Rigidity of piecewise convex surfaces of torus type

Closed genus-one surfaces pasted from finitely many pieces of convex $C^2$-surfaces are considered. Vertices and conical points are allowed. An algorithm for the construction of such surfaces is given. They are proved to be rigid outside at domains with respect to infinitesimal bendings of the first order with continuous bending fields belonging to the class $C^2$ on each $C^2$-smooth piece.