Let $M$ be an $n(\geq3)$-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form $F^{ n+p}(c)$ with $c+H^2>0$, where $H$ is the mean curvature of $M$. We prove that if the Ricci curvature of $M$ satisfies $Ric_{ M}\geq(n-2)(c+H^2),$ then $M$ is either a totally umbilic sphere, the Clifford hypersurface $S^{ m}\big(\frac{ 1}{ \sqrt{ 2(c+H^2)}}\big)\times S^{ m}\big(\frac{ 1}{ \sqrt{ 2(c+H^2)}}\big)$ in $S^{ n+1}(\frac{ 1}{ \sqrt{ c+H^2}})$ with $n=2m$, or $\mathbb{ C}P^{ 2}(4/3(c+H^2))$ in $S^7(\frac{ 1}{ \sqrt{ c+H^2}})$. In particular, if $Ric_{ M}>(n-2)(c+H^2),$ then $M$ is a totally umbilic sphere.
Categories: math.DG