A 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. Isot
hermic_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.
E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersur
faces in S^4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S^4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n-2)-spheres with flat normal bundle and constant multiplicities.
Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables
us to write down a natural set of simple elements. Using these simple elements we prove generator theorems for the fundamental representations of the remaining neo-classical groups and most of their symmetric spaces. In order to apply our theorems to submanifold geometry we also obtain explicit dressing and permutability formulae. We introduce a new submanifold geometry associated to G_2/SO(4) to which our theory applies.
