Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics (Q1651442)

There is a well-known theorem by E. Cartan stating that a conformally flat hypersurface of an \(n\)-dimensional space form, \(n\geq 5\) is a branched channel surface. In the case \(n=4\), the channel hypersurfaces are still examples of conformally flat hypersurfaces. However in that dimension there also exist generic conformally flat hypersurfaces and partial classifications were obtained by two of the authors in previous papers. In the present paper the authors reduce the generic conformally flat hypersurfaces to finding functions \(\varphi\) such that \(g=\cos^2(\varphi) (dx)^2+\sin^2(\varphi) (dy)^2 +(dz)^2\) is a conformally flat metric (Guichard condition). The authors show that these solutions in a natural way give a \(1\) parameter family of surfaces with constant negative curvature \(-1\). A one parameter family of conformally flat 3-metrics with the Guichard condition is determined as evolutions starting from a \(2\)-metric.