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.
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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.
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I don't know what a manifold or a non-vanishing differential form are... do you know another way of proving it? – Rachel May 26 '14 at 16:36
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Oh. What exactly is your definition of "orientable"? – May 26 '14 at 16:40
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my definition is that S is orientable if there is a differentiable function $N: S \rightarrow R^3$ such that $N(p)$ is orthogonal to the tangent plane on $p$ of $S$ for all p in S – Rachel May 26 '14 at 16:44