A rigidity theorem for submanifolds with parallel mean curvature in a sphere
It is proved that if \(M^ n\) is an \(n\)-dimensional compact submanifold with parallel mean curvature in a unit sphere \(S^{n+p}(1)\), and \(S\leq C(n,p,H)\), where \(H\) and \(S\) are the mean curvature and the square norm of the second fundamental form of \(M\) respectively, and \(C(n,p,H)\) is a positive constant depending on \(n\), \(p\) and \(H\), then either \(M\) is the totally umbilical sphere \(S^ n({1\over \sqrt{1+H^ 2}})\), the standard immersion of the product of two spheres or the Veronese surface in \(S^ 4({1\over \sqrt{1+H^ 2}})\).
scientific article; zbMATH DE number 493370