Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Couples contacto-symplectiques

Author: Gianluca Bande
Journal: Trans. Amer. Math. Soc. 355 (2003), 1699-1711
MSC (2000): Primary 53D10; Secondary 57R17
Published electronically: November 20, 2002
MathSciNet review: 1946411
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new geometric structure on differentiable manifolds. A contact-symplectic pair on a manifold $M$ is a pair $\left( \alpha ,\eta \right) $ where $\alpha $ is a Pfaffian form of constant class $2k+1$ and $\eta $ a $2$-form of constant class$ 2h$ such that $\alpha \wedge d\alpha ^{k}\wedge \eta ^{h}$ is a volume form. Each form has a characteristic foliation whose leaves are symplectic and contact manifolds respectively. These foliations are transverse and complementary. Some other differential objects are associated to it. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds and principal torus bundles. As a deep application of this theory, we give a negative answer to the famous Reeb's problem which asks if every vector field without closed 1-codimensional transversal on a manifold having contact forms is the Reeb vector field of a contact form.

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

  • 1. Bande G., On generalized contact forms, Differential Geom. Appl., 11, 257-263, 1999. MR 2001g:53135
  • 2. Bande G., Formes de contact généralisé, couples de contact et couples contacto-symplectiques, Thèse de Doctorat, Université de Haute Alsace, 2000.
  • 3. Bande G. et Hadjar A., Couples de contact, à paraître.
  • 4. Cartan E., Les systèmes différentiels et leurs applications géométriques, Hermann et Cie., Paris, 1945. MR 7:520d
  • 5. Cartan E., Leçons sur les invariants intégraux , Hermann, Paris, 1922.
  • 6. Goze M. and Khakimdjanov Y., Nilpotent Lie Algebras, Mathematics and its Applications 361, Kluwer Academic Publisher Group, Dordrecht, 1996. MR 97e:17017
  • 7. Gray J.W., Some global properties of contact structures, Ann. of Math., 69, 421-450, 1959. MR 22:3016
  • 8. Hadjar A., Sur un problème d'existence relatif de formes de contact invariantes en dimension trois, Ann. Inst. Fourier, Grenoble, 42, 891-904, 1992.
  • 9. Hadjar A., Sur les structure de contact régulières en dimension trois, Trans. Amer. Math. Soc., 347, 2473-2480, 1995. MR 95k:57030
  • 10. Kobayashi S., Principal fibre bundles with the 1-dimensional toroidal group, Tohoku Math. J., 8, 29-45, 1956. MR 18:328a
  • 11. Libermann P. et Marle C.M., Géométrie symplectique, bases théoriques de la mécanique, I, II, III, IV, Publications Mathématiques de l'Université Paris VII, 1987. MR 88g:58049a; MR 88g:58049b; MR 88g:58049c; MR 88g:58049d
  • 12. Libermann P., Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact , Colloque Géom. Diff. Globale, Bruxelles, 37-59, 1959. MR 22:9919
  • 13. Lutz R., Structures de contact sur les fibrés principaux en cercles en dimension trois, Ann. Inst. Fourier, Grenoble, 27, 1-15, 1977. MR 57:17668
  • 14. Lutz R., Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier, Grenoble, 29, 283-306, 1979. MR 82j:53067
  • 15. Reeb G., Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques, Acad. Roy. Belgique Cl. Sci. Mèm. Coll. in $8^0$, 27, 1952. MR 15:336b
  • 16. Tischler D., On fibering certain foliated manifolds over $\mathit{S}^{1}$, Topology, 9, 153-154, 1970. MR 41:1069

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D10, 57R17

Retrieve articles in all journals with MSC (2000): 53D10, 57R17

Additional Information

Gianluca Bande
Affiliation: Università degli studi di Cagliari, Dip. Mat., Via Ospedale 72, 09129 Cagliari, Italy

Keywords: Contact-Symplectic Pair, Reeb field, foliations, contact geometry, symplectic geometry
Received by editor(s): May 3, 2002
Received by editor(s) in revised form: September 26, 2002
Published electronically: November 20, 2002
Article copyright: © Copyright 2002 American Mathematical Society