I am Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma \in (0,1)$, where $M$ is a compact manifold.

Any references are appreciated.

PS I am also looking for a reference that gives interpolation inequalities that justify (for when $u_\epsilon \in C(I, H^k) \cap C^1(I,H^{k-1})$), that $\{ u_\epsilon : \epsilon \in (0,1] \} $ is bounded in $C^{\sigma}(I,H^{k-\sigma}(M))$.

where $k$ is some nonnegative integer, $\sigma$ as above.

Thanks.