Sobolev functions without compactly supported approximations

A basic property and useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space Wk,p(Rn) (i.e., the functions with weak derivatives of orders 0 to k in Lp). On Riemannian manifolds, it is well known that the same property remains valid under suitable geometric assumptions. However, on a complete noncompact manifold it can fail to be true in general, as we prove here. This settles an open problem raised for instance by E. Hebey (Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lect. Notes Math. 5 (1999), 48–49)