Simplices with fixed volumes of codimension 2 faces in a continuous deformation

TL;DR

This paper investigates whether preserving the volumes of codimension 2 faces in a simplex implies rigidity during continuous deformation, providing counterexamples in pseudo-Euclidean spaces for dimensions four and higher.

Contribution

It introduces counterexamples in pseudo-Euclidean spaces for the deformation of simplices with fixed codimension 2 face volumes, challenging the conjecture in Euclidean spaces.

Findings

Counterexamples in pseudo-Euclidean spaces for n ≥ 4

Supports the belief that Euclidean simplices are rigid under volume-preserving deformations

Raises questions about rigidity in non-Euclidean geometries

Abstract

For any $n$-dimensional simplex in the Euclidean space $\mathbb{R}^n$ with $n\ge 4$, it is asked that if a continuous deformation preserves the volumes of all the codimension 2 faces, then is it necessarily a \emph{rigid} motion. While the question remains open and the general belief is that the answer is affirmative, for all $n\ge 4$, we provide counterexamples to a variant of the question where $\mathbb{R}^n$ is replaced by a pseudo-Euclidean space $\mathbb{R}^{p,n-p}$ for some unspecified $p\ge 2$.