Contact foliations and generalised Weinstein conjectures

TL;DR

This paper introduces contact foliations as higher-dimensional generalizations of Reeb flows, proposes conjectures on their leaf topology akin to Weinstein's conjecture, and provides partial positive results under specific conditions.

Contribution

It formulates new conjectures about the topology of contact foliation leaves and extends existing results in Contact Dynamics to these higher-dimensional structures.

Findings

Partial positive results for the conjectures when holonomy preserves a Riemannian metric

Extension of known contact dynamics results to contact foliations

Connections between contact foliations and classical Weinstein conjecture

Abstract

We consider contact foliations: objects which generalise to higher dimensions the flow of the Reeb vector field on contact manifolds. We list a number of properties of such foliations, and propose two conjectures about the topological types of their leaves, both of which coincide with the classical Weinstein conjecture in the case of contact flows. We give positive partial results for our conjectures in particular cases -- when the holonomy of the contact foliation preserves a Riemannian metric, for instance -- extending already established results from the field of Contact Dynamics.