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 R^d 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 R^d 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.
Cruickshank et al. (2025) studied this question.