In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the ( n+ p)-dimensional Euclidean space E n + p are the totally geodesic Euclidean space E n , the totally umbilical sphere S n ( c) or the generalized cylinder S n − 1 ( c) × E 1 if the second fundamental form h satisfies < h>2≤ n 2| H|2/ ( n− 1).