Contact structures and cones of structure currents

In his paper Cycles for the dynamical study of foliated manifolds and complex manifolds, Denis Sullivan proves that a closed manifold supports a symplectic structure if and only if it admits a distribution of cones of bivectors that satisfies two conditions. We prove a similar result for contact structures. It relies on a suitable variant of the symplectization process that produces a $S^1$-invariant nondegenerate $2$-form on the closed manifold $S^1 \times M$ that is closed for a twisted differential.

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The first author’s work was supported by the Belgian Interuniversity Attraction Pole (IAP) within the framework Dynamics, Geometry and Statistical Physics (DYGEST). Chercheur Qualifié FNRS.

Received 20 August 2015

Accepted 9 January 2018

Published 11 February 2019