Towards the cosymplectic topology

Abstract In this article, the cosymplectic analogue of the symplectic flux homomorphism of a compact connected cosymplectic manifold ( M , η , ω ) \left(M,\eta ,\omega ) with ∂ M = ∅ \partial M=\varnothing is studied. This is a continuous map with respect to the C 0 {C}^{0} -metric, whose kernel is connected by smooth arcs and coincides with the subgroup of all weakly Hamiltonian diffeomorphisms. We discuss the cosymplectic analogue of the Weinstein’s chart, and derive that the group G η , ω ( M ) {G}_{\eta ,\omega }\left(M) of all cosymplectic diffeomorphisms isotopic to the identity map is locally contractible. A study of an analogue of Polterovich’s regularization process for co-Hamiltonian isotopies follows. Finally, we study Moser’s stability theorems for locally conformal cosymplectic manifolds.