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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Couples contacto-symplectiques
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by Gianluca Bande PDF
Trans. Amer. Math. Soc. 355 (2003), 1699-1711 Request permission

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
  • Email: gbande@unica.it
  • Received by editor(s): May 3, 2002
  • Received by editor(s) in revised form: September 26, 2002
  • Published electronically: November 20, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 1699-1711
  • MSC (2000): Primary 53D10; Secondary 57R17
  • DOI: https://doi.org/10.1090/S0002-9947-02-03209-9
  • MathSciNet review: 1946411