Conformally flat submanifolds in spheres and integrable systems

TL;DR

This paper links conformally flat hypersurfaces in spheres to integrable systems, demonstrating their Gauss-Codazzi equations are soliton equations and constructing transformations using soliton theory.

Contribution

It establishes the Gauss-Codazzi equation as a soliton equation for conformally flat hypersurfaces in S^4 and develops a geometric Ribaucour transform via soliton theory.

Findings

Gauss-Codazzi equation is a soliton equation in this context

Constructed geometric Ribaucour transforms using soliton theory

Described moduli and symmetries of hypersurfaces in S^4

Abstract

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 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 normal bundle and constant multiplicities.