A correspondence between a class of cone structures and contact forms

TL;DR

This paper establishes a correspondence between certain cone structures and contact forms on closed manifolds, linking geometric structures with dynamical and topological properties.

Contribution

It demonstrates that the existence of a contact structure on a closed manifold is equivalent to the presence of an invariant cone structure with specific properties, extending Sullivan's ideas.

Findings

Contact structures correspond to specific cone structures.

Existence of invariant cone structures implies contact structures.

Provides a new geometric criterion for contact structures.

Abstract

In the spirit of Sullivan's paper "Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds", existence of a contact structure on a closed manifold $M$ is shown to be equivalent to existence of an ample $S^1$-invariant cone structure with no nontrivial exact structure cycles on the manifold $S^1 \times M$.