suppose we are given a complete, non-compact Riemannian manifold $(M,g)$. Is it possible to embed (or just immerse) it isometrically into some $R^N$, such that the second fundamental form is bounded? Maybe under some additional assumptions on our manifold and/or metric on it?

This question is a follow-up question to this one: Riemannian manifold of bounded geometry has a normal bundle of bounded geometry.

Thanks, Alex

edit: With "second fundamental form" I mean the quadratic form on the tangent space defined via taking the covariant derivative in $R^N$ and then orthogonally project it onto the normal bundle. So it is defined not only for hypersurfaces.

Anton Petrunin claimed in his answer that bounded curvature of $M$ and bounded injectivity radius are sufficient for the existence of such an embedding. I this is true, I would be grateful for a reference.