Let $M$ be a connected orientable smooth manifold.
Is it true that $M$ must have only 2 orientations?
If yes, why?
differential-geometry
asked 2012-02-15
user id:8390
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See [this question](http://math.stackexchange.com/questions/15605/how-should-i-think-about-what-it-means-for-a-manifold-to-be-orientable) for more details. The answer is yes. Why just two orientations? Because you are picking out compatibility based on Jacobian determinants being positive. 2 choices since "positive, negative" is a list of 2 things. Or another way to look at it: Because $\mathrm{GL}_n(\mathbb{R})$ has two connected components (matrices with positive determinants and those with negative determinants). – 2012-02-15
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Alternatively, two orientations agree on an open set and disagree on an open set. – 2012-02-15
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@MarianoSuárez-Alvarez sorry I didn't understand what you mean – 2012-02-15
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@Jr.: What he means is that if I have two arbitrary orientations, then on any given open set, they either agree or they disagree. – 2012-02-15
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