Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-20T02:47:19.017Z Has data issue: false hasContentIssue false
  • English
  • Français

Dupin Hypersurfaces in ${{\mathbb{R}}^{5}}$

Published online by Cambridge University Press:  20 November 2018

Carlos M.C. Riveros  and
Keti Tenenblat
Carlos M.C. Riveros
Affiliation:
Departamento de Matemática, Universidade Federal do Tocantins, 77330-000, Arraias, TO, Brazil, e-mail: carlosc@uft.edu.br
Keti Tenenblat
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil, e-mail: keti@mat.unb.br
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study Dupin hypersurfaces in ${{\mathbb{R}}^{5}}$ parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties.

Keywords

53B2553C4235N1037K10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 6 , 01 December 2005 , pp. 1291 - 1313
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Cecil, T. E. and Chern, S. S., Dupin submanifolds in Lie sphere geometry. In: Differential Geometry and Topology, Lecture Notes in Math. 1369, Springer, Berlin, 1989, pp. 1–48,Google Scholar
[2] Cecil, T. E. and Ryan, P. J., Conformal geometry and the cyclides of Dupin. Canad. J. Math. 32(1980), 767–782.Google Scholar
[3] Cecil, T. E. and Ryan, P. J., Tight and Taut Immersions Of Manifolds. Research Notes in Mathematics 107, Pitman, Boston, 1985.Google Scholar
[4] Chern, S. S., An introduction to Dupin submanifolds. In: Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math. 52, Longman, Harlow, 1991, pp. 95–102.Google Scholar
[5] Cecil, T. E. and Jensen, G., Dupin hypersurfaces with three principal curvatures. Invent.Math. 132(1998), 121–178.Google Scholar
[6] Cecil, T. E. and Jensen, G., Dupin hypersurfaces with four principal curvatures. Geom. Dedicata 79(2000), 1–49.Google Scholar
[7] Kamran, N. and Tenenblat, K., Laplace transformation in higher dimensions. Duke Math. J. 84(1996), 237–266.Google Scholar
[8] Kamran, N. and Tenenblat, K., Periodic systems for the higher-dimensional Laplace transformation. Discrete Contin. Dynam. Systems, 4(1998), 359–378.Google Scholar
[9] Miyaoka, R., Compact Dupin hypersurfaces with three principal curvatures. Math. Z. 187(1984), 433–452.Google Scholar
[10] Niebergall, R., Dupin hypersurfaces in R5. Geom. Dedicata 40(1991), 1–22, and 41(1992), 5–38.Google Scholar
[11] Pinkall, U., Dupinsche Hyperflachen. Dissertation, Univ. Freiburg, 1981.Google Scholar
[12] Pinkall, U., Dupinsche Hyperflachen in E4. Manuscripta Math. 51(1985), 89–119.Google Scholar
[13] Pinkall, U., Dupin hypersurfaces. Math. Ann. 270(1985), 427–440.Google Scholar
[14] Pinkall, U. and Thorbergsson, G., Deformations of Dupin hypersurfaces. Proc. Amer.Math. Soc. 107(1989), 1037–1043.Google Scholar
[15] Riveros, C. M. C. and Tenenblat, K., On four dimensional Dupin hypersurfaces in Euclidean space. An. Acad. Brasil. Ciênc.75(2003), 1–7.Google Scholar
[16] Stolz, S.,. Multiplicities of Dupin hypersurfaces. Invent.Math. 138(1999), 253–279.Google Scholar
[17] Thorbergsson, G., Dupin hypersurfaces. Bull. LondonMath. Soc. 15(1983), 493–498.Google Scholar