Linings in violin building consists of little boards used to increase the surface area for gluing sections of the belly and back to the ribs thus increase stability. They are attached to the belly side and back side of the rib wreath from the inside. Just like the ribs, these boards are bent while hot and fitted into the upper and lower blocks along cutouts. They are tapered toward the rib and can be seen by looking into the f-holes. On the following photo the linings are shown clearly:

Linings of a cello
Cello ribs with linings