  In this paper, we obtain several new intrinsic and extrinsic differential
sphere theorems via Ricci flow.
  For intrinsic case, we show that a closed simply connected $n(\ge
4)$-dimensional Riemannian
  manifold $M$ is diffeomorphic to $S^n$
  if one of the following
  conditions holds pointwisely: $$
  (i)\ R_0>\left(1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right)K_{max};\quad
  \ (ii)\
\frac{Ric^{[4]}}{4(n-1)}>\left(1-\frac{6(\sqrt{10}-3)}{n-1}\right)K_{max}.$$
  Here $K_{max}$, $Ric^{[k]}$ and $R_0$ stand for the maximal sectional
curvature,
  the $k$-th weak Ricci curvature and
  the normalized scalar curvature.
  For extrinsic case, i.e., when $M$ is a closed simply connected $n(\ge
4)$-dimensional submanifold immersed in $\bar{M}$.
  We prove that $M$ is diffeomorphic to $S^n$ if it satisfies some pinching
curvature conditions. The only involved extrinsic quantities in our pinching
conditions are the maximal sectional curvature $\bar K_{max}$ and the squared
norm of mean curvature vector $\vert H\vert^2$. More precisely,
  we show that $M$ is diffeomorphic to $S^n$
  if one of the following conditions
  holds: \begin{itemize}
  \item[(1)] $R_0\ge \left(1-\frac{2}{n(n-1)}\right)\bar{K}_{max}
+\frac{n(n-2)}{(n-1)^2}\vert H\vert^2$, and strict
  inequality is achieved at some point;
  \item[(2)] $\dfrac{Ric^{[2]}}{2}\ge (n-2)\bar K_{max}+\frac{n^2}{8}\vert
H\vert^2,$ and strict
  inequality is achieved at some point;
  \item[(3)] $\dfrac{Ric^{[2]}}{2} \ge\frac{n(n-3)}{n-2}\left(\bar
K_{max}+\vert H\vert^2\right),$ and strict
  inequality is achieved at some point.
  \end{itemize}
  It is worth pointing out that, in the proof of extrinsic case, we apply
suitable complex orthonormal frame
  and simplify the calculations considerably.
  We also emphasize that both of the pinching constants in (2) and (3) are
optimal for $n=4$.
