMA.2026.043
Contribution to the Special Issue on Geometry and Dynamics in memory of Will Merry

Finiteness of Totally Magnetic Hypersurfaces

James Marshall Reber ^a and Ivo Terek ^b
a) Department of Mathematics, University of Chicago, Chicago IL 60637, USA
^b) Department of Mathematics, University of California Riverside, Riverside CA 92521, USA

Received January 31, 2026, in final form April 26, 2026; Published online May 02, 2026

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.

Key words: magnetic flows; real-analytic manifolds; hyperbolic metrics.

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References

1. Albers P., Benedetti G., Maier L., The Hopf-Rinow theorem and the Mañé critical value for magnetic geodesics on odd-dimensional spheres, J. Geom. Phys. 214 (2025), 105521, 21 pages, arXiv:2503.02406.
2. Assenza V., Magnetic curvature and existence of a closed magnetic geodesic on low energy levels, Int. Math. Res. Not. 2024 (2024), 13586-13610, arXiv:2309.03159.
3. Assenza V., Marshall Reber J., Terek I., Magnetic flatness and E. Hopf's theorem for magnetic systems, Comm. Math. Phys. 406 (2025), 24, 20 pages, arXiv:2404.17726.
4. Cekić M., Lefeuvre T., Moroianu A., Semmelmann U., On the ergodicity of the frame flow on even-dimensional manifolds, Invent. Math. 238 (2024), 1067-1110, arXiv:2111.14811.
5. Contreras G., Gambaudo J.M., Iturriaga R., Paternain G.P., The asymptotic Maslov index and its applications, Ergodic Theory Dynam. Systems 23 (2003), 1415-1443.
6. Dairbekov N.S., Paternain G.P., Stefanov P., Uhlmann G., The boundary rigidity problem in the presence of a magnetic field, Adv. Math. 216 (2007), 535-609, arXiv:math.DG/0611788.
7. Dajczer M., Tojeiro R., Submanifold theory: Beyond an introduction, Universitext, Springer, New York, 2019.
8. Echevarría Cuesta J., Marshall Reber J., Thermostats without conjugate points, Ergodic Theory Dynam. Systems, to appear, arXiv:2501.01923.
9. Echevarría Cuesta J., Marshall Reber J., On the creation of conjugate points for thermostats, Nonlinearity 39 (2026), 045003, 13 pages, arXiv:2504.17153.
10. Filip S., Fisher D., Lowe B., Finiteness of totally geodesic hypersurfaces, arXiv:2408.03430.
11. Fisher T., Hasselblatt B., Hyperbolic flows, Zur. Lect. Adv. Math., EMS Publishing House, Berlin, 2019.
12. Foulon P., Géométrie des équations différentielles du second ordre, Ann. Inst. H. Poincaré Phys. Théor. 45 (1986), 1-28.
13. Grognet S., Flots magnétiques en courbure négative, Ergodic Theory Dynam. Systems 19 (1999), 413-436.
14. Lefeuvre T., Isometric extensions of Anosov flows via microlocal analysis, Comm. Math. Phys. 399 (2023), 453-479, arXiv:2112.05979.
15. Narasimhan R., Introduction to the theory of analytic spaces, Lecture Notes in Math., Vol. 25, Springer, Berlin, 1966.
16. Paternain G.P., Geodesic flows, Progr. Math., Vol. 180, Birkhäuser, Boston, MA, 1999.
17. Paternain G.P., Magnetic rigidity of horocycle flows, Pacific J. Math. 225 (2006), 301-323, arXiv:math.DS/0409528.
18. Paternain G.P., Paternain M., On Anosov energy levels of convex Hamiltonian systems, Math. Z. 217 (1994), 367-376.
19. San Martin L.A.B., Lie groups, Lat. Amer. Math. Ser., Springer, Cham, 2021.
20. Terek I., The submanifold compatibility equations in magnetic geometry, Ann. Global Anal. Geom. 69 (2026), 4, 15 pages, arXiv:2506.22990.