Ramified Covering Maps and Stability of Pulled-back Bundles

Abstract Let $f\,:\,C\,\longrightarrow \,D$ be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ the maximal semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, induced homomorphism $f_*\,:\, \pi _1^{\textrm{et}}(C)\,\longrightarrow \, _1^{\textrm{et}}(D)$ étale fundamental groups surjective). prove pullback $f^*E\,\longrightarrow C$ stable for every vector bundle $E$ on $D$ and only ramified.