Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry).
Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions $u \in H^1(M)$: $$\nabla_g f(u) = f'(u)\nabla_g u \quad\in L^2(M)?$$
Seems to me true by density of smooth functions but I wonder if I have missed anything.
Since $f'$ is bounded, it is clear that $\nabla_gf(u)$ is in $L^2$ if $\nabla_gu$ is. It just remains to check that your formula indeed gives the weak gradient of $f(u)$. Weak derivatives on manifolds are often define on the coordinate charts (it suffices to consider test functions supported in a single chart), and it does not matter for such calculations whether $M$ is compact or not. It sounds like you already know the result is true on compact manifolds, and the noncompact case is not different.
