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.