  We apply results derived in other contexts for the spectrum of the Laplace
operator in curved geometries to the study of an ideal polymer chain confined
to a spherical annulus in arbitrary space dimension D and conclude that the
free energy compared to its value for an uncurved box of the same thickness and
volume, is lower when $D < 3$, stays the same when $D = 3$, and is higher when
\mbox{$D > 3$}. Thus confining an ideal polymer chain to a cylindrical shell,
lowers the effective bending elasticity of the walls, and might induce
spontaneous symmetry breaking, i.e. bending. (Actually, the above mentioned
results show that {\em {any}} shell in $D = 3$ induces this effect, except for
a spherical shell). We compute the contribution of this effect to the bending
rigidities in the Helfrich free energy expression.
