Existence of conformal symplectic foliations on closed manifolds

TL;DR

This paper investigates the existence and construction of conformal symplectic foliations on closed manifolds, demonstrating their prevalence in high dimensions and providing explicit examples in dimension five.

Contribution

It proves that in dimensions at least 7, any almost contact structure can be homotoped to a conformal symplectic foliation, and constructs explicit foliations in dimension five.

Findings

Any almost contact structure in dimension ≥7 is homotopic to a conformal symplectic foliation.

Explicit conformal symplectic foliations are constructed on all closed, simply-connected, almost contact 5-manifolds.

Examples of conformal symplectic foliations deforming to contact structures are provided via round-connected sums.

Abstract

We consider the existence of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5. Our main theorem, based on a recent result by Bertelson-Meigniez, states that in dimension at least 7 any almost contact structure is homotopic to a conformal symplectic foliation. In dimension 5 we construct explicit conformal symplectic foliations on every closed, simply-connected, almost contact manifold, as well as honest symplectic foliations on a large subset of them. Lastly, via round-connected sums, we obtain, on closed manifolds, examples of conformal symplectic foliations which admit a linear deformation to contact structures.