Global pinching theorems of submanifolds in spheres (Q1607933)

Let \(M\) be an \(m\)-dimensional (\(m \geq 3\)) compact submanifold of the unit sphere \(S^n\) having parallel mean curvature and positive Ricci curvature. Denote by \(\sigma\) the length of the second fundamental form. In the present paper the author shows that there is a constant \(C\), depending only on \(m\), the mean curvature and \(k\), where \((m-1)k\) is the lower bound of the Ricci curvature, such that if \((\int_M \sigma^{m/2})^{2/m} < C\), then \(M\) is a totally umbilical hypersurface in a sphere \(S^{m+1}\). A similar result for surfaces, with Gaussian curvature playing the role of Ricci curvature, is also stated.