\(L_{n/2}\)-pinching theorems for submanifolds with parallel mean curvature in a sphere (Q1890835)

Two pinching theorems for \(n\)-dimensional submanifolds with parallel mean curvature vector in \(S^{n + p} (1)\) are proved. Both characterize totally umbilical submanifolds and depend on the estimation of \[ \int_ M (S - nH^ 2)^{n/2}, \] where \(S\) is the square of the length of the second fundamental form and \(H\) is the mean curvature.