An Application of Convex Integration to Contact Geometry

Abstract

We prove that every closed, orientable 3 3 -manifold M M admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form. Our proof is based on Gromov’s convex integration technique and the h h -principle. Similar methods can be used to show that M M admits a parallelization by contact forms with everywhere linearly independent Reeb vector fields. We also prove a generalization of this latter result to higher dimensions. If M M is a closed ( 2 n + 1 ) (2n+1) -manifold with contact form ω \omega whose contact distribution ker ⁡ ω \ker \omega admits k k everywhere linearly independent sections, then M M admits k + 1 k+1 linearly independent contact forms with linearly independent Reeb vector fields.