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 ${\mathbb{R}}^3$ which preserves its edge lengths results in a polyhedron which is congruent to $P$. This result was extended to convex poytopes in ${\mathbb{R}}^d$ for all $d\ge 3$ by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in ${\mathbb{R}}^d$ by Kalai for $d\ge 4$ and Fogelsanger 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 $d=4,5,6$.