Rigidity of certain polyhedra in R3

We extend the Cauchy theorem stating rigidity of convex polyhedra in R-3. We do not require that the polyhedron be convex nor embedded, only that the realization of the polyhedron in R-3 be linear and isometric on each face. We also extend the topology of the surfaces to include the projective plane in addition to the sphere. Our approach is to choose a convenient normal to each face in such a way that as we go around the star of a vertex the chosen normals are the vertices of a convex polygon on the unit sphere. When we can make such a choice at each vertex we obtain rigidity. For example, we can prove that the heptahedron is rigid.