Isothermic hypersurfaces in Rn+1
Abstract
A diagonal metric sumi=1n gii dxi2 is termed Guichardk if sumi=1n-kgii-sumi=n-k+1n gii=0. A hypersurface in Rn+1 is isothermick if it admits line of curvature co-ordinates such that its induced metric is Guichardk. Isothermic1 surfaces in R3 are the classical isothermic surfaces in R3. Both isothermick hypersurfaces in Rn+1 and Guichardk orthogonal co-ordinate systems on Rn are invariant under conformal transformations. A sequence of n isothermick hypersurfaces in Rn+1 (Guichardk orthogonal co-ordinate systems on Rn 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 Guichardk orthogonal co-ordinate systems on Rn and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence between Combescure sequences of isothermick hypersurfaces in Rn+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.