On cosymplectic dynamics

Abstract

Cosymplectic geometry can be viewed as an odd dimensional counterpart of symplectic geometry. Likely in the symplectic case, a related property which is preservation of closed forms $\omega$ and $\eta$, refers to the theoretical possibility of further understanding a cosymplectic manifold $(M, \omega, \eta)$ from its group of diffeomorphisms. In this paper we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold $(M, \omega, \eta)$ in threefold:first of all, we study cosymplectics counterpart of the Moser isotopy method, a proof of a cosymplectic version of Darboux theorem follows, and we present the features of the space of almost cosymplectic vector fields, this set forms a Lie group whose Lie algebra is the group of all almost cosymplectic diffeomorphisms; $(II)$ we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is $C^0-$closed in the group $Diff^\infty(M)$, while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the $C^0-$limit $\phi$ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition (based on Reeb's vector field) which guarantees that $\phi$ is a cosymplectic diffeomorphism is given (a flexibility condition), and also an attempt to the study cosymplectic counterpart of flux geometry follows: this gives rise to a group homomorphism whose kernel is path connected; and $(III)$ we study the almost cosymplectic analogues of Hofer geometry and Hofer-like geometry: the group of almost co-Hamiltonian diffeomorphisms carries two bi-invariant norms, the cosymplectic analogues of the usual symplectic capacity-inequality are derived and the cosymplectic analogues of a result that was proved by Hofer-Zehnder follow.