PINCHING THEOREMS FOR TOTALLY REAL MINIMAL SUBMANIFOLDS OF C{P^n}(c)

Abstract

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