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I need to prove that, being $S$ a regular surface,

If exists a two-form not null in $S$, so $S$ is orientable.

Here, I need to use estricly the basic definition for orientability, ie, any two parametrizations $\psi:U\to \psi(U)\subset S$ and $\varphi:V\to \varphi(V)\subset S$ are s.t. $\psi(U)\cap\varphi(V)=\emptyset$ or the jacobian of the parametrization changing is always $>0$.

I was trying to follow this line of thought:

In a point of $S$ bellowing to two parametrizations, if I have some $w=w_1\wedge w_2$ not null, so when I take the vectors of the basis induced by these parametrizations, $w$ may be positive or negative, so the basis have same orientation.

Maybe it can be right, but in this case the two-form must be not null in any point. I'd like to know how can I do this to a two-form not null (ie, not null in all points).

Many thanks in advance.

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In this context, "not null" means "not null at any point", not "not null at every point" (the result would not be true with the latter meaning). In other words, you are meant to assume that $w$ is nonzero at every point, and so your argument (with more details filled in) works and you are done.

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  • $\begingroup$ Many thanks, Eric. $\endgroup$ Oct 13, 2019 at 14:25

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