Regular parallelisms on $\mathrm{PG}(3,\mathbb R)$ admitting a 2-torus action

Abstract

A regular parallelism of real projective 3-space $\mathrm{PG}(3,\mathbb R)$ is an equivalence relation on the line space such that every class is equivalent to the set of 1-dimensional complex subspaces of $\mathbb C^2 = \mathbb R^4$. We shall assume that the set of classes is compact, and characterize those regular parallelisms that admit an action of a 2-dimensional torus group. We prove that there is a one-dimensional subtorus fixing every parallel class. From this property alone we deduce that the parallelism is a 2- or 3-dimensional regular parallelism in the sense of Betten and Riesinger [6]. If a 2-torus acts, then the parallelism can be described using a so-called generalized line star or \it gl star \rm which admits a 1-torus action. We also study examples of such parallelisms by constructing gl stars. In particular, we prove a claim which was presented in [6] with an incorrect proof. The present article continues a series of papers by the first author on parallelisms with large groups.