On maximal tori in the contactomorphism groups of regular contact manifolds

Abstract

By a theorem of Banyaga the group of diffeomorphisms of a manifold $P$ preserving a regular contact form $\alpha$ 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 $\Diff(P, \alpha)$ of diffeomorphisms of $P$ preserving $\alpha$ but also in the much bigger group of contactomorphisms $\Diff (P, \xi)$, the group of diffeomorphism of $P$ preserving the contact distribution $\xi = \ker \alpha$. We use this (and the work of Hausmann, and Tolman on polygon spaces) to give examples of contact manifolds $(P, \xi = \ker \alpha)$ with maximal tori of different dimensions in their group of contactomorphisms.