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If $M \subset \mathbb R^n$ is a compact smooth manifold with boundary, and ${M_\varepsilon }$ is the closed $\varepsilon$-neighborhood of $M$ in $\mathbb R^n$, then whether for sufficiently small $\varepsilon$, ${M_\varepsilon }$ is a smooth manifold?

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Yes. This follows from the tubular neighborhood theorem, which you may find in many differential geometry/topology books. See e.g.

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Wouldn't this be rather a comment than an answer? – t.b. Nov 13 '11 at 0:46
@t.b.: what's the difference? – Damian Sobota Nov 13 '11 at 1:04
@Damian: look at an answer e.g. by Arturo Magidin and look at this which basically says: it is a consequence of the tubular neighborhood theorem, look, here's what Google gives you. Then you should see a fundamental difference (in quality, content, helpfulness, etc...) – t.b. Nov 13 '11 at 1:11
I wouldn't have used such a strong word. The link didn't work for me, (it still doesn't), so I stripped out the 's' from https and the rest of the Google noise, sorry that I overlooked the &tbm=bks modifier. – t.b. Nov 13 '11 at 1:27
Now it works. Thank you. – t.b. Nov 13 '11 at 1:45

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