Conformally flat submanifolds in spheres and integrable systems

Abstract

É. 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 hypersurfaces in S4 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 S4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n - 2)-spheres with flat and non-degenerate normal bundle.