On maximal tori in the contactomorphism groups of regular contact manifolds

Details On maximal tori in the contactomorphism groups of regular contact manifolds On maximal tori in the contactomorphism groups of regular contact manifolds

2002-12-03

arxiv.org/abs/math/0212043v1

Mathematics

Symplectic Geometry

3 pages

Scientific paper

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.

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