Submanifolds of a Euclidean space with parallel mean curvature vector (Q2710513)

Let \(M^n\) be a compact, positively curved submanifold with parallel mean curvature vector \(H\) in a Euclidean space \(\mathbb{R}^{n+p}\), \(p\geq 2\). The author proves the following result. If the normal connection is flat and for each unit normal vector field \(N\), the Weingarten map \(A_N\) satisfies \(\|A_N\|^2 <2n\|H\|^2\), then \(M^n\) is a sphere of constant curvature \(\|H\|^2\).