Remarks on complete deformable hypersurfaces in \(\mathbb{R}^ 4\) (Q1336251)

It is shown that, for each pair \(\{k_ 1(u), k_ 2(v)\}\) of smooth functions on \(\mathbb{R}\) with some conditions, there exists a family of complete nonruled deformable hypersurfaces \(M(\lambda, k_ 1, k_ 2)\), \(-1/2 < \lambda < 1/2\), in Euclidean space \(\mathbb{R}^ 4\) with rank \(\rho = 2\) almost everywhere. This is an answer to one of the problems posed by \textit{M. Dajczer} and \textit{D. Gromoll} [J. Differ. Geom. 31, No. 2, 401- 416 (1990; Zbl 0686.53009)].