An $h$-principle for embeddings transverse to a contact structure

TL;DR

This paper establishes an $h$-principle for embeddings transverse to contact structures, providing new flexibility results and applications in contact and symplectic geometry, including Hamiltonian dynamics.

Contribution

It introduces a sufficient condition called isocontact realization for the $h$-principle and applies it to embeddings transverse to contact structures, with new cases of full $h$-principle.

Findings

Embeddings transverse to contact structures satisfy a full $h$-principle under certain conditions.

The framework applies to classical results and new applications in symplectic and contact geometry.

Demonstrates universality of Hamiltonian dynamics on level sets via embeddings.

Abstract

Given a class of embeddings into a contact or a symplectic manifold, we give a sufficient condition, that we call isocontact or isosymplectic realization, for this class to satisfy a general $h$-principle. The flexibility follows from the $h$-principles for isocontact and isosymplectic embeddings, it provides a framework for classical results, and we give two new applications. Our main result is that embeddings transverse to a contact structure satisfy a full $h$-principle in two cases: if the complement of the embedding is overtwisted, or when the intersection of the image of the formal derivative with the contact structure is strictly contained in a proper symplectic subbundle. We illustrate the general framework on symplectic manifolds by studying the universality of Hamiltonian dynamics on regular level sets via a class of embeddings.