Abstract
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.
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Notes
It appears to be likely that this duality should have been known to the classical geometers; however, we have thus far been unable to detect it in the classical literature.
As \(T\) is \(\mathrm{II}\)-symmetric and \(\mathrm{II}^*=\mathrm{II}\circ (\mathrm{id},T)\), the eigenspaces of \(T\) form a common “conjugate system” for \(f\) and \(f^*\): they are simultaneously \(\mathrm{II}\)-orthogonal and \(\mathrm{II}^*\)-orthogonal. Thus we obtain a generalization of the classical Combescure transformation, cf. Darboux (1868).
The first part (Burstall and Calderbank 2010) of this work is publicly available, the second part, containing an analysis of the Guichard dual is in preparation at the time of this writing.
Note that this yields two classes of Guichard surfaces, depending on the sign \(\varepsilon \): pseudospherical surfaces fall into the class given by \(\varepsilon =-1\) while surfaces of constant positive Gauss curvature fall into the one given by \(\varepsilon =+1\), cf. Calapso (1905).
If the rank of \({ A}\) drops we obtain only two non-trivial solutions, as \(({ S}-\kappa { A}+{1\over 2}\kappa ^2)^2\parallel { S}^2\) if and only if \(\kappa \) is a root of the (cubic) characteristic polynomial of \({ A}\) by (3.6).
This argument is spoiled, to some extent, by the fact that the Guichard dual is only defined up to translation; however, this deficiency of the computation does clearly not affect the claim.
Equivalently, \(\mathcal{S}^{m+1}\) can be thought of as \(\mathbb {R}^{m+1}\cup \{\infty \}\) equipped with its standard conformal structure.
Thus \(g\) and \(g^*\) provide conformal coordinates for \((M^3,\mathrm{I})\) and \((M^3,\mathrm{I}^*)\), respectively, via stereographic projection.
Two of the Lamé functions are, with the forms \(dx_i\), imaginary. In particular, \(l_1^2l_2^2l_3^2>0\).
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Acknowledgments
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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Hertrich-Jeromin, U., Suyama, Y., Umehara, M. et al. A duality for conformally flat hypersurfaces. Beitr Algebra Geom 56, 655–676 (2015). https://doi.org/10.1007/s13366-014-0225-3
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DOI: https://doi.org/10.1007/s13366-014-0225-3
Keywords
- Conformally flat hypersurface
- Combescure transformation
- Guichard dual
- Goursat transformation
- Guichard net
- Ribaucour transformation