  Let $X$ be a smooth projective curve of genus $g(X)\geq 1$ over an
algebraically closed field $k$ of characteristic $p>0$ and
$F_{X/k}:X\rightarrow X^{(1)}$ be the relative Frobenius morphism. Let
$\mathfrak{M}^{s(ss)}_X(r,d)$ (resp. $\mathfrak{M}^{s(ss)}_{X^{(1)}}(r\cdot
p,d+r(p-1)(g-1))$) be the moduli space of (semi)-stable vector bundles of rank
$r$ (resp. $r\cdot p$) and degree $d$ (resp. $d+r(p-1)(g-1)$) on $X$ (resp.
$X^{(1)}$). We show that the set-theoretic map
$S^{ss}_{\mathrm{Frob}}:\mathfrak{M}^{ss}_X(r,d)\rightarrow\mathfrak{M}^{ss}_{X^{(1)}}(r\cdot
p,d+r(p-1)(g-1))$ induced by $[\E]\mapsto[{F_{X/k}}_*(\E)]$ is a proper
morphism. Moreover, if $g(X)\geq 2$, the induced morphism
$S^s_{\mathrm{Frob}}:\mathfrak{M}^s_X(r,d)\rightarrow\mathfrak{M}^s_{X^{(1)}}(r\cdot
p,d+r(p-1)(g-1))$ is a closed immersion. As an application, we obtain that the
locus of moduli space $\mathfrak{M}^{s}_{X^{(1)}}(p,d)$ consists of stable
vector bundles whose Frobenius pull back have maximal Harder-Narasimhan Polygon
is isomorphic to Jacobian variety $\Jac_X$ of $X$.
