TITLE: Submanifolds of $\mathbb{R}^N$ whose local charts have uniformly bounded derivatives
QUESTION [9 upvotes]: Working on a problem in Differential Geometry, which is quite far away from my area of expertise, I was recently led to consider the class of those smooth, $n$-dimensional embedded submanifolds $M \subset \mathbb{R}^N$ such that the following condition is satisfied:

the manifold $M$ possesses an oriented atlas $\{(U_{\alpha}, \, \phi_{\alpha})\}$, where $\phi_{\alpha} \colon U_{\alpha} \stackrel{\cong}{\longrightarrow} V_{\alpha} \subset \mathbb{R}^{n}$, such that, denoting by $i_{\alpha} \colon U_{\alpha} \hookrightarrow \mathbb{R}^N$ the inclusion map, the composition $$i_{\alpha} \circ \phi_{\alpha}^{-1} \colon \, V_{\alpha} \longrightarrow \mathbb{R}^N$$
  has bounded partial derivatives up to order $k$, uniformly on $\alpha$.

My question is now the following:

Does the condition above have a name? In this case, is there any characterization of the manifolds $M$ satisfying it in terms of the usual Riemannian invariants (for instance, curvature)? 

Any answer or reference to the relevant literature will be greatly appreciated.

REPLY [4 votes]: Posting this a an answer since it's too long for a comment.
I have a feeling that the question as formulated might be ill posed, or at least not quite what you'd expect in the sense that all embedded manifolds $M \subset \mathbb{R}^N$ satisfy it, as already noted by Deane Yang and Fan Zheng: you didn't fix an atlas on $M$, so you can always stretch charts.
Possibly a better condition would be to consider if the embedding $\iota: M \to \mathbb{R}^N$ is a $C^k$ bounded map in the sense of bounded geometry, e.g. with Definition 2.9 from my book (that Thomas Rot already linked to) and with $M$ having its metric induced by the Euclidean metric of $\mathbb{R}^N$. Modulo some issues with loss of smoothness due to curvature conditions in (that Rbega seems able to address) this should be equivalent to $C^k$ boundedness of $M$ in terms of local graphs.
The fact that your original condition would be satisfied by any $C^k$ embedding $M$ seems to match with the fact that any smooth manifold possesses a metric of bounded geometry (see Greene, "Complete metrics of bounded curvature on noncompact manifolds", Arch. Math. (31) 1978), as mentioned by Deane Yang.