Translation hypersurfaces whose curvature depends partially on its variables

Abstract

We investigate translation hypersurfaces in the (n+1)-dimensional Euclidean space whose Gauss-Kronecker curvature depends on either its first p or on its second q variables. These hypersurfaces are the graph of the sum of two functions in p and q independent variables respectively and with n=p+q. We prove that under this condition, the Gauss-Kronecker curvature is constant zero. On other side, if the mean curvature is nowhere zero and it depends on either its first p or on its second q variables, we get again that the Gauss-Kronecker curvature is constant zero.