An Application of Convex Integration to Contact Geometry

We prove that every closed, orientable $3$-manifold $M$ 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 $h$-principle. Similar methods can be used to show that $M$ 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 $M$ is a closed $(2n+1)$-manifold with contact form $\omega$ whose contact distribution $\ker \omega$ admits $k$ everywhere linearly independent sections, then $M$ admits $k+1$ linearly independent contact forms with linearly independent Reeb vector fields.