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Let $X$ a codimension 1 smooth submanifold of the n-dimensional smooth manifold $Y$. Assume $Y$ is oriented. We want to show that $X$ is orientable if and only if it admits a global smooth normal vector field (in Y).

How can we prove this? I have no idea how to even begin...

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    Also, what do you mean by "normal vector field"? Are you assuming that $Y$ has a Riemannian metric? Or do you just mean that the vector field is transverse to $X$?2012-12-10
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    Yes, I meant $X$, sorry for that.2012-12-10
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    Um, I guess so, although I don't know too much about Riemannian metrics. Basically I'm thinking of $Y$ being embedded in some Euclidean space.2012-12-10
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    The concept of "normal" only makes sense when a Riemannian metric is present; the one from euclidean space should work. The concept of "transverse" always makes sense. Anyway, where are you getting this problem from?2012-12-10
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    Guillemin and Pollack problem 18 p. 1062012-12-10
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    What's your definition of "oriented"?2012-12-10
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    That you can smoothly orient the tangent space at each point; i.e. for each point there is a local parametrization around it such that its differential at each point preserves orientation.2012-12-10

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