It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ for $k=0, 1, 2$.
My question: Is this an if and only if? That is, if $C^\infty_c(M)$ is dense in these Sobolev spaces, does $M$ necessarily have bounded curvature and injectivity radius bounded away from zero?
Best Answer
The general answer to this question is no. Global bound on the Ricci curvature is not necessary for the density of smooth functions with compact supports.
Indeed, when $(M,g)$ is a smooth complete Riemannian manifold with positive injectivity radius and lower bound for the Ricci curvature, then the smooth functions with compact support are density in the Sobolev spaces for $p$ equals 2. This can be found for instance on Emmanuel Hebey's book: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities.