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A zoo of coisotropic embeddings

Rubén Izquierdo-López, Manuel de León, Luca Schiavone, Pablo Soto

Abstract

The aim of this paper is to extend the coisotropic embedding theorem obtained by M. J. Gotay for pre-symplectic manifolds to more general geometric settings: cosymplectic, contact, cocontact, $k$-symplectic, $k$-cosymplectic, $k$-contact, and multisymplectic manifolds. The results are obtained by applying a generic methodology, which gives more relevance to the potential applications. In that sense, this paper gives the fundamental basis to be able to apply the results to the so-called regularization problem of singular Lagrangian systems, both in mechanics and in classical field theories.

A zoo of coisotropic embeddings

Abstract

The aim of this paper is to extend the coisotropic embedding theorem obtained by M. J. Gotay for pre-symplectic manifolds to more general geometric settings: cosymplectic, contact, cocontact, -symplectic, -cosymplectic, -contact, and multisymplectic manifolds. The results are obtained by applying a generic methodology, which gives more relevance to the potential applications. In that sense, this paper gives the fundamental basis to be able to apply the results to the so-called regularization problem of singular Lagrangian systems, both in mechanics and in classical field theories.

Paper Structure

This paper contains 46 sections, 50 theorems, 316 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributions and Frobenius theorem
  4. Moser's trick and relative Poincaré Lemma
  5. Differential calculus and almost product structures
  6. Pre-symplectic manifolds
  7. Coisotropic submanifolds
  8. Coisotropic embeddings
  9. Existence
  10. Uniqueness
  11. Pre-cosymplectic manifolds
  12. Coisotropic submanifolds
  13. Coisotropic embeddings
  14. Existence
  15. Uniqueness
  16. ...and 31 more sections

Key Result

Theorem 2.1

Let $\mathcal{D}$ be a smooth rank-$r$ distribution on a smooth manifold $M$. The following statements are equivalent:

Theorems & Definitions (191)

  • Definition 1: Distribution
  • Remark 1
  • Definition 2: Integral manifold
  • Definition 3: Integrable distribution
  • Definition 4: Smooth foliation
  • Remark 2
  • Definition 5: Completely integrable distribution
  • Theorem 2.1: Frobenius Theorem
  • proof
  • Theorem 2.2
  • ...and 181 more