On a maximal torus in the volume-preserving diffeomorphism group of the finite cylinder
type: Item
instance of: scholarly article
title: On a maximal torus in the volume-preserving diffeomorphism group of the finite cylinder
zbMATH Open document ID: 0885.58006
DOI: 10.1016/S0926-2245(96)00040-X
author: D. Bao
author: T. S. Ratiu
published in: Differential Geometry and its Applications
publication date: 1997-09-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58B25
Mathematics Subject Classification ID: 22E65
zbMATH DE Number: 1060718
zbMATH Keywords: maximal torus
zbMATH Keywords: diffeomorphism group
MaRDI profile type: Publication
Wikidata QID: Q115337619
cites work: Groups of diffeomorphisms and the motion of an incompressible fluid
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cites work: Matrix Analysis
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full work available at URL: s0926-2245(96)00040-x
OpenAlex ID: W2149750295
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description: scientific article; zbMATH DE number 1060718