Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the rigidity of magnetic systems with the same magnetic geodesics


Authors: Keith Burns and Vladimir S. Matveev
Journal: Proc. Amer. Math. Soc. 134 (2006), 427-434
MSC (2000): Primary 37Dxx, 37D40, 37Jxx, 53B10
DOI: https://doi.org/10.1090/S0002-9939-05-08196-7
Published electronically: September 20, 2005
MathSciNet review: 2176011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the analogue for magnetic flows of the classical question of when two different metrics on the same manifold share geodesics, which are the same up to reparametrization.


References [Enhancements On Off] (What's this?)

  • 1. E. Beltrami, Resoluzione del problema: riportari i punti di una superficie sopra un piano in modo che le linee geodetische vengano rappresentante da linee rette, Ann. Mat. 1 (1865), no. 7.
  • 2. K. Burns, G. P. Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity 15 (2002), no. 2, 281-314. MR 1888853 (2004d:37076)
  • 3. N. Gouda, Magnetic flows of Anosov type, Tôhoku Math. J. 49 (1997), 165-183. MR 1447180 (98e:58129)
  • 4. S. Grognet, Flots magnétiques en courbure négative, Ergod. Th. and Dynam. Syst. 19 (1999), 413-436. MR 1685401 (2000d:37032)
  • 5. S. Grognet, Entropies de flots magnétiques, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 395-424. MR 1721559 (2000k:37035)
  • 6. A. Lichnerowicz, Sur la transformation des équations de la dynamique, C. R. Acad. Sci. Paris 223 (1946), 649-651. MR 0018491 (8:293h)
  • 7. A. Lichnerowicz, D. Aufenkamp, The general problem of the transformation of the equations of dynamics, J. Rational Mech. Anal. 1 (1952), 499-520. MR 0051051 (14:421a)
  • 8. L. Macarini, Entropy rigidity and harmonic fields, Nonlinearity 13 (2000), 1761-1774. MR 1781817 (2001h:37061)
  • 9. V. S. Matveev, P. J. Topalov, Trajectory equivalence and corresponding integrals, Regular and Chaotic Dynamics 3 (1998), no. 2, 30-45. MR 1693470 (2000d:37068)
  • 10. Vladimir S. Matveev, Peter J. Topalov, Metric with ergodic geodesic flow is completely determined by unparametrized geodesics, AMS Elect. Res. Announcements 6 (2000), 98-104. MR 1796527 (2001i:37043)
  • 11. Vladimir S. Matveev, Hyperbolic manifolds are geodesically rigid, Invent. Math. 151 (2003), 579-609. MR 1961339 (2004f:53044)
  • 12. G. P. Paternain, M. Paternain, Anosov geodesic flows and twisted symplectic structures, International Conference on Dynamical Systems in Montevideo (a tribute to Ricardo Mañé), F. Ledrappier, J. Lewowicz, S. Newhouse, eds., Pitman Research Notes in Math. 362 (1996), 132-145. MR 1460801 (98h:58145)
  • 13. G. P. Paternain, On the regularity of the Anosov splitting for twisted geodesic flows, Math. Res. Lett. 4 (1997), 871-888. MR 1492126 (98h:58137)
  • 14. G. P. Paternain, M. Paternain, First derivative of topological entropy for Anosov geodesic flows in the presence of magnetic fields, Nonlinearity 10 (1997), 121-131. MR 1430743 (97j:58118)
  • 15. G. P. Paternain, On two noteworthy deformations of negatively curved Riemannian metrics, Discrete Contin. Dynam. Systems. 5 (1999), 639-650. MR 1696335 (2000g:37032)
  • 16. N. Peyerimhoff, K. F. Siburg, The dynamics of magnetic flows for energies above Mañé's critical value, Israel J. Math. 135 (2003), 269-298. MR 1997047 (2004e:37098)
  • 17. P. J. Topalov and V. S. Matveev, Geodesic equivalence via integrability, Geometriae Dedicata 96 (2003), 91-115. MR 1956835 (2003k:53043)
  • 18. H. Weyl, Geometrie und Physik, Die Naturwissenschaftler 19 (1931), 49-58. Can be found in ``Hermann Weyl Gesammelte Abhandlungen", Band 3, Springer Verlag, 1968.
  • 19. H. Weyl, Zur Infinitisimalgeometrie: Einordnung der projektiven und konformen Auffassung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 99-112, 1921. Can be found in ``Selecta Hermann Weyl", Birkhaüser Verlag, 1956.
  • 20. M. P. Wojtkowski, Convexly hyperbolic flows on unit tangent bundles of surfaces, Tr. Mat. Inst. Steklova 216 (1997), 373-383. MR 1632202 (99i:58123)
  • 21. M. P. Wojtkowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature, Fund. Math. 163 (2000), no. 2, 177-191. MR 1752103 (2001b:37038)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37Dxx, 37D40, 37Jxx, 53B10

Retrieve articles in all journals with MSC (2000): 37Dxx, 37D40, 37Jxx, 53B10


Additional Information

Keith Burns
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: burns@math.northwestern.edu

Vladimir S. Matveev
Affiliation: Mathematisches Institut, Universität Freiburg, 79104 Germany
Email: matveev@email.mathematik.uni-freiburg.de

DOI: https://doi.org/10.1090/S0002-9939-05-08196-7
Received by editor(s): September 5, 2004
Published electronically: September 20, 2005
Additional Notes: The first author was supported by NSF grants DMS-9803346 and DMS-0100416
The second author was supported by DFG-programm 1154 (Global Differential Geometry) and Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg (Eliteförderprogramm Postdocs 2003).
Communicated by: Michael Handel
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society