Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $a,b \in R^n$.

Show that there is a diffeomorphism of $R^n$ carrying $a$ to $b$ which is the identity outside of an open set.

I think I have a proof of this using integral curves but I would like to see if there's another proof; and more specifically if such a diffeomorphism can be described explicitly (using bump functions or otherwise).

Any help appreciated.

share|cite|improve this question
up vote 2 down vote accepted

Let $\Psi$ be a bump function such as the one described here:

Let $f(x)=ub+(1-u)x$, where $u=\Psi(|x-a|^2)/\Psi(0)$.

[EDIT] Bernard has a good point, which is that it's not obvious that $f^{-1}$ is smooth. In fact, I think the real issue is whether or not $f$ is invertible. So here's a patched-up version of the above map.




The specific bump function $\Psi(z)$ given in the WP article is clearly smooth, and has a smooth inverse, everywhere except possibly at $z=0$ and $z\ge 1$. The factor of $1/1000$ ensures that $f$ is always invertible, since $\Psi$ never has a slope as large as 1000. I don't think we have an issue at $z=0$, since the constancy of $\Psi$ there just means that $f$ looks like an identity map at $a$. Ditto at $z\ge1$.

share|cite|improve this answer
OK, clearly f is smooth, but how would you show that $f^{-1}$ is smooth? – Bernard Oct 12 '12 at 3:38

Find an open ball containing both points, then since the line joining $a$ and $b$ is closed, we can use Urysohns lemma to produce a smooth function $\rho$ equaling 1 on the line and having compact support in our big ball. Now let $X$ be the constant vector field on $R^n$ which points in the same direction as the line joining the two points. so that $\rho(x) X$ is a smooth, compactly supported vector field. The flow of $\rho(x)X$ will give the diffeomorphism you want.

share|cite|improve this answer
Yes, that's the solution I meant above when I mentioned integral curves. I am asking for other solutions (not using flows). – Bernard Oct 12 '12 at 3:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.