On some Closed Magnetic Curves on a 3-torus

  • Marian Ioan Munteanu
  • Ana Irina Nistor


We consider two magnetic fields on the 3-torus obtained from two different contact forms on the Euclidean 3-space and we study when their corresponding normal magnetic curves are closed. We obtain periodicity conditions analogues to those for the closed geodesics on the torus.


Magnetic field Closed curve Periodicity Elliptic function 

Mathematics Subject Classification (2010)

53C15 53C25 37J45 37C27 53C80 


  1. 1.
    Adachi, T.: Kähler Magnetic fields on a complex projective space. Proc. Japan Acad. 70 Ser. A, 12–13 (1994)Google Scholar
  2. 2.
    Adachi, T.: Kähler Magnetic flow for a manifold of constant holomorphic sectional curvature. Tokyo J. Math. 18, 473–483 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Armitage, J.V., Eberlein, W.F.: Elliptic functions, London Math. Soc. Student Texts 67, Cambridge Univ Press (2006)Google Scholar
  4. 4.
    Barros, M., Cabrerizo, J.L., Fernández, M., Romero, A.: Magnetic vortex filament flows, J. Math. Phys. 48, art 082904 (2007)Google Scholar
  5. 5.
    Barros, M., Romero, A., Cabrerizo, J.L., Fernández, M.: The Gauss-Landau-Hall problem on Riemannian surfaces, J. Math. Phys. 46, art 112905 (2005)Google Scholar
  6. 6.
    Barros, M., Romero, A.: Magnetic vortices, EPL 77, art. 34002 (2007)Google Scholar
  7. 7.
    Bennequin, D.: Entrelacements et équations de Pfaff. Astérisque 107–108, 87–161 (1983). English translation: Linkings and Pfaff’s equations, Russian Math. Surveys 44, 1–65, (1989)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Blair, D.E.: Riemannian geometry of contact and symplectic manifolds Progress in Math, vol. 203. Birkhäuser, Boston-Basel-Berlin (2002)CrossRefGoogle Scholar
  9. 9.
    Cabrerizo, J.L., Fernández, M., Gómez, J.: On the existence of almost contact structure and the contact magnetic field. Acta. Math. Hungar. 125, 191–199 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cabrerizo, J.L., Fernández, M., Gómez, J.: The contact magnetic flow in 3D Sasakian manifolds, J. Phys. A: Math. Theor. 42, art 195201 (2009)Google Scholar
  11. 11.
    Comtet, A.: On the Landau levels on the hyperbolic plane. Ann. Phys. 173, 185–209 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Druţă-Romaniuc, S.L., Inoguchi, J., Munteanu, M.I., Nistor, A.I.: Magnetic curves in Sasakian manifolds. J. Nonlinear Math. Phys. 22, 428–447 (2015)Google Scholar
  13. 13.
    Druţă-Romaniuc, S.L., Munteanu, M.I.: Magnetic curves corresponding to Killing magnetic fields in \(\mathbb {E}^{3}\), J. Math. Phys. 52, art 113506 (2011)Google Scholar
  14. 14.
    Ikawa, O.: Motion of charged particles in Sasakian manifolds. SUT J. Math. 43(2), 263–266 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Itoh, M.: Odd dimensional tori and structure, contact. Proc. Japan Acad. 72, 58–59 (1997)CrossRefGoogle Scholar
  16. 16.
    Inoguchi, J., Munteanu, M.I.: Periodic magnetic curves in Berger spheres, to appear in Tohoku. J. Math., 69 (2017)Google Scholar
  17. 17.
    Jantzen, R.T.: Geodesics on the torus and other surfaces of revolution clarified using undergraduate physics tricks with bonus: nonrelativistic and relativistic Kepler problems. arXiv:1212.6206v1 [math.DG] (2012)
  18. 18.
    Munteanu, M.I.: Magnetic curves in the Euclidean space: one example, several approaches. Publ. Inst. Math. (Beograd) 94(108), 141–150 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Munteanu, M.I., Nistor, A.I.: The classification of Killing magnetic curves in \(\mathbb {S}^{2}\times \mathbb {R}\). J. Geom. Phys. 62, 170–182 (2012)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Sunada, T.: Magnetic flows on a Riemann surface. In: Proceedings KAIST Mathematics Workshop: Analysis and Geometry, (KAIST, Taejeon, Korea 1993), pp. 93–108 (1993)Google Scholar

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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Faculty of MathematicsAlexandru Ioan Cuza University of IaşiIaşiRomania
  2. 2.Department of Mathematics and InformaticsGh. Asachi Technical University of IaşiIasiRomania

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