  In this paper, we consider the rigidity for an $n(\geq 4)$-dimensional
submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb
M}^{n+p}_c$ when the integral Ricci curvature of $M$ has some bound. We prove
that, if $c+H^2>0$ and $\|\mathrm{Ric}_{-}^\lambda\|_{n/2}< \epsilon(n,c,
\lambda, H)$ for $\lambda$ satisfying $ \frac{n-2}{n-1} (c+H^2) < \lambda \le
c+H^2$, then $M$ is the totally umbilical sphere
$\mathbb{S}^n(\tfrac{1}{\sqrt{c+H^2}})$. Here $H$ is the norm of the parallel
mean curvature of $M$, and $\epsilon(n,c,\lambda, H)$ is a positive constant
depending only on $n, c,\lambda$ and $H$. This extends some of the earlier work
of [15] from pointwise Ricci curvature lower bound to inetgral Ricci curvature
lower bound.

Differential Geometry