On the minimal submanifolds in CP^m(c) and S^N(1)

Abstract

Let M be an n-dimensional compact totally real submanifold minimally immersed in CP^m(c). Let σ be the second fundamental form of M. A known result states that if m=n and |σ|²{≤}(n(n+1)c)/(4(2n−1)), then M is either totally geodesic or a finite Riemannian covering of the unique flat torus minimally imbedded in CP²(c). In this paper, we improve the above pinching constant to (n+1)c/6 and prove a pinching theorem for |σ|² without the assumption on the codimension. We have also some pinching theorems for δ(u):=|σ(u, u)|², u∈UM, M→CP^m(c) and the Ricci curvature of a minimal submanifold in a sphere. In particular, a simple proof of a Gauchman's result is given.