Conformally flat hypersurfaces of \(E^4\) (Q5932748)

A Riemannian manifold \((M^n,g)\) is called conformally flat if every point has a neighbourhood which is conformal to an open set in the Euclidean space \(E^n\). For hypersurfaces \(M^n\) of Euclidean space \(E^{n+1}\) exists in dimensions \(n\geq 4\) a classical result by Cartan-Schouten. The induced metric of a hypersurface \(M^n\) of \(E^{n+1}\) \((n\geq 4)\) is conformally flat if and only if at least \(n-1\) of the principal curvatures coincide at each point, see e.g. \textit{B. Y. Chen} [Geometry of submanifolds, Marcel Dekker, New York (1973; Zbl 0262.53036)], \textit{M. Do Carmo, M. Dajczer} and \textit{F. Mercuri} [Trans. Am. Math. Soc. 288, 189-203 (1985; Zbl 0554.53040)]. Also \textit{G. M. Lancester} [Duke Math. J. 40, 1-8 (1973; Zbl 0256.53024)] presents examples of conformally flat hypersurfaces with three different principal curvatures and \textit{U. Hertrich-Jeronim} [Über konform flache Hyperflächen in vierdimensionalen Raumformen, PhD Thesis, TU Berlin (1994)] considers examples of such conformally flat hypersurfaces and proves a structural theorem. In this note the authors treat conformally flat hypersurfaces of \(E^4\) which allow these Guichard coordinates. Also they consider an immersion with 2 different principal curvatures and having constant mean curvature. In the Preliminaries an \(n\)-dimensional Riemannian manifold \((M^n,g)\) of class \(C^\infty\) is introduced. \((M^n,g)\) is said to be conformally flat if locally there exists a function such that \(g=e^u \tilde g\) where \(\tilde g\) is a flat metric on \(E^n\).