On a maximal torus in the volume-preserving diffeomorphism group of the finite cylinder

Abstract

Let T be the collection of all H s ( s > 2) diffeomorphisms η φ of the cylindrical surface M ≔ S 1 × [ p, q], where η φ rotates each “horizontal” circle S 1 × z rigidly by an angle φ = φ( z) which is a real valued H s function. Let M be given the flat metric g induced from the Euclidean metric of R 3. We shall prove in this paper that (1) Topologically, T is a real, infinite-dimensional, smooth, path-connected and closed submanifold of Diff vol relative to the H s topology. (2) Algebraically, T is a maximal Abelian subgroup of Diff vol, it is equal to its centralizer in Diff, and its Weyl group in Diff vol is Z 2. (3) Geometrically, with respect to the g-induced right-invariant L 2 metric 〈·, ·〉, T is a totally geodesic and flat Riemannian submanifold of Diff vol; we also identify its normal bundle.