The topological rigidity theorem for submanifolds in space forms

Let $M$ be an $n(\ge 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\ge 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2\sigma_n}{2n-\sigma_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. Here ${\sigma }_n=sgn(n-4)((-1{)}^n+3)$, and $sgn(\cdot )$ is the standard sign function. This improves our previous sphere theorem \cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new topological sphere theorems for submanifolds with pinched scalar curvature and Ricci curvature.