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 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 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.