A duality for conformally flat hypersurfaces

  • U. Hertrich-JerominEmail author
  • Y. Suyama
  • M. Umehara
  • K. Yamada
Original Paper


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.


Conformally flat hypersurface Combescure transformation  Guichard dual Goursat transformation Guichard net Ribaucour transformation 

Mathematics Subject Classification

53C42 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.


  1. Burstall, F., Calderbank, D.: Conformal submanifold geometry (complete); Manuscript (2007)Google Scholar
  2. Burstall, F.: Conserved quantities in geometric integrable systems; Talk 20 Dec 2008, 16th Osaka City Univ. Int. Acad. Symp. 2008 “Riemann Surfaces, Harmonic Maps and Visualization”Google Scholar
  3. Burstall, F., Calderbank, D.: Conformal submanifold geometry I-III; EPrint arXiv:1006.5700v1 (2010)
  4. Calapso, P.: Alcune superficie di Guichard e le relative trasformazioni. Ann. Mat. Pura. Appl. 11, 201–251 (1905)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cartan, E.: La déformation des hypersurfaces dans l’éspace conforme réel à \(n\ge 5\) dimensions. Bull. Soc. Math. France 45, 57–121 (1917)MathSciNetzbMATHGoogle Scholar
  6. Dajczer, M., Tojeiro, R.: Commuting Codazzi tensors and the Ribaucour transformation for submanifolds. Res. Math. 44, 258–278 (2003)MathSciNetCrossRefGoogle Scholar
  7. Darboux, G.: Sur les systèmes de surfaces orthogonales. C. R. 67, 1101–1103 (1868)Google Scholar
  8. Dussan, M., Magid, M.: Conformally flat Lorentzian hypersurfaces in the conformal compactification of Lorentz space. J. Geom. Phys. 57, 2466–2482 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Espinar, J., Gálvez, J., Mira, P.: Hypersurfaces in \(H^{n+1}\) and conformally invariant equations: the generalized Christoffel and Nirenberg problems. J. Eur. Math. Soc. 11, 903–939 (2009)CrossRefzbMATHGoogle Scholar
  10. Ferus, D., Pedit, F.: Curved flats in symmetric spaces. Manuscr. Math. 91, 445–454 (1996)MathSciNetCrossRefGoogle Scholar
  11. Guichard, C.: Sur les surfaces isothermiques. C. R. 130, 159–162 (1900)Google Scholar
  12. Hertrich-Jeromin, U.: Introduction to Möbius Differential Geometry; London Math Soc Lect Note Ser 300. Cambridge Univ Press, Cambridge (2003)CrossRefGoogle Scholar
  13. Hertrich-Jeromin, U., Suyama, Y.: Conformally flat hypersurfaces with cyclic Guichard net. Int. J. Math. 18, 301–329 (2007)MathSciNetCrossRefGoogle Scholar
  14. Hertrich-Jeromin, U., Suyama, Y.: Conformally flat hypersurfaces with Bianchi-type Guichard net. Osaka. Math. J. 50, 1–30 (2013)MathSciNetzbMATHGoogle Scholar
  15. Izumiya, S.: Legendrian dualities and spacelike hypersurfaces in the light cone. Moscow Math. J. 9, 325–357 (2009)MathSciNetGoogle Scholar
  16. Lafontaine, J.: Conformal geometry from the Riemannian viewpoint. Aspects Math. E 12, 65–92 (1988)MathSciNetGoogle Scholar
  17. Liu, H., Jung, S.: Hypersurfaces in lightlike cone. J. Geom. Phys. 58, 913–922 (2008)MathSciNetCrossRefGoogle Scholar
  18. Liu, H., Umehara, M., Yamada, K.: The duality of conformally flat manifolds. Bull. Braz. Math. Soc. 42, 131–152 (2011)MathSciNetCrossRefGoogle Scholar
  19. McCune, C.: Rational minimal surfaces. Ph.D. thesis, University of Massachusetts (1999)Google Scholar
  20. Opozda, B.: Parallel submanifolds. Res. Math. 56, 231–244 (2009)MathSciNetCrossRefGoogle Scholar
  21. Simon, U., Schwenk-Schellschmidt, A., Vrancken, L.: Codazzi-equivalent Riemannian metrics. Asian J. Math. 14, 291–302 (2010)MathSciNetCrossRefGoogle Scholar
  22. Suyama, Y.: Conformally flat hypersurfaces in Euclidean \(4\)-space. Nagoya Math. J. 158, 1–42 (2000)MathSciNetzbMATHGoogle Scholar
  23. Wang, C.-P.: Möbius geometry for hypersurfaces in \(S^4\). Nagoya Math. J. 139, 1–20 (1995)MathSciNetGoogle Scholar

Copyright information

© The Managing Editors 2014

Authors and Affiliations

  • U. Hertrich-Jeromin
    • 1
    Email author
  • Y. Suyama
    • 2
  • M. Umehara
    • 3
  • K. Yamada
    • 4
  1. 1.Technische Universität Wien, E104WienAustria
  2. 2.Department of Applied MathematicsFukuoka UniversityFukuokaJapan
  3. 3.Department of Mathematics and Computer SciencesTokyo Institute of TechnologyTokyoJapan
  4. 4.Department of MathematicsTokyo Institute of TechnologyTokyoJapan

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