Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly bounded together with their first two derivatives. Or do we need assumptions on the curvature of $M$?
If the curvature is bounded, then it is (somewhat) well-known that $C^\infty_c(M)$ is dense in $W^{2, p}(M)$, hence in particular the space $C^2_b(M)\cap W^{2,p}(M)$. But what if we drop this assumption?
