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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): http://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbb2012-08-20
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    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)2012-08-20
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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.2012-08-20
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    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 $\Gamma_x(t)$.2012-08-20
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    Thanks again. It will take me a bit to digest this.2012-08-20
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    One answer in [this related thread](http://math.stackexchange.com/q/165629) claims to give a fairly complete proof (I haven't read it myself). Some references are also given in that thread.2012-08-20
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    Thanks t.b. A little more googling turned up this http://mathforum.org/kb/thread.jspa?forumID=13&threadID=1962874&messageID=6777047 which to my amazement seems to be an extremely elementary proof of the statement. I feel like I must be missing something.2012-08-20
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    @CarlWienecke The mathforum thread does not look like a proof to me. In the thread linked by t.b. I like the answer by Bruce Evans. It implicitly uses the idea of flow by radial fields, but states it in a more elementary language. Computing the flow of a stationary **radial** field boils down to integrating a scalar function, and this is what Bruce does. I just gave +1 to his answer.2012-08-21
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    Thanks LVK I'll have a look at that answer. The mathforum thread does not provide a full proof but (I believe) I was able to trivially verify smoothness, bijectivity, and nonsingularity of the function $x \mapsto x \int_0^1 \frac{dt}{f(xt)}$2012-08-21
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    @CarlWienecke If the verification was easy, perhaps you can post it as an answer?2012-09-01

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