Lifting degenerate simplices with a single volume constraint

Abstract

Let $M^d$ be the spherical, Euclidean, or hyperbolic space of dimension $d\ge n+1$. Given any degenerate $(n+1)$-simplex $\mathbf{A}$ in $M^d$ with non-degenerate $n$-faces $F_i$, there is a natural partition of the set of $n$-faces into two subsets $X_1$ and $X_2$ such that $\sum_{X_1}V_n(F_i)=\sum_{X_2}V_n(F_i)$, except for a special spherical case where $X_2$ is the empty set and $\sum_{X_1}V_n(F_i)=V_n(\mathbb{S}^n)$ instead. For all cases, if the vertices vary smoothly in $M^d$ with a \emph{single} volume constraint that $\sum_{X_1}V_n(F_i)-\sum_{X_2}V_n(F_i)$ is preserved as a constant (0 or $V_n(\mathbb{S}^n)$), we prove that if a \emph{stress} invariant $c_{n-1}(\alpha^{n-1})$ of the degenerate simplex is non-zero, then the vertices will be confined to a lower dimensional $M^n$ for any sufficiently small motion. This answers a question of the author and we also show that in the Euclidean case, $c_{n-1}(\alpha^{n-1})=0$ is equivalent to the vertices of a \emph{dual} degenerate $(n+1)$-simplex lying on an $(n-1)$-sphere in $\mathbb{R}^n$.