The generalized Lu rigidity theorem for submanifolds with parallel mean curvature

Abstract

Let M be an n-dimensional oriented compact submanifold with parallel mean curvature in the unit sphere $$S^{n+p}$$ . Denote by H and S the mean curvature and the squared length of the second fundamental form of M, respectively. We obtain a classification theorem of M if it satisfies $$S+\lambda _2\le \alpha (n,H)$$ , where $$\lambda _{2}$$ is the second largest eigenvalue of the fundamental matrix and $$\alpha (n,H)$$ is defined as in Theorem B.