  Let $\Omega\subset M$ be an open subset of a Riemannian manifold $M$ and let
$V:M\to \IR$ be a Kato decomposable potential. With $W^{1,2}_{0}(M;V)$ the
natural form domain of the Schr\"odinger operator $-\Delta+V$ in $L^2(M)$, in
this paper we study systematically the following question: Under which
assumption on $\Omega$ is the statement $$ \text{ for all $f\in
W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus \Omega$ one has $f|_\Omega\in
W^{1,2}_{0}(\Omega;V)$} $$ true for every such $V$? We prove that without any
further assumptions on $V$, the above property is satisfied, if $\Omega$ is Kac
regular, a probabilistic property which means that the first exit time of
Brownian motion on $M$ from $\Omega$ is equal to its first penetration time to
$M\setminus \Omega$. In fact, we treat more general covariant Schr\"odinger
operators acting on sections in metric vector bundles, allowing new results
concerning the harmonicity of Dirac spinors on singular subsets. Finally, we
prove that locally Lipschitz regular $\Omega$'s are Kac regular.
