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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 $X$? –  Jesse Madnick Dec 10 '12 at 2:19
    
Yes, I meant $X$, sorry for that. –  Bernard Dec 10 '12 at 2:22
    
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. –  Bernard Dec 10 '12 at 2:23
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Guillemin and Pollack problem 18 p. 106 –  Bernard Dec 10 '12 at 2:27
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What's your definition of "oriented"? –  Jason DeVito Dec 10 '12 at 2:39

1 Answer 1

Hint: For $p\in X$, let $U_p$, with coordinates $(x_1,...,x_n,t)$, be a slice chart around $p$ (meaning around $p$, $X$ corresponds to points where $t=0$).

Now, given your normal vector field $V$, orient $X\cap U_p$ by declaring the ordered basis $\{\partial_{x_i}\}$ to be positively oriented iff the ordered basis $\{\partial_{x_i}, v\}$ is positively oriented in $Y$.

Conversely, if $X$ is oriented, define $V = \partial_t$.

I'll leave it to you to prove that all this works.

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I see... but don't you need to pick the slice chart so that its differential (on TY) preserves orientation? How can we do that? –  Bernard Dec 10 '12 at 4:45
    
On each chart, you pick an individual orientation. If you can pick them so that on overlaps they agree, you've oriented $X$. I'm telling you how to pick them on slice charts - you still have to verify that on overlaps, the choice agrees. –  Jason DeVito Dec 10 '12 at 5:08
    
I was asking about the converse. –  Bernard Dec 10 '12 at 5:44
    
Oh, I see! You start with slice charts whose differential (on $TX$) preserves orientation. Then the question is, do you pick $V = \partial_t$ or $V = -\partial_t$? Pick it in such a way that $\{\partial_{x_i}, \pm\partial_t\}$ is positively oriented (in $TY$). (I definitely could have given a better hint for that part.) –  Jason DeVito Dec 10 '12 at 13:18
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@gofvonx: $V$ is a coordinate vector field and coordinate vector fields are always smooth: If $f$ is smooth then, $\partial_i f = \frac{d}{dx^1} f\circ x^{-1}$ is the derivative of a compositioin of smooth functions. (You pick $V = \partial_t$ or $V = -\partial_t$ once for the whole chart: Using an argument relying on the disconnectedness of $GL_n$, one shows the choice at one point uniquely determines the choice at every point.) –  Jason DeVito Jun 3 '13 at 14:33

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