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Cosymplectic Lagrangian-like submanifolds

S. Tchuiaga, F. Balibuno, E. Djoukeng

TL;DR

The paper develops a comprehensive cosymplectic analogue of Lagrangian geometry by defining Lagrangian-like subspaces and the cosymplectic Lagrangian-like Grassmannian, then builds the foundational toolkit—cosymplectic vector spaces, orthogonality, and co-complex structures. It extends to smooth manifolds with cosymplectomorphisms, Lagrangian-like submanifolds, and normal bundle theory, culminating in Weinstein-like tubular neighborhoods and a Weinstein-like chart that links cosymplectomorphisms to closed 1-forms via the co-flux framework. The main contributions include a relative Moser trick in the cosymplectic setting, fixed-point results for near-identity cosymplectomorphisms, and the co-flux homomorphism that encodes deformation data in $H^1_{\mathrm{Reeb}}(M,\mathbb{R})$. These results provide local-to-global control over deformations in odd-dimensional cosymplectic manifolds and have potential implications for time-dependent Hamiltonian systems and related geometric structures. Overall, the work unifies local cosymplectic geometry with global isotopy invariants, offering new tools for studying stability, normal forms, and canonical charts in odd dimensions.

Abstract

This paper highlights the similarities between even-dimensional geometry (symplectic) and odd-dimensional geometry (cosymplectic). We study the Lagrangian Grassmannian in the cosymplectic setting. The space of compatible co-complex structures is introduced and analyzed. A study of Moser's trick and Lagrangian neighborhood theorems in the cosymplectic context follows. The corresponding Weinstein $1-$form is derived, and its de Rham class is a co-flux.

Cosymplectic Lagrangian-like submanifolds

TL;DR

The paper develops a comprehensive cosymplectic analogue of Lagrangian geometry by defining Lagrangian-like subspaces and the cosymplectic Lagrangian-like Grassmannian, then builds the foundational toolkit—cosymplectic vector spaces, orthogonality, and co-complex structures. It extends to smooth manifolds with cosymplectomorphisms, Lagrangian-like submanifolds, and normal bundle theory, culminating in Weinstein-like tubular neighborhoods and a Weinstein-like chart that links cosymplectomorphisms to closed 1-forms via the co-flux framework. The main contributions include a relative Moser trick in the cosymplectic setting, fixed-point results for near-identity cosymplectomorphisms, and the co-flux homomorphism that encodes deformation data in . These results provide local-to-global control over deformations in odd-dimensional cosymplectic manifolds and have potential implications for time-dependent Hamiltonian systems and related geometric structures. Overall, the work unifies local cosymplectic geometry with global isotopy invariants, offering new tools for studying stability, normal forms, and canonical charts in odd dimensions.

Abstract

This paper highlights the similarities between even-dimensional geometry (symplectic) and odd-dimensional geometry (cosymplectic). We study the Lagrangian Grassmannian in the cosymplectic setting. The space of compatible co-complex structures is introduced and analyzed. A study of Moser's trick and Lagrangian neighborhood theorems in the cosymplectic context follows. The corresponding Weinstein form is derived, and its de Rham class is a co-flux.
Paper Structure (18 sections, 22 theorems, 30 equations)

This paper contains 18 sections, 22 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cosymplectic vector space
  4. Cosymplectic orthogonality
  5. Cosymplectic Lagrangian-like Grassmanian
  6. Co-complex structure
  7. On smooth manifolds
  8. The $C^r$ topology on $Diff^k(M)$
  9. Cosymplectic manifolds
  10. Cosymplectomorphisms
  11. Lagrangian-like submanifolds
  12. The graph of a closed $1-$form and Lagrangian-like submanifolds
  13. Neighborhood of a Lagrangian-like submanifold
  14. The normal space to a submanifold :
  15. Normal subspace of a cosymplectic manifold.
  16. ...and 3 more sections

Key Result

Proposition 2.5

S-C-M Let $(V, b, \psi)$ be a cosymplectic vector space with Reeb vector $\xi$, and $F$ be a proper subspace of $V$ different from $\langle \xi \rangle$. Then, $V \neq F\oplus F^{b, \psi}$.

Theorems & Definitions (65)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Example 2.9
  • Example 2.10
  • ...and 55 more