Tohoku Mathematical Journal

Conformally flat submanifolds in spheres and integrable systems

Neil Donaldson and Chuu-Lian Terng

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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.

Article information

Tohoku Math. J. (2), Volume 63, Number 2 (2011), 277-302.

First available in Project Euclid: 6 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A30: Conformal differential geometry
Secondary: 37K25: Relations with differential geometry 37K35: Lie-Bäcklund and other transformations 53B25: Local submanifolds [See also 53C40]


Donaldson, Neil; Terng, Chuu-Lian. Conformally flat submanifolds in spheres and integrable systems. Tohoku Math. J. (2) 63 (2011), no. 2, 277--302. doi:10.2748/tmj/1309952090.

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