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We generalize Cauchy's celebrated theorem on the global rigidity of convexpolyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circlepolyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex andproper non-unitary c-polyhedra with M\"obius-congruent faces that areconsistently oriented are M\"obius-congruent. Our result implies the globalrigidity of convex inversive distance circle packings in the Riemann sphere aswell as that of certain hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$.

Rigidity of Circle Polyhedra in the 2-Sphere and of Hyperideal Polyhedra in Hyperbolic 3-Space