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