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](https://mathoverflow.net/questions/57109/).


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.



Existence of an isometric embedding into Euclidean space with bounded second fundamental form
The curvature tensor can be expressed in terms of second fundamental form.
Therefore bounded curvature is a necessary condition.


Yet injectivity radius has to be bounded below.


These two conditions might be sufficient.
If we assume a bit better regularity (say a bound on covariant derivatives of the curvature tensor), then this could be proved along the same lines as the [Nash embedding theorem](http://en.wikipedia.org/wiki/Nash_embedding_theorem#Ck_embedding_theorem).


**Postscript.** In the formulation, you had to say what you mean by "second fundamental form".
Most people think it is only defined for hypersurfaces, but you mean a quadratic form on tangent space with values in the normal space.


