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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Paires de structures de contact sur les variétés de dimension trois

Vincent Colin and Sebastião Firmo

Algebraic & Geometric Topology 11 (2011) 2627–2653
Abstract

On introduit une notion de paire positive de structures de contact sur les variétés de dimension trois qui généralise celle de Eliashberg et Thurston [Confoliations, Univ. Lecture Series 13, Amer. Math. Soc. (1998)] et Mitsumatsu [Ann. Inst. Fourier (Grenoble) 45 (1995) 1407–1421 ; Foliations : geometry and dynamics (Warsaw, 2000) World Sci. Publ., River Edge, NJ (2002) 75–125]. Une telle paire “normale” donne naissance à un champ de plans continu et localement intégrable λ. On montre que si λ est uniquement intégrable et si les structures de contact sont tendues, alors le feuilletage intégral de λ est sans composante de Reeb d’âme homologue à zéro. De plus, dans ce cas, la variété ambiante porte un feuilletage sans composante de Reeb. On démontre également un théorème de stabilité “à la Reeb” pour les paires positives de structures tendues.

We introduce the notion of a positive pair of contact structures of a three dimensional manifold which generalises that of Eliashberg, Thurston and Mitsumatsu. A normal such pair gives rise to a continuous, locally integrable plane field λ. We show that if lambda is uniquely integrable and if the contact structures are tight then the integral foliation of λ has no Reeb component whose core is homologous to zero. Moreover, in this case, the ambient manifold carries a foliation without a Reeb component. We also show a Reeb stability theorem for positive pairs of tight structures.

Keywords
structure de contact, pair of contact structure, paire, feuilletage, foliation, tendu, tight, composante de Reeb, reeb component
Mathematical Subject Classification 2000
Primary: 57M50, 57R17, 57R30
References
Publication
Received: 2 June 2009
Accepted: 25 July 2011
Published: 15 September 2011
Authors
Vincent Colin
Laboratoire de Mathématiques Jean Leray
UMR 6629 du CNRS
Université de Nantes
2 rue de la Houssinière
F-44322 Nantes
France
http://www.math.sciences.univ-nantes.fr/~vcolin/
Sebastião Firmo
Instituto de Matemática
Universidade Federal Fluminense
Rua Mário Santos Braga
Valonguinho 24.020-140
Niterói RJ
Brazil