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

Lizhao Zhang

TL;DR

The paper investigates rigidity of $n$-simplices under deformations that preserve all $(n-2)$-face volumes, with a focus on pseudo-Euclidean spaces. It develops a Gram-matrix framework and duality to construct explicit counterexamples: for $n=4$ in $\\mathbb{R}^{3,1}$ there is a continuous family of noncongruent $4$-simplices with fixed $2$-face areas, and for all $n\ge 5$ in suitable spaces $\mathbb{R}^{p,n-p}$ there are continuous families with fixed $(n-2)$-volumes and Euclidean dihedral angles; a related symmetric-matrix counterexample shows constant $2\times 2$ minors is not enough to enforce rigidity. The methods combine block-structured constructions, determinant analysis of Gram data, and duality between simplices and their polars, yielding a constructive view that informs the limits of rigidity in non-Euclidean signatures and suggests avenues for addressing the Euclidean case. The results highlight the nuanced boundary between rigidity and flexibility for higher-dimensional simplices and provide a toolkit for exploring related invariants and potential counterexamples.

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 $p\ge 2$.

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

TL;DR

The paper investigates rigidity of -simplices under deformations that preserve all -face volumes, with a focus on pseudo-Euclidean spaces. It develops a Gram-matrix framework and duality to construct explicit counterexamples: for in there is a continuous family of noncongruent -simplices with fixed -face areas, and for all in suitable spaces there are continuous families with fixed -volumes and Euclidean dihedral angles; a related symmetric-matrix counterexample shows constant minors is not enough to enforce rigidity. The methods combine block-structured constructions, determinant analysis of Gram data, and duality between simplices and their polars, yielding a constructive view that informs the limits of rigidity in non-Euclidean signatures and suggests avenues for addressing the Euclidean case. The results highlight the nuanced boundary between rigidity and flexibility for higher-dimensional simplices and provide a toolkit for exploring related invariants and potential counterexamples.

Abstract

For any -dimensional simplex in the Euclidean space with , 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 , we provide counterexamples to a variant of the question where is replaced by a pseudo-Euclidean space for some unspecified .
Paper Structure (18 sections, 20 theorems, 30 equations, 3 figures)

This paper contains 18 sections, 20 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Main results
  4. Why pseudo-Euclidean spaces
  5. Preliminaries
  6. Some notions
  7. Basic properties
  8. Proofs of Theorem \ref{['theorem_flexible_pseudo_Euclidean_n_4']} and \ref{['theorem_flexible_simplex_5_face_2']}
  9. Proof of Theorem \ref{['theorem_flexible_pseudo_Euclidean_n_4']}
  10. Proof of Theorem \ref{['theorem_flexible_simplex_5_face_2']}
  11. Equivalent questions of Question \ref{['question_simplex_rigid']}
  12. Proof of Theorem \ref{['theorem_symmetric_matrix']}
  13. An outline of the proof
  14. For $n$ odd
  15. For $n$ even
  16. ...and 3 more sections

Key Result

Theorem 1.2

For $n=4$ in $\mathbb{R}^{3,1}$, there exists a continuous family of non-congruent $4$-simplices $Q$ with fixed areas of all the 2-faces, and all the 2-faces are in Euclidean planes.

Figures (3)

  • Figure 1: Pseudo 3-simplex $Q_0$ with squares of edge lengths
  • Figure 2: Pseudo 4-simplices $Q_1$ and $Q_2$ with squares of edge lengths
  • Figure 3: Pseudo 5-simplices $Q_1$ and $Q_2$ with squares of edge lengths

Theorems & Definitions (51)

  • Theorem 1.2: Main Theorem 1
  • Theorem 1.3: Main Theorem 2
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 41 more