Isothermic hypersurfaces in R^{n+1}

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dc.creatorDonaldson, Neil
dc.creatorTerng, Chuu-Lian
dc.date2008-09-21
dc.date2009-01-13
dc.date.accessioned2026-07-07T12:28:14Z
dc.date.available2026-07-07T12:28:14Z
dc.descriptionA diagonal metric sum_{i=1}^n g_{ii} dx_i^2 is termed Guichard_k if sum_{i=1}^{n-k}g_{ii}-sum_{i=n-k+1}^n g_{ii}=0. A hypersurface in R^{n+1} is isothermic_k if it admits line of curvature co-ordinates such that its induced metric is Guichard_k. Isothermic_1 surfaces in R^3 are the classical isothermic surfaces in R^3. Both isothermic_k hypersurfaces in R^{n+1} and Guichard_k orthogonal co-ordinate systems on R^n are invariant under conformal transformations. A sequence of n isothermic_k hypersurfaces in R^{n+1} (Guichard_k orthogonal co-ordinate systems on R^n resp.) is called a Combescure sequence if the consecutive hypersurfaces (orthogonal co-ordinate systems resp.) are related by Combescure transformations. We give a correspondence between Combescure sequences of Guichard_k orthogonal co-ordinate systems on R^n and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence between Combescure sequences of isothermic_k hypersurfaces in R^{n+1} and solutions of the O(2n+1-k,k)/O(n+1)xO(n-k,k)-system, both being integrable systems. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.
dc.description21 pages, reworked definition to correct error in theorem
dc.identifierhttps://arxiv.org/abs/0809.3608
dc.identifierhttp://arxiv.org/abs/0809.3608
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/215469
dc.subjectDifferential Geometry
dc.subject53A30
dc.titleIsothermic hypersurfaces in R^{n+1}
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