On the Group of Diffeomorphis Preserving an Exact Symplectic Form

Abstract

Let M be a smooth paracompact connected manifold and let ${\text{Diff}}_{\text{K}}^\infty \left( {\text{M}} \right)$ (M) be the group of all C^∞ -diffeomorphisms of M, supported in a compact subset K of M, equipped with the C^∞ -topology. Denote by Diff^∞ (M) the group $\mathop {{\text{lim}}} \limits_{\rightarrow {\text{K}} } \,{\text{Diff}}_{\text{K}}^ \infty \left( {\text{M}} \right)$ with the direct limit topology.

A symplectic manifold is a couple (M,Ω) where M is a smooth manifold of dimension even 2n and Ω is a closed 2-form such that ΩⁿΩ̂.…Ω is everywhere non zero. Let (M,Ω) be a symplectic manifold. We shall denote by ${\text{Diff}}_\Omega ^\infty \left( {\text{M}} \right)$ the subgroup of Diff^∞ (M) whose elements are those diffeomorphisms h such that h*Ω = Ω. Denote by G_Ω(M) its identity Component.