Tohoku Mathematical Journal

Contact pairs

Gianluca Bande and Amine Hadjar

Full-text: Open access

Abstract

We introduce a new geometric structure on differentiable manifolds. A Contact Pair on a $2h+2k+2$-dimensional manifold $M$ is a pair $(\alpha,\eta) $ of Pfaffian forms of constant classes $2k+1$ and $2h+1$, respectively, whose characteristic foliations are transverse and complementary and such that $\alpha$ and $\eta$ restrict to contact forms on the leaves of the characteristic foliations of $\eta$ and $\alpha$, respectively. Further differential objects are associated to Contact Pairs: two commuting Reeb vector fields, Legendrian curves on $M$ and two Lie brackets on the set of differentiable functions on $M$. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds, bundles over the circle and principal torus bundles.

Article information

Source
Tohoku Math. J. (2), Volume 57, Number 2 (2005), 247-260.

Dates
First available in Project Euclid: 27 June 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1119888338

Digital Object Identifier
doi:10.2748/tmj/1119888338

Mathematical Reviews number (MathSciNet)
MR2137469

Zentralblatt MATH identifier
1084.53064

Subjects
Primary: 53D10: Contact manifolds, general
Secondary: 57R17: Symplectic and contact topology

Keywords
Contact geometry Reeb vector field complementary foliations invariant forms

Citation

Bande, Gianluca; Hadjar, Amine. Contact pairs. Tohoku Math. J. (2) 57 (2005), no. 2, 247--260. doi:10.2748/tmj/1119888338. https://projecteuclid.org/euclid.tmj/1119888338


Export citation

References

  • G. Bande, On generalized contact forms, Differential Geom. Appl. 11 (1999), 257--263.
  • G. Bande, Formes de contact généralisé, couples de contact et couples contacto-symplectiques, Thèse de Doctorat, Université de Haute Alsace, 2000.
  • G. Bande, Couples contacto-symplectiques, Trans. Amer. Math. Soc. 355 (2003), 1699--1711.
  • E. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922.
  • J. Feldbau, Sur la classification des espaces fibrés, C. R. Acad. Sci. Paris, 208, 1936.
  • C. Godbillon, Géométrie différentielle et mécanique analytique, Hermann, Paris, 1969.
  • M. Goze and Y. Khakimdjanov, Nilpotent Lie algebras, Math. Appl. 361, Kluwer Acad. Publ., Dordrecht, 1996.
  • A. Hadjar, Sur un problème d'existence relatif de formes de contact invariantes en dimension trois, Ann. Inst. Fourier (Grenoble) 42 (1992), 891--904.
  • A. Hadjar, Sur les structures de contact régulières en dimension trois, Trans. Amer. Math. Soc. 347 (1995), 2473--2480.
  • S. Kobayashi and K. Nomizu, Foundations of differential geometry I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.
  • P. Libermann and C. M. Marle, Géométrie symplectique, bases théoriques de la mécanique, Publ. Math. Univ. Paris VII, vol. I, II, III and IV, 1986.
  • R. Lutz, Structures de contact sur les fibrés en cercles en dimension trois, Ann. Inst. Fourier (Grenoble) 27 (1977), 1--15.
  • R. Lutz, Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble) 29 (1979), 283--306.
  • G. Reeb, Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques, Acad. Roy. Belgique Cl. Sci. Mém. Coll. in $8\sp \circ$, 27, 1952.
  • D. Tischler, On fibering certain foliated manifolds over $S^1$, Topology 9 (1970), 153--154.