Local diffeoorphism and orientability of surfaces

Question

I need some help to prove this: Let $S_2$ be an orientable regular surface and $f : S1 \rightarrow S2$ be a local diffeomorphism. Then $S_1$ is an orientable surface. Thanks.

Answer 1

A manifold is orientable if and only if it admits a volume form (a non-vanishing differential form). Since $S_2$ is orientable, it has a volume form $\omega$. To show that $S_1$ is orientable, consider the pullback of $\omega$ by $f$. You should be able to show that this is non-vanishing at every point using the fact that $f$ is a local diffeomorphism.