  We prove a Hardy inequality for uniformly elliptic operators subject to
Dirichlet or mixed boundary conditions on domains $\Omega$ with piecewiese
smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach
of E.B. Davies for the euclidean case, we show that it implies a sufficient
geometric criterion under which the Laplace- Beltrami operator with Dirichlet
boundary conditions $\Delta^D$ has purely discrete spectrum on $\Omega$. We
proceed to classify all non-compact $\Omega$ with discrete spectrum up to a
boundary regularity condition and show that these include for example polygons
with ideal vertices in manifolds of negative curvature. This a new result for
non-constant curvature.
