  A previously introduced method to renormalize the one-loop stress tensor and
the one-loop vacuum fluctuations in a curved background by a direct local
$\zeta$-function approach is checked in some thermal and nonthermal cases.
First the method is checked in the case of a conformally coupled massless field
in the static Einstein universe where all hypotheses initially requested by the
method hold true. Secondly, dropping the hypothesis of a closed manifold, the
method is checked in the open static Einstein universe. Finally, the method is
checked for a massless scalar field in the presence of a conical singularity in
the Euclidean manifold (i.e. Rindler spacetimes/ large mass black hole
manifold/cosmic string manifold). In all cases, a complete agreement with other
approaches is found. Concerning the last case in particular, the method is
proved to give rise to the stress tensor already got by the point-splitting
approach for every coupling with the curvature regardless of the presence of
the singular curvature. In the last case, comments on the measure employed in
the path integral, the use of the optical manifold and the different approaches
to renormalize the Hamiltonian are made.
