  We obtain several essential self-adjointness conditions for a Schroedinger
type operator D*D+V acting in sections of a vector bundle over a manifold M.
Here V is a locally square-integrable bundle map. Our conditions are expressed
in terms of completeness of certain metrics on M; these metrics are naturally
associated to the operator. We do not assume a priori that M is endowed with a
complete Riemannian metric. This allows us to treat e.g. operators acting on
bounded domains in the euclidean space.
  For the case when the principal symbol of the operator is scalar, we
establish more precise results. The proofs are based on an extension of the
Kato inequality which modifies and improves a result of Hess, Schrader and
Uhlenbrock.
