Rigidity and Volume Preserving Deformation on Degenerate Simplices

Abstract

Given a degenerate \((n+1)\)-simplex in a d-dimensional space \(M^d\) (Euclidean, spherical or hyperbolic space, and \(d\ge n\)), for each k, \(1\le k\le n\), Radon’s theorem induces a partition of the set of k-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in \(M^d\) for \(d=n\), and the volumes of k-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all k-faces; and this property still holds in \(M^d\) for \(d\ge n+1\) if an invariant \(c_{k-1}(\alpha ^{k-1})\) of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant \(c_k(\omega )\) we discovered for any k-stress \(\omega \) on a cell complex in \(M^d\). We introduce a characteristic polynomial of the degenerate simplex by defining \(f(x)=\sum _{i=0}^{n+1}(-1)^{i}c_i(\alpha ^i)x^{n+1-i}\), and prove that the roots of f(x) are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.