Let $M$ be a n-dimensional closed submanifold in $\mathbb{R}^m.$ I was looking for a version of Sobolev's lemma saying that for $f \in {W}^{k,2}$ we find a representative of $f \in C^{r}$ satisfying $k>r + \frac{n}{2}.$ Does this still hold for manifolds or is this no longer true in this context?


I should add that maybe the Sobolev lemma does not exactly hold in this version, but I assumed that this is the most probable one for it to hold.



Sobolev's lemma on manifolds
This follows from its counterpart in the Euclidean space by local charts. 


If you want to have an estimate on the derivative $D^r f$, then you should impose some bounds on derivatives of the curvature and a lower bound on the injectivity radius.


