Nagoya Mathematical Journal

Conformally flat hypersurfaces in Euclidean 4-space

Yoshihiko Suyama

Full-text: Open access

Abstract

We study generic and conformally flat hypersurfaces in Euclidean four-space. What kind of conformally flat three manifolds are really immersed generically and conformally in Euclidean space as hypersurfaces? According to the theorem due to Cartan [1], there exists an orthogonal curvature-line coordinate system at each point of such hypersurfaces. This fact is the first step of our study. We classify such hypersurfaces in terms of the first fundamental form. In this paper, we consider hypersurfaces with the first fundamental forms of certain specific types. Then, we give a precise representation of the first and the second fundamental forms of such hypersurfaces, and give exact shapes in Euclidean space of them.

Article information

Source
Nagoya Math. J., Volume 158 (2000), 1-42.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631373

Mathematical Reviews number (MathSciNet)
MR1766177

Zentralblatt MATH identifier
1003.53043

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]

Citation

Suyama, Yoshihiko. Conformally flat hypersurfaces in Euclidean 4-space. Nagoya Math. J. 158 (2000), 1--42. https://projecteuclid.org/euclid.nmj/1114631373


Export citation

References

  • E. Cartan, La déformation des hypersurfaces dans L'espace conforme á $n \geq 5$ dimensions , Oeuvres complétes III, 1, 221–286.
  • U. Hertrich-Jeromin, On conformally flat hypersurfaces and Guichard's nets , Beitr. Alg. Geom., 35 (1994), 315–331.
  • J. Lafontaine, Conformal geometry from Riemannian viewpoint , Conformal Geometry (R. S. Kulkarni and U. Pinkall, eds.), Aspects of Math. Vol. E12, Max-Plank-Ins. für Math. (1988), 65–92.
  • G. M. Lancaster, Canonical metrics for certain conformally Euclidean spaces of dimension three and codimension one , Duke Math. J., 40 (1973), 1–8.
  • Y. Suyama, Explicit representation of compact conformally flat hypersurfaces , Tôhoku Math. J., 50 (1998), 179–196.