## Nagoya Mathematical Journal

### Conformally flat hypersurfaces in Euclidean 4-space

Yoshihiko Suyama

#### 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

https://projecteuclid.org/euclid.nmj/1114631373

Mathematical Reviews number (MathSciNet)
MR1766177

Zentralblatt MATH identifier
1003.53043

Subjects
• E. Cartan, La déformation des hypersurfaces dans L'espace conforme á $n \geq 5$ dimensions , Oeuvres complétes III, 1, 221–286.