  When quantum supergravity is studied on manifolds with boundary, one may
consider local boundary conditions which fix on the initial surface the whole
primed part of tangential components of gravitino perturbations, and fix on the
final surface the whole unprimed part of tangential components of gravitino
perturbations. This paper studies such local boundary conditions in a flat
Euclidean background bounded by two concentric 3-spheres. It is shown that, as
far as transverse-traceless perturbations are concerned, the resulting
contribution to $\zeta(0)$ vanishes when such boundary data are set to zero,
exactly as in the case when non-local boundary conditions of the spectral type
are imposed. These properties may be used to show that one-loop finiteness of
massless supergravity models is only achieved when two boundary 3-surfaces
occur, and there is no exact cancellation of the contributions of gauge and
ghost modes in the Faddeev-Popov path integral. In these particular cases,
which rely on the use of covariant gauge-averaging functionals, pure gravity is
one-loop finite as well.
