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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.

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    $\begingroup$ Yes, this is true. For the n-torus (where one can use Fourier series) a proof is given in Theorem 5.7 of maths.ed.ac.uk/~aar/papers/roeindex.pdf The general case should be in many textbooks on global analysis. $\endgroup$
    – ThiKu
    Commented Mar 9, 2015 at 15:10
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    $\begingroup$ You should do this for an abstract manifold yourself using a partition of unity. $\endgroup$
    – Deane Yang
    Commented Mar 9, 2015 at 15:38
  • $\begingroup$ This is done in Griffiths-Harris's book on Algebraic geometry, along the lines checked by ThiKu. $\endgroup$
    – ACL
    Commented Mar 9, 2015 at 20:24

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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.

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