Contact foliations and generalised Weinstein conjectures

Abstract

We consider contact foliations: objects which generalise to higher dimensions the flow of the Reeb vector field on contact manifolds. We list several 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 establish results in the field of Contact Dynamics. Mathematics Subject Classification (2010). Primary 37C85; Secondary 37C86, 53D10, 53E50.