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Riemannian geometry on the diffeomorphism group of the circle
Lund University, Sweden .ORCID iD: 0000-0001-6191-7769
2007 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 45, no 2, 297-325 p.Article in journal (Refereed) Published
Abstract [en]

The topological group Dk(S) of diffeomorphisms of the unit circle S of Sobolev class H k , for k large enough, is a Banach manifold modeled on the Hilbert space Hk(S) . In this paper we show that the H 1 right-invariant metric obtained by right-translation of the H 1 inner product on TidDkbbS)\simeq Hk(S) defines a smooth Riemannian metric on Dk(S) , and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa-Holm equation - a nonlinear wave equation that has attracted much attention in recent years - and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the H 1 right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa-Holm equation.

Place, publisher, year, edition, pages
2007. Vol. 45, no 2, 297-325 p.
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URN: urn:nbn:se:kth:diva-163831DOI: 10.1007/s11512-007-0047-8ISI: 000250460200008ScopusID: 2-s2.0-38149115470OAI: diva2:802360

QC 20150416

Available from: 2015-04-12 Created: 2015-04-12 Last updated: 2015-04-16Bibliographically approved

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Lenells, Jonatan
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