  In classical surface theory there are but few known examples of surfaces
admitting nontrivial isometric deformations and fewer still
non-simply-connected ones. We consider the isometric deformability question for
an immersion x: M \to R^3 of an oriented non-simply-connected surface with
constant mean curvature H. We prove that the space of all isometric immersions
of M with constant mean curvature H is, modulo congruences of R^3, either
finite or a circle. When it is a circle then, for the immersion x, every cycle
in M has vanishing force and, when H is not 0, also vanishing torque. Our work
generalizes a rigidity result for minimal surfaces to constant mean curvature
surfaces. Moreover, we identify closed vector-valued 1-forms whose periods give
the force and torque.
