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After googling this fact (which is used in Bott and Tu to show the existence of good covers of manifolds) I've gotten the impression this is somewhat difficult to prove. But I also came across this homework problem (problem 0) with a hint: http://www.math.columbia.edu/~thaddeus/geometry/hw10.pdf

which gives me the impression that it is maybe not so difficult as I thought. If anyone has any thoughts about how to prove this result I'd be very appreciative. Thanks for your time.

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Related thread (but without a self-contained proof): mathoverflow.net/questions/4468/… – user31373 Aug 20 '12 at 1:39
Thanks, LVK. Yeah that was something that led me to believe that the proof was long and complicated which makes me puzzled by the fact that it was assigned as a homework problem (albeit for extra credit and at Columbia) – Carl Wienecke Aug 20 '12 at 1:45
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I suppose that the suitable $f$ is $f(q)=\mathrm{dist}(q,\partial U)$ (or a smooth version of it, i.e., a $C^\infty$ positive function comparable to the distance. Such a function is constucted, e.g., in Stein's 1970 book Singular Integrals). This gives you the flow of $fX$ which stays within $U$, and the flow of $X$ on $\mathbb R^n$. It remains to think of a neat way to map one onto the other. – user31373 Aug 20 '12 at 1:49
How about this: pick a small sphere $S\subset U$ centered at $0$. For each point $x\in S$ let $n_x$ be the outward unit vector. Let $\gamma_x$ be the solution of $\dot\gamma=fX(\gamma)$ with $\gamma(0)=x$, $\dot \gamma(0)=n_x$. Also, let $\Gamma_x$ be the solution of $\dot\Gamma=X(\Gamma)$ with $\Gamma(0)=x$, $\dot \Gamma(0)=n_x$. Now map $\gamma_x(t)$ to $\Gamma_x(t)$. – user31373 Aug 20 '12 at 1:59
Thanks again. It will take me a bit to digest this. – Carl Wienecke Aug 20 '12 at 2:16
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