Let $M$ be a compact Riemannian manifold without boundary.
Suppose $f \in H^s(M,M)$, where $H^s$ denotes the ($L^2$-based) Sobolev space. Assume $s > n/2 + 1$, so that by the Sobolev Embedding Theorem all maps in $H^s(M, M)$ are at least $C^1$.
Problem: If $f$ is a $C^1$ diffeomorphism (in addition to being $H^s$), can you show that $f^{-1}$ is also in $H^s(M, M)$?
I know that something along the lines of differentiating $f \circ f^{-1}(x) = x$ and seeing what bounds can you get from there should work, but I can't quite do the details.
Thank you very much.