# Symplectic Submanifolds of Contact Manifolds and Contact Submanifolds of Symplectic Manifolds

We know every contact manifold admits a symplectic submanifold, e.g., by Giroux's bijection : if $\omega$ is a contact form for a 3-manifold $M^3$ , then $d \omega$ is symplectic on the fibers of the open book associated to the contact structure $(M^3 , \omega)$. Now, if we are given a symplectic manifold $M^{2n}$: when does $M^{2n}$ admit a contact submanifold? I think it has to see with the existence of a Liouville vector field, but I am not sure. Thanks for any answers, refs., etc.

• You can hope for a hypersurface and a chosen primitive for $\omega$ in a neighborhood of it. This is in the spirit of the "canonical" example $(M,\lambda)\subset (\mathbb{R}\times M,d(e^s\lambda))$. Dec 11 '14 at 5:35
• Thanks, can this be extended to higher dimensions, i.e. 4-or-higher? Dec 11 '14 at 5:42