Volume Rigidity of Simplicial Manifolds

Abstract Classical results of Cauchy [4] and Dehn [5] imply that the 1-skeleton of a convex simplicial polyhedron P is rigid, i.e. every continuous motion of the vertices of P in $${\mathbb R}^3$$ R 3 which preserves its edge lengths results in a polyhedron which is congruent to P . This result was extended to convex simplicial polytopes in $${\mathbb R}^d$$ R d for all $$d\ge 3$$ d ≥ 3 by Whiteley [16], and to generic realisations of 1-skeletons of simplicial $$(d-1)$$ ( d - 1 ) -manifolds in $${\mathbb R}^{d}$$ R d by Kalai [8] for $$d\ge 4$$ d ≥ 4 and Fogelsanger [6] for $$d\ge 3$$ d ≥ 3 . We will generalise Kalai’s result by showing that, for all $$d\ge 4$$ d ≥ 4 and any fixed $$1\le k\le d-3$$ 1 ≤ k ≤ d - 3 , every generic realisation of the k -skeleton of a simplicial $$(d-1)$$ ( d - 1 ) -manifold in $${\mathbb R}^{d}$$ R d is volume rigid, i.e. every continuous motion of its vertices in $${\mathbb R}^d$$ R d which preserves the volumes of its k -faces results in a congruent realisation. In addition, we conjecture that our result remains true for $$k=d-2$$ k = d - 2 and verify this conjecture when $$d=4,5,6$$ d =