Conformally Flat Submanifolds in Spheres and Integrable Systems

نویسنده

  • NEIL DONALDSON
چکیده

É. Cartan proved that conformally flat hypersurfaces in Sn+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 themoduli 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 normal bundle and constant multiplicities.

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تاریخ انتشار 2008