Here is an assignment problem:
$f:\mathbb{S}^2 \longrightarrow \mathbb{S}^2$ is smooth and surjective. Prove $\exists$ open subset $ U $ of $\mathbb{S}^2$, such that $f|_U$ is a diffeomorphism.
I've tried to relate it to covering maps but failed.
Can someone help me with this problem? Thanks for help.
Use Sard's Theorem. It gives a regular value of $f$, and since $f$ is surjective, this means that the regular value isn't vacuous (i.e. it's not just something not in the image of $f$). This gives us a point at which the derivative of $f$ is an isomorphism. Then hit it with the Inverse Function Theorem to get your diffeomorphism.
http://en.wikipedia.org/wiki/Inverse_function_theorem