A duality for conformally flat hypersurfaces
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We discuss the Guichard duality for conformally flat hypersurfaces in a Euclidean ambient space. This duality gives rise to a Goursat-type transformation for conformally flat hypersurfaces, which is generically essential. Using a suitable representation of the associated family of a conformally flat hypersurface in Euclidean space, its dual as well as conformal images of their canonical principal Guichard net(s) are recovered from the family. It is shown that the hypersurface and its dual can be reconstructed from a Ribaucour pair of Guichard nets.
KeywordsConformally flat hypersurface Combescure transformation Guichard dual Goursat transformation Guichard net Ribaucour transformation
Mathematics Subject Classification53C42 53B25 53A30 37K35 37K25
We would like to thank F. Burstall, D. Calderbank, U. Simon for fruitful and enjoyable discussions around the subject.
This work has been partially supported by: Fukuoka University Graduate School of Science, Fellowship grant 2012; Japan Society for the Promotion of Science, Grant-in-Aid for Research (C) No. 21540102, Grant-in-Aid for Scientific Research (A) No. 22244006 and Grant-in-Aid for Scientific Research (B) No. 21340016.
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