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Transactions of the American Mathematical Society

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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
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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.

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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