Clifford Parallelisms and External Planes to the Klein quadric

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.

References in corpus (1)

• A note on Clifford parallelisms in characteristic two

Cited by in corpus (4)

• Pencilled regular parallelisms
• Clifford-like parallelisms
• Characterising Clifford parallelisms among Clifford-like parallelisms
• Automorphisms of a Clifford-like parallelism