Transformations for hypersurfaces with vanishing Gauss-Kronecker curvature (Q2568864)

The authors provide a method of constructing families of hypersurfaces of a space form with zero Gauss-Kronecker curvature, from a given such hypersurface, based on Ribaucour transformations. Applications provide a \(1\)-parameter family of complete, non-cylindrical hypersurfaces of \(\mathbb R^4\), with zero Gauss-Kronecker curvature, a \(5\)-parameter family of compact Dupin hypersurfaces of \(S^4\), with vanishing Gauss-Kronecker curvature, infinite families of hypersurfaces of \( \mathbb R^n+1\) and of the hyperbolic space \(H^4\), with flat Gauss-Kronecker curvature.