Existence of an isometric embedding into Euclidean space with bounded second fundamental form 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.
 A: The Gauss Equations are able to give you some coarse information immediately, see for instance the wikipedia article http://en.wikipedia.org/wiki/Gauss–Codazzi_equations here. For instance, if  $M$ is $n$ dimensional, and you have such an isometric embedding, then about some point $p \in M\cap \mathbb{R}^N$ there is a basis of $N-n$ vectorfields, say $\{e_i\}$ normal to $M$ in $R^N$. Then for each $e_i$ one gets an operator $X \to \nabla_X e_i$ so in this sense there are $N- n$ second fundamental forms. Now the gauss equation writes the curvature of $M$ in terms of the sum of these operators, so if $M$ has unbounded curvature, so must this sum. In particular if $N = n + 1$  then a necessary condition for an embedding (with bounded second fundamental form) is that the curvature of $M$ is bounded.
A: 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.
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.
