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 $S^4$ 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 and non-degenerate normal bundle.