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On conformally flat hypersurfaces, curved flats and cyclic systems

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Abstract

It will be shown that suitable “Gauß maps” associated to a conformally flat hypersurface inS n+1 (n≥3) yield normal congruences of circles having a whole 1-parameter family of conformally flat orthogonal hypersurfaces. However such a “cyclic system” is not uniquely associated to a conformally flat hypersurface. The key idea is to show that these Gauß maps are “curved flats” in a pseudo Riemannian symmetric space. Additionally, in this context some characterizations of 3-dimensional conformally flat hypersurfaces arise with a new flavour. The curved flat approach allows us to handle conformally flat hypersurfaces in the context of integrable system theory.

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Authors and Affiliations

  1. Forschungsinstitut für Mathematik, ETH Zürich, CH-8092, Zürich, (Switzerland)

    U. Hertrich-Jeromin

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  1. U. Hertrich-Jeromin
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Supported by DFG grant He 2490/1-1

This article was processed by the author using the LATEX style filecljour 1 from Springer-Verlag.

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Hertrich-Jeromin, U. On conformally flat hypersurfaces, curved flats and cyclic systems. Manuscripta Math 91, 455–466 (1996). https://doi.org/10.1007/BF02567966

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  • DOI: https://doi.org/10.1007/BF02567966

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