Clifford Parallelisms and External Planes to the Klein quadric

TL;DR

This paper establishes a one-to-one correspondence between Clifford parallelisms in a projective space and external planes to the Klein quadric, providing new characterizations that do not rely on the ambient space.

Contribution

It introduces a novel correspondence between Clifford parallelisms and external planes to the Klein quadric, with characterizations independent of the ambient space.

Findings

Established a one-to-one correspondence between Clifford parallelisms and external planes.

Provided two new characterizations of Clifford parallelisms avoiding the Klein quadric's ambient space.

Abstract

For any three-dimensional projective space ${\mathbb P}(V)$, where $V$ is a vector space over a field $F$ of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of ${\mathbb P}(V)$ and those planes of ${\mathbb P}(V\wedge V)$ that are external to the Klein quadric representing the lines of ${\mathbb P}(V)$. We also give two characterisations of a Clifford parallelism of ${\mathbb P}(V)$, both of which avoid the ambient space of the Klein quadric.