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Let $B$ is a ball in $\mathbb R^n$, let $f$ is a diffeomorphism, mapping $B$ to a bounded region in $\mathbb R^n$. Is there a diffeomorphism $g$ of $\mathbb R^n$ such that $g=f$ when restricted to $B$?

Remark: this is a followup of the question The extension of diffeomorphism where the problem for small $B\subset Dom(f)$ has been solved. Can the problem be solved for $B=Dom(f)$?

Igor
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1 Answers1

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In the question that you refer to, the problem has not been solved for small $B$, but locally in $B$. The global problem you pose cannot be solveable in general, for example in the case that $f$ is a diffeomorphism from $B$ onto $\mathbb R^n$ (which always exists, regardless of the size of $B$).

Andreas Cap
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  • Thanks for contra example. I am mainly interested by the case when the image is a bounded region and will edit the question accordingly. – Igor May 19 '18 at 10:53