On some isotropic submanifolds in spheres (Q1609943)

In a joint paper with \textit{P. Verheyen} in Math. Ann. 269, 417-429 (1984; Zbl 0536.53053), \textit{B.-Y. Chen} proved the following result: Let \(M^n\) be a connected compact isotropic submanifold of a sphere \(S^{n+p}\) of constant curvature \(\widetilde c\). If \(M^n\) has constant mean curvature \(H\), and if the sectional curvature function \(K\) of \(M^n\) satisfies \(K \geq (H^2+\widetilde c)/2\), then \(M^n\) is totally umbilical in \(S^{n+p}\). In the present paper the author constructs counterexamples to the above result.