Journal of Symplectic Geometry

Volume 16 (2018)

Number 4

Contact structures and cones of structure currents

Pages: 1021 – 1040



Mélanie Bertelson (Département de Mathématique, Université Libre de Bruxelles, Belgium)

Cédric De Groote (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


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.

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