Curvatures of complete hypersurfaces in space forms

In this paper we investigate three-dimensional complete minimal hypersurfaces with, 0). We prove that if constant Gauss-Kronecker curvature in a space form M-4(c) (c less than or equal to 0 the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space E-4 and the hyperbolic space H-4(c) with vanishing Gauss-Kronecker curvature are also presented. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M-4(c) (c less than or equal to 0) if the scalar curvature is bounded from below. In particular, we classify complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M-4(c) (c less than or equal to 0) if the scalar curvature r satisfies r greater than or equal to 2/3c.