On a maximal torus in the volume-preserving diffeomorphism group of the finite cylinder (Q1366595)

Let \({\mathcal T}\) be the collection of all \(H^s\) \((s>2)\) diffeomorphisms \(\eta_\varphi\) of the cylindrical surface \(M:= S^1\times [p,q]\), where \(\eta_\varphi\) rotates each ``horizontal'' circle \(S^1\times z\) rigidly by an angle \(\varphi=\varphi(z)\) which is a real valued \(H^s\) function. Let \(M\) be given the flat metric \(g\) induced from the Euclidean metric of \(\mathbb{R}^3\). We prove that: (1) Topologically, \({\mathcal T}\) is a real, infinite-dimensional, smooth, path-connected and closed submanifold of \(\text{Diff}_{\text{vol}}\) relative to the \(H^s\) topology. (2) Algebraically, \({\mathcal T}\) is a maximal Abelian subgroup of \(\text{Diff}_{\text{vol}}\), it is equal to its centralizer in Diff, and its Weyl group in \(\text{Diff}_{\text{vol}}\) is \(\mathbb{Z}_2\). (3) Geometrically, with respect to the \(g\)-induced right-invariant \(L^2\) metric \(\langle \cdot,\cdot\rangle\), \({\mathcal T}\) is a totally geodesic and flat Riemannian submanifold of \(\text{Diff}_{\text{vol}}\); we also identify its normal bundle.