Conformally flat submanifolds in spheres and integrable systems

Details Conformally flat submanifolds in spheres and integrable systems Conformally flat submanifolds in spheres and integrable systems

2008-03-19

arxiv.org/abs/0803.2754v2

Mathematics

Differential Geometry

24 pages, 1 figure. Minor changes

Scientific paper

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.

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