Tohoku Mathematical Journal
2011
June
SECOND SERIES VOL. 63, NO. 2

Tohoku Math. J.
63 (2011), 277-302

Title CONFORMALLY FLAT SUBMANIFOLDS IN SPHERES AND INTEGRABLE SYSTEMS

Author Neil Donaldson and Chuu-Lian Terng

(Received May 6, 2010, revised December 7, 2010)
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.

2000 Mathematics Subject Classification. Primary 53A30; Secondary 37K25, 37K35, 53B25.

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