Complete submanifolds in \(E^{n+p}\) with parallel mean curvature (Q1067620)

Complete submanifolds in Euclidean space can be classified completely by a pinching condition on the second fundamental form. More precisely the author proves : Theorem 1. Let M be an n-dimensional (n\(\geq 3)\) complete and connected submanifold in \(E^{n+p}\) with parallel mean curvature vector. If the second fundamental form \(\sigma\) of M satisfies the condition \(| \sigma |^ 2\leq (trace \sigma)^ 2/(n-1),\) then M is an n-sphere, an n-plane or a circular cylinder \(S^{n+1}\times E\). Theorem 2 and Corollary in this paper are not complete. As an example of incompleteness of these propositions, stands the Veronese surface which satisfies the hypotheses of these propositions, but is not contained in the results.