  We generalize some fundamental results for noncompact Riemannian manfolds
without boundary, that only require completeness and no curvature assumptions,
to manifolds with boundary: let $M$ be a smooth Riemannian manifold with
boundary $\partial M$ and let $\hat{C}^\infty_c(M)$ denote the space of smooth
compactly supported cut-off functions with vanishing normal derivative, Neumann
cut-offs. We show, among other things, that under completeness:
  - $\hat{C}^\infty_c(M)$ is dense in $W^{1,p}(\mathring{M})$ for all $p\in
(1,\infty)$; this generalizes a classical result by Aubin [2] for $\partial
M=\emptyset$.
  - $M$ admits a sequence of first order cut-off functions in
$\hat{C}^\infty_c(M)$; for $\partial M=\emptyset$ this result can be traced
back to Gaffney [7].
  - the Laplace-Beltrami operator with domain of definition
$\hat{C}^\infty_c(M)$ is essentially self-adjoint; this is a generalization of
a classical result by Strichartz [20] for $\partial M=\emptyset$.
