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Clifford-like parallelisms

Hans Havlicek, Stefano Pasotti, Silvia Pianta

TL;DR

This work characterises the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms and establishes necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

Abstract

Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space $(\mathbb{P},\parallel_\ell,\parallel_r)$ over a quaternion skew field we characterise the "Clifford-like" parallelisms, i.e. the blends of the Clifford parallelisms $\parallel_\ell$ and $\parallel_r$, in a geometric and an algebraic way. Finally, we establish necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

Clifford-like parallelisms

TL;DR

This work characterises the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms and establishes necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

Abstract

Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space over a quaternion skew field we characterise the "Clifford-like" parallelisms, i.e. the blends of the Clifford parallelisms and , in a geometric and an algebraic way. Finally, we establish necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

Paper Structure

This paper contains 4 sections, 17 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Blends of equivalence relations
  3. Blends of parallelisms
  4. Clifford-like parallelisms from quaternion skew fields

Key Result

Lemma 2.1

Let $\mathrel{\pi_{}}$ be an equivalence relation on ${\mathcal{L}}$ and ${\mathcal{B}}\subseteq{\mathcal{L}}$. Then the following are equivalent.

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • ...and 29 more