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I, ask my question as a comment in this post. Without answer I post a more detailed version.

I am looking for a reference about $C^\infty$ Nash isometric embedding for non compact manifold.

My question is what are exactly the hypothesis needed on a complete manifold $M$ in order to be properly isometrically embedded into some $\mathbb{R}^n$ (I am not very interested by the optimal dimension $n$) and which admits a nice projection (or equivalently a tubular neighborhood of fixed width). Any modern reference will appreciated. Thx in advance

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    $\begingroup$ There are two very different Nash embedding theorems; One for $C^1$ embeddings and one for $C^\infty$ embeddings. Which one are you interested in? $\endgroup$
    – Piotr Hajlasz
    Commented Oct 15, 2018 at 13:25
  • $\begingroup$ Good remark, I have edited the post with $C^\infty$ in fact at least $C^2$ should be good, I just want to preserve curvature... $\endgroup$
    – Paul
    Commented Oct 15, 2018 at 13:52
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    $\begingroup$ The answer mathoverflow.net/a/124878/26935 shows that it is not possible in general to get a a tubular neighborhood of fixed width. $\endgroup$
    – Peter Michor
    Commented Oct 15, 2018 at 18:15
  • $\begingroup$ Yes, indeed I was suspected such a problem. But can I assume a lower bound on the curvature and the injectivity radius, for instance? $\endgroup$
    – Paul
    Commented Oct 15, 2018 at 18:33

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Your nice projection is usually called positive reach. In order to have it, one has to have both curvature bounds and a lower bound on injectivity radius. Plus the volume growth must be at most polynomial. Say the Lobachevsky plane does not admit a smooth embedding positive reach; see this question: Does a Riemannian manifold with bounded geometry... I suspect that there are more conditions.

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