A generalization of Gauchman's rigidity theorem (Q996156)

\textit{H. Gauchman} [Trans. Am. Math. Soc. 298, 779--791 (1986; Zbl 0608.53056)] proved that if \(M\) is an \(n\)-dimensional closed minimal submanifold in the unit sphere \(S^{n+p}\) and if \(\sigma(u)\leq\tfrac 13\) for any unit vector \(u\in TM\), where \(\sigma(u)=\|h(u,u)\|^2\) for the second fundamental form \(h\) of \(M\), then either \(\sigma(u)\equiv 0\) and \(M\) is a totally geodesic sphere, or \(\sigma(u) \equiv\tfrac 13\). In the present paper this result is generalized to the case of \(n\)-dimensional closed submanifolds \(M\) with parallel mean curvature vector field of norm \(H\) in \(S^{n+p}\). It is proved that if \(\sigma(u)\leq\tfrac 13\), then either \(\sigma(u)=H^2\) and \(M\) is a totally umbilical sphere, or \(\sigma(u) =\tfrac 13\). For the last case a geometrical classification is given using embeddings of the projective planes over the real numbers, complex numbers, quaternions and octonions into a sphere. For the case \(H=0\) this classification goes back to Gauchman's.