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

Abstract

Betten and Riesinger constructed Parallelisms of $\mathrm{PG}(3,\mathbb{R})$ with automorphism group $\mathrm{SO}(3,\mathbb{R})$ by applying the reducible $\mathrm{SO}(3,\mathbb{R})$-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 $\mathrm{SO}(3,\mathbb{R})$ is Clifford.

References in corpus (2)

• Compactness of the automorphism group of a topological parallelism on real projective 3-space
• Rotational spreads and rotational parallelisms and oriented parallelisms of PG(3,R)

Cited by in corpus (2)

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