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The Simplicity of the Group of Weakly Hamiltonian Diffeomorphisms on Cosymplectic Manifolds

S. Tchuiaga, P. Bikorimana

TL;DR

This work extends Banyaga’s framework to cosymplectic geometry by proving that the group of weakly Hamiltonian diffeomorphisms $ ext{Ham}_{ ext{eta,omega}}(M)$ on any closed cosymplectic manifold is simple. A key structural result is the periodicity of the Reeb flow, which enforces discreteness of the flux group $ extGamma_{ ext{eta,omega}}$ and enables fragmentation and transitivity arguments that yield perfectness and simplicity. The authors also connect the cosymplectic setting to a Liouville-type integrability theory for Reeb-invariant dynamics, showing that closed cosymplectic manifolds arise as symplectic mapping tori with invariant $(n+1)$-dimensional tori in the full system. Furthermore, they identify the commutator subgroup of the full cosymplectomorphism group with $ ext{Ham}_{ ext{eta,omega}}(M)$ and establish flux-group discreteness as a cosymplectic invariant, with explicit computations in key examples, highlighting structural obstructions to cosymplectomorphism between manifolds. These results fuse symplectic, cosymplectic, and foliation techniques to illuminate the algebraic structure of transformation groups in a mixed geometric setting.

Abstract

We establish a cosymplectic counterpart of Banyaga's theorem by proving that the group of weakly Hamiltonian diffeomorphisms, $\Ham_{η,ω}(M)$, is simple on any closed cosymplectic manifold. A key structural result, derived from Lie group theory, provides the foundation for our argument: the Reeb flow on any closed cosymplectic manifold is always periodic. This property, in turn, forces the associated flux group to be discrete. Building on this discrete invariant, we develop the essential fragmentation and transitivity principles needed to prove perfectness and simplicity. Beyond this algebraic framework, we recover Li's result realizing closed cosymplectic manifolds as symplectic mapping tori, and we establish a Liouville-type integrability theorem for Hamiltonian systems invariant under the Reeb flow, producing $(n+1)$-dimensional invariant tori. Finally, we characterize the commutator subgroup of the full cosymplectomorphism group as $\Ham_{η,ω}(M)$.

The Simplicity of the Group of Weakly Hamiltonian Diffeomorphisms on Cosymplectic Manifolds

TL;DR

This work extends Banyaga’s framework to cosymplectic geometry by proving that the group of weakly Hamiltonian diffeomorphisms on any closed cosymplectic manifold is simple. A key structural result is the periodicity of the Reeb flow, which enforces discreteness of the flux group and enables fragmentation and transitivity arguments that yield perfectness and simplicity. The authors also connect the cosymplectic setting to a Liouville-type integrability theory for Reeb-invariant dynamics, showing that closed cosymplectic manifolds arise as symplectic mapping tori with invariant -dimensional tori in the full system. Furthermore, they identify the commutator subgroup of the full cosymplectomorphism group with and establish flux-group discreteness as a cosymplectic invariant, with explicit computations in key examples, highlighting structural obstructions to cosymplectomorphism between manifolds. These results fuse symplectic, cosymplectic, and foliation techniques to illuminate the algebraic structure of transformation groups in a mixed geometric setting.

Abstract

We establish a cosymplectic counterpart of Banyaga's theorem by proving that the group of weakly Hamiltonian diffeomorphisms, , is simple on any closed cosymplectic manifold. A key structural result, derived from Lie group theory, provides the foundation for our argument: the Reeb flow on any closed cosymplectic manifold is always periodic. This property, in turn, forces the associated flux group to be discrete. Building on this discrete invariant, we develop the essential fragmentation and transitivity principles needed to prove perfectness and simplicity. Beyond this algebraic framework, we recover Li's result realizing closed cosymplectic manifolds as symplectic mapping tori, and we establish a Liouville-type integrability theorem for Hamiltonian systems invariant under the Reeb flow, producing -dimensional invariant tori. Finally, we characterize the commutator subgroup of the full cosymplectomorphism group as .

Paper Structure

This paper contains 17 sections, 13 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Cosymplectic settings
  3. The co-flux geometry revisited
  4. The co-flux homomorphism
  5. New results
  6. Integrability on compact cosymplectic manifolds with periodic Reeb flows
  7. Mapping torus structure of closed cosymplectic manifolds
  8. Commutator subgroup of cosymplectic diffeomorphism groups
  9. The group $\Gamma_{\eta,\omega}$
  10. Contact Hamiltonian Simplicity vs. Cosymplectic Invariants
  11. Perfectness and fragmentation in cosymplectic settings
  12. Cohomological implications and explicit computation of the flux group
  13. Flux group as a cosymplectic invariant
  14. Appendix
  15. Periodicity of the Reeb flow
  16. ...and 2 more sections

Key Result

Proposition 2.6

T-al Let $(M, \eta, \omega)$ be a compact cosymplectic manifold. Suppose that $\Phi_F = \{\phi_t\}_{t\in [0,1]}$ is a weakly Hamiltonian (resp. cosymplectic) isotopy on $M$ satisfying $\widetilde{I}_{\eta,\omega}(\dot\phi_t) = dF_t, \quad \forall\, t\in[0,1].$ Then the isotopy with $\mathcal{R}_{\Lambda_t(\Phi_F)}(x,\theta) := \theta - \int_0^t \eta\bigl(\dot\phi_s\bigr)\bigl(\phi_s(x)\bigr) ds \

Theorems & Definitions (37)

  • Example 2.1: The Standard Product Manifold
  • Example 2.2: Symplectic Mapping Torus
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 27 more