On maximal tori in the contactomorphism groups of regular contact manifolds
Abstract
Description
By a theorem of Banyaga the group of diffeomorphisms of a manifold $P$ preserving a regular contact form $α$ is a central $S^1$ extension of the commutator of the group of symplectomorphisms of the base $B = P/S^1$. We show that if $T$ is a Hamiltonian maximal torus in the group of symplectomorphism of $B$, then its preimage under the extension map is a maximal torus not only in the group $\Diff(P, α)$ of diffeomorphisms of $P$ preserving $α$ but also in the much bigger group of contactomorphisms $\Diff (P, ξ)$, the group of diffeomorphism of $P$ preserving the contact distribution $ξ= \ker α$. We use this (and the work of Hausmann, and Tolman on polygon spaces) to give examples of contact manifolds $(P, ξ= \ker α)$ with maximal tori of different dimensions in their group of contactomorphisms.
3 pages
3 pages