Conformally flat submanifolds in spheres and integrable systems
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 oneparameter family of (n1)spheres. We prove that the GaussCodazzi 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 nimmersions in (2n2)spheres with flat normal bundle and constant multiplicities.
 Publication:

arXiv eprints
 Pub Date:
 March 2008
 DOI:
 10.48550/arXiv.0803.2754
 arXiv:
 arXiv:0803.2754
 Bibcode:
 2008arXiv0803.2754D
 Keywords:

 Mathematics  Differential Geometry;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 53A30
 EPrint:
 24 pages, 1 figure. Minor changes