Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Robert Cardona, Francisco Presas

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.

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

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 -principle. The flexibility follows from the -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 -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.
Paper Structure (20 sections, 23 theorems, 53 equations)

This paper contains 20 sections, 23 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Flexibility in contact and symplectic geometry
  4. Flexibility of even contact and Hamiltonian structures
  5. Embeddings admitting isocontact/isosymplectic realizations
  6. Embeddings in contact manifolds
  7. Embeddings in symplectic manifolds
  8. Transverse embeddings into contact manifolds
  9. Ambient overtwisted contact manifold
  10. A flexible subclass in the general case
  11. Hamiltonian energy level sets with arbitrary dynamics
  12. The cotangent lift
  13. Arbitrary dynamics on Hamiltonian structures
  14. Hamiltonian embeddings and perturbations of the unit sphere
  15. Hamiltonian and generalized isosymplectic embeddings
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(M,\xi)$ be a contact manifold. Embeddings transverse to the contact structure satisfy an $h$-principle in the following cases: The $h$-principle holds in the parametric, relative to the domain and relative to the parameter versions. In the small case, it is also $C^0$-dense.

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 49 more