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

Abstract

We establish a cosymplectic counterpart of Banyaga's theorem by proving that the group of weakly Hamiltonian diffeomorphisms, $\Ham_{\eta,\omega}(M)$, 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 $\Ham_{\eta,\omega}(M)$.