Characterising Clifford parallelisms among Clifford-like parallelisms

TL;DR

This paper investigates the automorphism groups of Clifford and Clifford-like parallelisms in 3D projective double spaces, providing new characterizations of Clifford parallelisms based on group actions.

Contribution

It offers a novel analysis of how automorphism groups act on parallel classes, distinguishing Clifford from Clifford-like parallelisms.

Findings

Automorphism group actions characterize Clifford parallelisms.

The linear part of the automorphism group is consistent across Clifford-like parallelisms.

Main results provide criteria to identify Clifford parallelisms among Clifford-like ones.

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.