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Isomorphism of cosymplectomorphism groups implies diffeomorphism of manifolds

Etienne Djoukeng, Stephane Tchuiaga

TL;DR

The paper addresses whether the cosymplectomorphism group determines the underlying manifold for regular cosymplectic spaces. It identifies the Reeb flow as the center of the cosymplectomorphism group and uses this to descend the isomorphism to the base symplectic manifolds, then matches the monodromies via conjugacy of centralizers to reconstruct a diffeomorphism of the total spaces. The main result proves that isomorphic cosymplectomorphism groups imply identical circle-bundle classes in $H^2(B,\mathbb{Z})$ and yield a diffeomorphism of the total spaces, with a precise description of how the base and fiber data align up to scaling of the transverse form; an extension to coKähler manifolds is also provided. Together, these findings establish a strong rigidity principle in cosymplectic geometry, showing that symmetry groups encode the manifold topology and bundle structure.

Abstract

We prove that if two closed, connected, regular cosymplectic manifolds have isomorphic groups of cosymplectomorphisms (as topological groups), then the underlying manifolds are diffeomorphic. The proof proceeds by characterizing the Reeb flow as the center of the group and descending the isomorphism to the symplectic base manifolds. We show that the isomorphism preserves the conjugacy class of the monodromy of the mapping torus, which ensures that the bundle structures, and thus the total spaces are equivalent.

Isomorphism of cosymplectomorphism groups implies diffeomorphism of manifolds

TL;DR

The paper addresses whether the cosymplectomorphism group determines the underlying manifold for regular cosymplectic spaces. It identifies the Reeb flow as the center of the cosymplectomorphism group and uses this to descend the isomorphism to the base symplectic manifolds, then matches the monodromies via conjugacy of centralizers to reconstruct a diffeomorphism of the total spaces. The main result proves that isomorphic cosymplectomorphism groups imply identical circle-bundle classes in and yield a diffeomorphism of the total spaces, with a precise description of how the base and fiber data align up to scaling of the transverse form; an extension to coKähler manifolds is also provided. Together, these findings establish a strong rigidity principle in cosymplectic geometry, showing that symmetry groups encode the manifold topology and bundle structure.

Abstract

We prove that if two closed, connected, regular cosymplectic manifolds have isomorphic groups of cosymplectomorphisms (as topological groups), then the underlying manifolds are diffeomorphic. The proof proceeds by characterizing the Reeb flow as the center of the group and descending the isomorphism to the symplectic base manifolds. We show that the isomorphism preserves the conjugacy class of the monodromy of the mapping torus, which ensures that the bundle structures, and thus the total spaces are equivalent.
Paper Structure (8 sections, 7 theorems, 23 equations)

This paper contains 8 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Cosymplectic settings
  3. Statement of the main theorem
  4. Structure interchange via the induced diffeomorphism
  5. Invariance of the pull-back forms
  6. Behaviour on the Reeb field
  7. Comparison on the transverse symplectic structure
  8. Comparison of the Reeb 1-forms

Key Result

Theorem 1

Let $(M_1,\eta_1,\omega_1)$ and $(M_2,\eta_2,\omega_2)$ be two compact, connected, regular cosymplectic manifolds without boundary. Suppose there exists an isomorphism of topological groups: where the groups are equipped with the $C^\infty$ compact-open topology. Then the circle bundle classes of $M_1$ and $M_2$ in $H^2(B,\mathbb{Z})$ coincide, where $B$ denotes the common symplectic base. Moreov

Theorems & Definitions (18)

  • Example 1: The Standard Product Manifold
  • Example 2: Symplectic Mapping Torus
  • Definition 1
  • Theorem 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • Corollary 1
  • ...and 8 more