  The aim of this paper is to prove a qualitative property, namely the
preservation of positivity, for Schr\"odinger-type operators acting on $L^p$
functions defined on (possibly incomplete) Riemannian manifolds. A key
assumption is a control of the behaviour of the potential of the operator near
the Cauchy boundary of the manifolds. As a by-product, we establish the
essential self-adjointness of such operators, as well as its generalization to
the case $p\neq 2$, i.e. the fact that smooth compactly supported functions are
an operator core for the Schr\"odinger operator in $L^p$.
