The orientation-preserving diffeomorphism of $\mathbb R^n$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:07:23Z http://mathoverflow.net/feeds/question/95423 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95423/the-orientation-preserving-diffeomorphism-of-mathbb-rn The orientation-preserving diffeomorphism of $\mathbb R^n$ Adterram 2012-04-28T09:12:31Z 2012-04-29T00:10:41Z <p>If $f$ is an orientation-preserving diffeomorphism of $\mathbb R^n$ and $K$ is a compact set in $\mathbb R^n$, can we find another diffeomorphism $\tilde f$ of $\mathbb R^n$ such that:</p> <p>(1)$f=\tilde f$ on a neighborhood of $K$. (2)There is a bounded set $V$ and $\tilde f=id$ outside $V$?</p> http://mathoverflow.net/questions/95423/the-orientation-preserving-diffeomorphism-of-mathbb-rn/95425#95425 Answer by Alberto Abbondandolo for The orientation-preserving diffeomorphism of $\mathbb R^n$ Alberto Abbondandolo 2012-04-28T10:16:02Z 2012-04-28T10:16:02Z <p>Yes. You may use the fact that <strong><em>f</em></strong> is isotopic to the identity to see it as the time-1 flow of a time-dependent vector field. Then you just have to modify the vector field so that it vanishes outside from a large ball.</p> http://mathoverflow.net/questions/95423/the-orientation-preserving-diffeomorphism-of-mathbb-rn/95474#95474 Answer by Igor Rivin for The orientation-preserving diffeomorphism of $\mathbb R^n$ Igor Rivin 2012-04-29T00:10:41Z 2012-04-29T00:10:41Z <p>I believe that the result is actually Theorem 5.5 in R. S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. vol. 92 (1959) pp. 125-141, after a bit of massaging.</p>