Large-area ordered arrays can be created by the self-assembly of linear defects in thin Smectic-A films deposited in air on crystalline substrates. Such structures could find applications in nanoparticle assembly and soft lithography. The smectic layers are bent in concentric cylinders 10 under the effect of conflicting strong anchoring conditions at the substrate surface and free interface, and neighboring cylindrical domains are separated by curvature walls. We have studied the internal structure of such domains near the surface of muscovite mica using synchrotron x-ray diffraction. Our findings show that the domains are centered on virtual singularities, running below the substrate plane, and rest upon a surface region of submicrometric thickness where layers are 15 flat and vertical, satisfying the planar anchoring condition imposed by the substrate.