A classification of certain 3-dimensional conformally flat Euclidean hypersurfaces (Q1318478)

The classification of conformally flat hypersurfaces of the four dimensional Euclidean space \(\mathbb{R}^ 4\), whose mean curvature vector \(H\) is an eigenvector of their Laplacian, is obtained: They have to be either minimal or the Riemannian product \(S^ p\times \mathbb{R}^{3-p}\), \(0\leq p\leq 3\). This result extends a previous theorem of \textit{A. Ferrández}, \textit{O. J. Garay} and \textit{P. Lucas} [Lect. Notes Math. 1481, 48-54 (1991; Zbl 0738.53035)] to the case of 3-dimensional hypersurfaces. The main idea of the proof is to use a method of \textit{B.-Y. Chen} [Nagyoa Math. J. 122, 139-148 (1991; Zbl 0702.53035)] to show that the classical Cartan- Schouten result is still valid for this kind of hypersurfaces and then one can use similar computations to those of the paper by Ferrández- Garay-Lucas quoted above.