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On Dupin Hypersurfaces in R 5 Parametrized by Lines of Curvature

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Abstract

We provide a complete local characterization of Dupin hypersurfaces in R 5, with four distinct principal curvatures, parametrized by lines of curvature. Such hypersurfaces are given in terms of the principal curvatures and vector valued functions that describe plane curves. We include explicit examples of such Dupin hypersurfaces which are irreducible and have nonconstant Lie curvature.

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References

  1. Cartan E.: La déformation des hypersurfaces dans l’espace conforme reel à \({n\geq 5}\) dimensions. Bull. Soc. Math. Fr. 45, 57–121 (1917)

    Article  Google Scholar 

  2. Cecil T.E.: Lie Sphere Geometry. Springer, New York (1992)

    Book  Google Scholar 

  3. Cecil, T.E.: Isoparametric and Dupin Hypersurfaces. Symmetry, Integrability and Geometry: Methods and Applications, vol. 4, p. 062 (2008)

  4. Cecil, T.E., Chern, S.S.: Dupin submanifolds in Lie sphere geometry. Differential geometry and topology, pp. 1–48, Lecture Notes in Math., vol. 1369. Springer, Berlin (1989)

    Google Scholar 

  5. Cecil T.E., Chi Q., Jensen G.: Dupin hypersurfaces with four principal curvatures II. Geom. Dedicata 128, 55–95 (2007)

    Article  MathSciNet  Google Scholar 

  6. Cecil T.E., Jensen G.: Dupin hypersurfaces with three principal curvatures. Invent. Math. 132, 121–178 (1998)

    Article  MathSciNet  Google Scholar 

  7. Cecil T.E., Jensen G.: Dupin hypersurfaces with four principal curvatures. Geom. Dedicata 79, 1–49 (2000)

    Article  MathSciNet  Google Scholar 

  8. Cecil T.E., Ryan P.J.: Conformal geometry and the cyclides of Dupin. Can. J. Math. 32, 767–782 (1980)

    Article  MathSciNet  Google Scholar 

  9. Chern S.S.: Laplace transforms of a class of higher-dimensional varieties in projective space of n dimensions. Proc. Natl. Acad. Sci. 30, 95–97 (1944)

    Article  MathSciNet  Google Scholar 

  10. Chern, S.S.: Sur une classe remarquable de variétés sans l’espace projectif à n dimensions. Science Reports Tsing Hua Univ., vol. 4, pp. 328–336 (1947) [reprinted in S.S. Chern, Selected Papers, vol. 1. Springer, pp. 138–146 (1978)]

  11. Corro A.M.V., Ferreira W.P., Tenenblat K.: On Ribaucour transformations for hypersurfaces. Mat. Contemp. 17, 137–160 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Darboux, G.: Leçons sur la théorie générale des surfaces. Gauthier-Villars, Paris (1896) [reprinted by Chelsea, New York (1988)]

  13. Ferro M.L., Rodrigues L.A., Tenenblat K.: On a class of Dupin hypersurfaces in R 5 with nonconstant Lie curvature. Geom. Dedicata 169, 301–321 (2014)

    Article  MathSciNet  Google Scholar 

  14. Forsyth E.: Theory of Differential Equations, vol. 6. Dover, New York (1959)

    MATH  Google Scholar 

  15. Goursat E.: Lecons sur l’intégration des équations aux dérivées partielles du second ordre à deux variables indépendantes. Hermann, Paris (1896)

    MATH  Google Scholar 

  16. Kamran N., Tenenblat K.: Laplace transformation in higher dimensions. Duke Math. J. 84, 237–266 (1996)

    Article  MathSciNet  Google Scholar 

  17. Kamran N., Tenenblat K.: Periodic systems for the higher-dimensional Laplace transformation. Discrete Contin. Dyn. Syst. 4, 359–378 (1998)

    Article  MathSciNet  Google Scholar 

  18. Miyaoka R.: Compact Dupin hypersurfaces with three principal curvatures. Math. Z. 187, 433–452 (1984)

    Article  MathSciNet  Google Scholar 

  19. Miyaoka R.: Dupin hypersurfaces and a Lie invariant. Kodai Math. J. 12, 228–256 (1989)

    Article  MathSciNet  Google Scholar 

  20. Niebergall R.: Dupin hypersurfaces in \({\mathbb{R}^{5}}\). Geom. Dedicata 40, 1–22 (1991)

    Article  MathSciNet  Google Scholar 

  21. Niebergall R.: Dupin hypersurfaces in \({\mathbb{R}^{5}}\). Geom. Dedicata 41, 5–38 (1992)

    Article  MathSciNet  Google Scholar 

  22. Pinkall, U.: Dupin’sche Hyperflachen. Dissertation, Univ. Freiburg (1981)

  23. Pinkall U.: Dupinsche Hyperflachen in E 4. Manuscr. Math. 51, 89–119 (1985)

    Article  MathSciNet  Google Scholar 

  24. Pinkall U.: Dupin hypersurfaces. Math. Ann. 270, 427–440 (1985)

    Article  MathSciNet  Google Scholar 

  25. Pinkall U., Thorbergsson G.: Deformations of Dupin hypersurfaces. Proc. Am. Math. Soc. 107, 1037–1043 (1989)

    Article  MathSciNet  Google Scholar 

  26. Riveros C.M.C.: Dupin hypersurface with four principal curvatures in R 5 with principal coordinates. Rev. Mat. Comput. 23, 341–354 (2010)

    Article  MathSciNet  Google Scholar 

  27. Riveros C.M.C., Tenenblat K.: On four dimensional Dupin hypersurfaces in Euclidean space. Ann. Acad. Bras. Ci. 75, 1–7 (2003)

    Article  MathSciNet  Google Scholar 

  28. Riveros C.M.C., Tenenblat K.: Dupin hypersurfaces in \({\mathbb{R}^{5}}\). Can. J. Math. 57, 1291–1313 (2005)

    Article  MathSciNet  Google Scholar 

  29. Riveros C.M.C., Rodrigues L.A., Tenenblat K.: On Dupin hypersurfaces with constant Moebius curvature. Pac. J. Math. 236, 89–103 (2008)

    Article  Google Scholar 

  30. Rodrigues L.A., Tenenblat K.: A characterization of Moebius isoparametric hypersurfaces of the sphere. Monatsh. Math. 158, 321–327 (2009)

    Article  MathSciNet  Google Scholar 

  31. Stolz S.: Multiplicities of Dupin hypersurfaces. Invent. Math. 138, 253–279 (1999)

    Article  MathSciNet  Google Scholar 

  32. Thorbergsson G.: Dupin hypersurfaces. Bull. Lond. Math. Soc. 15, 493–498 (1983)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. IME, Universidade Federal de Goiás, Caixa Postal 131, 74001-970, Goiânia, GO, Brazil

    Marcelo Lopes Ferro

  2. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

    Luciana Ávila Rodrigues & Keti Tenenblat

Authors
  1. Marcelo Lopes Ferro
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  2. Luciana Ávila Rodrigues
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  3. Keti Tenenblat
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Corresponding author

Correspondence to Keti Tenenblat.

Additional information

M. L. Ferro was partially supported by CAPES PROCAD. L. Á. Rodrigues was partially supported by CAPES PET, CAPES PROCAD. K. Tenenblat was partially supported by CNPq, CAPES PROCAD.

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Ferro, M.L., Rodrigues, L.Á. & Tenenblat, K. On Dupin Hypersurfaces in R 5 Parametrized by Lines of Curvature. Results. Math. 70, 499–531 (2016). https://doi.org/10.1007/s00025-016-0577-0

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  • DOI: https://doi.org/10.1007/s00025-016-0577-0

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