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