  We consider complete non-compact manifolds with either a sub-quadratic growth
of the norm of the Riemann curvature, or a sub-quadratic growth of both the
norm of the Ricci curvature and the squared inverse of the injectivity radius.
We show the existence on such a manifold of a distance-like function with
bounded gradient and mild growth of the Hessian. As a main application, we
prove that smooth compactly supported functions are dense in $W^{2,p}$. The
result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor,
and the injectivity radius assumption. As further applications we prove new
disturbed Sobolev and Calder\'on-Zygmund inequalities on manifolds with
possibly unbounded curvature and highlight consequences about the validity of
the full Omori-Yau maximum principle for the Hessian.
