  A basilar property and a useful tool in the theory of Sobolev spaces is the
density of smooth compactly supported functions in the space $W^{k,p}(\R^n)$
(i.e. the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On
Riemannian manifolds, it is well known that the same property remains valid
under suitable geometric assumptions. However, on a complete non-compact
manifold it can fail to be true in general, as we prove in this paper. This
settles an open problem raised for instance by E. Hebey [\textit{Nonlinear
analysis on manifolds: Sobolev spaces and inequalities}, Courant Lecture Notes
in Mathematics, vol. 5, 1999, pp. 48-49].
