Volume Rigidity of Simplicial Manifolds
James Cruickshank, Bill Jackson, Shin-ichi Tanigawa
TL;DR
The paper addresses when generic realizations of higher-order hypergraphs embedded in ${\mathbb R}^d$ are volume rigid. It develops a volume-rigidity framework using the volume map and a rigidity matrix, proving the equivalence between volume rigidity and infinitesimal volume rigidity for generic configurations, and extends core rigidity techniques by introducing gluing and vertex-splitting lemmas for hypergraphs. The main results show that for $d\ge 4$ and $1\le k\le d-3$, the $(k+1)$-uniform hypergraph of $k$-faces of a connected simplicial $(d-1)$-manifold is volume rigid under generic realizations, with complete $k$-uniform hypergraphs also volume rigid in broad settings; the $k=d-2$ case is conjectured and verified in small dimensions ($d=4,5,6$). These findings broaden rigidity theory to volume-preserving deformations of higher-dimensional simplicial structures, with potential implications for geometric modeling and combinatorial topology.
Abstract
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 $d\geq 3$ by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in $\mathbb R^{d}$ by Kalai for $d\geq 4$ and Fogelsanger for $d\geq 3$. We will generalise Kalai's result by showing that, for all $d\geq 4$ and any fixed $1\leq k\leq d-3$, 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 $d=4,5,6$.