Pinching theorems for totally real minimal submanifolds of \(CP^ n(c)\) (Q1113471)

Let h be the second fundamental form of a compact minimal totally real submanifold M of a complex space form \({\mathbb{C}}P^ n(c)\) of holomorphic curvature c. For any \(u\in TM\), set \(\delta (u)=\| h(u,u)\|^ 2.\) It is proved that if \(\delta\) (u)\(\leq c/12\) for any unit vector \(u\in TM\), then either \(\delta\) (u)\(\equiv 0\) (i.e. M is totally geodesic) or \(\delta\) (u)\(\equiv c/12\). All compact minimal totally real submanifolds of \({\mathbb{C}}P^ n(c)\) satisfying \(\delta\) (u)\(\equiv c/12\) are determined.