I, ask my question as a comment [in this post](https://mathoverflow.net/questions/311630/nash-isometric-embedding-for-noncompact-manifolds). 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



Nash embedding for complete manifold
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..](https://mathoverflow.net/q/124840).
I suspect that there are more conditions.


