  We prove essential self-adjointness for semi-bounded below magnetic
Schr\"odinger operators on complete Riemannian manifolds with a given positive
smooth measure which is fixed independently of the metric. Some singularities
of the scalar potential are allowed.
  This is an extension of the Povzner--Wienholtz--Simader theorem. The proof
uses the scheme of Wienholtz but requires a refined invariant integration by
parts technique, as well as a use of a family of cut-off functions which are
constructed by a non-trivial smoothing procedure due to Karcher.
