Let $M$ be a $C^k$ manifold for some integer $k$. How does one show that $$H^1(M) \subset H^s(M)$$ is continuous, where $s \in (0,1)$?
I was planning to pull back the norms onto a subset $D_i$ of Euclidean space via chart maps, and apply the result for open sets in Euclidean space and transfer back. But my problem is that the open set the chart map pulls back to, $D_i$, I need it to be at least a Lipschitz domain to apply the result for Euclidean space. How do I ensure that it is Lipschitz? AFAIK I know nothing about the $D_i$.
Or is it the case that a $C^k$ manifold implies that the chart maps are actually $C^k-$diffeomorphism? In this case, maybe we can say that the chart maps map from an open ball around the boundary onto $D_i$, and because it's a diffeomorphism, perhaps $D_i$ also shares the same smoothness as the open ball?
If $M$ is $C^k$, $k\ge 1$, then the charts are also $C^k$. Thus you can prove your inclusion as you suggest.