Volume Rigidity of Simplicial Manifolds

Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex polyhedron P is rigid i.e. every continuous motion of the vertices of P in R³ which preserves its edge lengths results in a polyhedron which is congruent to P. This result was extended to convex poytopes in Rᵈ for all d≥ 3 by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in Rᵈ by Kalai for d≥ 4 and Fogelsanger for d≥ 3. We will generalise Kalai's result by showing that, for all d≥ 4 and any fixed 1≤ k≤ d-3, every generic realisation of the k-skeleton of a simplicial $(d-1)$-manifold in Rᵈ is volume rigid, i.e. every continuous motion of its vertices in Rᵈ which preserves the volumes of its k-faces results in a congruent realisation. In addition, we conjecture that our result remains true for $k=d-2$ and verify this conjecture when $d=4,5,6$.