关键词: Euclidean space, the unit sphere, submanifolds with parallel mean curvature, global rigidity theorem

Abstract: In this paper, we mainly study the global rigidity theorem of Riemannian submanifolds in space forms. Let $M^n(n \geq 3)$ be a complete minimal submanifold in the unit sphere $ S^{n+p}(1) $. For $\lambda \in [0,\frac{n}{2-1/p})$, there is an explicit positive constant $C(n, p, \lambda )$, depending only on $n,p,\lambda $, such that, if $ \int_{M} S^{n/2} {\rm d}M < \infty, \int_{M}(S-\lambda )_{+}^{n/2}{\rm d}M<C(n, p, \lambda), $ then $M^n$ is a totally geodetic sphere, where $S$ denotes the square of the second fundamental form of the submanifold and $f_+=\max\{0,f\}$. Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space $ R^{n+p}$.

Key words: Euclidean space, the unit sphere, submanifolds with parallel mean curvature, global rigidity theorem