A note on groups of symplectomorphisms (Q2732690)

The goal of this note is to prove the following theorem which generalizes a result of \textit{A. Banyaga} [J. Differ. Geom. 28, 23-35 (1988; Zbl 0632.53035)].NEWLINENEWLINENEWLINETheorem. Let \((M_i,\omega_i)\) \((i=1,2)\) be a connected symplectic manifold, and let \(G_0(M_i)\) (resp. \(G_0^*(M_i)\); resp. \(G_1(M_i)\)) denote the subgroup of \(\text{Symp}(M_i)\), the group of all symplectomorphisms of \((M_i,\omega_i)\), generated by all \(\exp(X)\), where \(X\) is a locally (resp. globally; resp. special globally) Hamiltonian vector field with compact support. If there is a group isomorphism \(\Phi: G_0(M_1) \rightarrow G_0(M_2)\) (resp. \(\Phi: G_0^*(M_1) \rightarrow G_0^*(M_2)\); resp \(\Phi: G_1(M_1) \rightarrow G_1(M_2)\)), then there is a unique diffeomorphism \(\phi: M_1 \rightarrow M_2\) such that \(\phi^*(\omega_2)=c\omega_1\), \(c\) being a constant, and \(\Phi(f)=\phi\circ f \circ \phi^{-1}\), \(\forall f\in G_0(M_1)\) (resp. \(G_0^*(M_1)\); resp \(G_1(M_1)\)).NEWLINENEWLINENEWLINEIn the paper of A. Banyaga the following is additionally supposed: either \(M_i\) are compact or the symplectic pairings of \(\omega_i\) are identically zero.NEWLINENEWLINENEWLINECorollary. Let \((M,\omega)\) be a compact connected symplectic manifold. Then every automorphism of either \(G_0(M)\), \(G_0^*(M)\), or \(G_1(M)\) is inner.