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 ${\mathbb{R}}^n$.

17 pages. To appear in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry