  Let $M$ be an $n(\geq 4)$-dimensional compact submanifold
  in the simply connected space form $F^{n+p}(c)$ with constant curvature
$c\geq 0$, 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.
