Parallelisms of $\mathop{\rm PG}(3,\mathbb R)$ admitting a 3-dimensional group

TL;DR

This paper classifies topological parallelisms of projective 3-space with automorphism groups of dimension three or more, showing that only Clifford parallelism admits an irreducible SO(3,R) action.

Contribution

It extends previous constructions of parallelisms with SO(3,R) symmetry to include oriented cases and proves the uniqueness of Clifford parallelism under certain symmetry conditions.

Findings

Clifford parallelism is the only topological parallelism with an irreducible SO(3,R) action.

Parallelisms with automorphism group dimension ≥ 3 are characterized, excluding all but Clifford.

The paper generalizes earlier constructions to oriented parallelisms, broadening the class of known examples.

Abstract

Betten and Riesinger constructed Parallelisms of $\mathop{\rm PG}(3,\mathbb R)$ with automorphism group $\mathop{\rm SO}(3,\mathbb R)$ by applying the reducible $\mathop{\rm SO}(3,\mathbb R)$-action to a rotational Betten spread. This was generalized by the present author so as to include oriented parallelisms (i.e., parallelisms of oriented lines). In this way, a much larger class of examples was produced. Here we show that, apart from Clifford parallelism, these are the only topological parallelisms admitting an automorphism group of dimension 3 or larger. In particular, we show that a topological parallelism admitting the irreducible action of $\mathop{\rm SO}(3,\mathbb R)$ is Clifford.