  The known ground state of ultrathin smectic films confined to the surface of
a sphere is described by four +1/2 defects assembled on a great circle and a
director which follows geodesic lines. Using a simple perturbative approach we
show that for thick smectic films on a sphere with planar anchoring this
solution breaks down, distorting the smectic layers. The instability happens
when the bend elastic constant exceeds the anchoring strength times the radius
of the inner sphere. Above this threshold, the formation of a periodic
chevron-like structure, observed experimentally as well, relieves geometric
frustration. We quantify the effect of thickness and curvature of smectic
shells and provide insight into the wavelength of the observed texture.
