On maximal tori in the contactomorphism groups of regular contact manifolds

2002/12/03 by Eugene Lerman, Lerman, Eugene
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG

paper · pdf · doi:10.48550/arxiv.math/0212043

3 pages

arxiv created 2002/12/03 · arxiv updated 2009/11/30

Abstract

By a theorem of Banyaga the group of diffeomorphisms of a manifold P preserving a regular contact form α is a central S¹ extension of the commutator of the group of symplectomorphisms of the base B = P/S¹. 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.

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