Let $M$ be a compact Riemannian manifold without boundary. Is it true that the embedding $$ W^{1,p}(M)\subset L^q(M) $$ is compact for $1\le q < p^*$?
According to wikipedia and this question, the result is true for manifold with boundary. However, I cannot find a reference for the case $M$ is without boundary.
This paper uses the result that I stated but doesn't give a reference so I assume that the result is standard. Could anyone please help me find a nice reference to this?