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Characterising Clifford parallelisms among Clifford-like parallelisms

Hans Havlicek, Stefano Pasotti, Silvia Pianta

Abstract

We recall the notions of Clifford and Clifford-like parallelisms in a $3$-dimensional projective double space. In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it. In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones.

Characterising Clifford parallelisms among Clifford-like parallelisms

Abstract

We recall the notions of Clifford and Clifford-like parallelisms in a -dimensional projective double space. In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it. In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones.

Paper Structure

This paper contains 7 sections, 12 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Clifford and Clifford-like parallelisms
  3. Automorphisms, their orbits and actions
  4. Automorphisms
  5. Orbits under the group of inner automorphisms
  6. Parallel classes fixed by automorphisms
  7. Main results

Key Result

Proposition 2.4

If Clifford parallelisms $\parallel$ and $\parallel'$ on a three-dimensional projective space have two distinct parallel classes in common, then these parallelisms coincide.

Figures (2)

  • Figure 1: $\mathop{\mathrm{Char}}\nolimits F\neq 2$
  • Figure 2: $\mathop{\mathrm{Char}}\nolimits F= 2$

Theorems & Definitions (31)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 21 more