The Gauss-Kronecker curvature of hypersurfaces is an important geometric invariant.In this paper, we investigate complete minimal hypersurfaces in the Euclidean space R~4 with vanishing Gauss-Kronecker curvature. This is related with the following well-known question: Is it true that any complete minimal hypersurface with vanishing Gauss-Kronecker curvature in R~4 should be a cylinder over a minimal surface in R~3? A partial answer to this question is given by the following theorem in [1]: If f : M~3→R~4 is a complete minimal hypersurface with Gauss-kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then f(M~3) splits as a Euclidean product L~2×R, where L~2 is a complete minimal surface in R~3 with Gaussian curvature bounded from below.In this paper, by checking some natural examples, we show that some conditions in the above theorem are in fact not necessary. |