I, ask my question as a comment [in this post][1]. 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



  [1]: https://mathoverflow.net/questions/311630/nash-isometric-embedding-for-noncompact-manifolds