Regular \(3\)-dimensional parallelisms of \(\mathrm{PG}(3,\mathbb{R})\)
Every regular topological parallelism \(\mathbf{P}\) of \(\mathrm{PG}(3,\mathbb{R})\) has two natural numbers associated with it: the dimension \(m\) of \(\mathbf{P}\), that is, the dimension of the span in \(\mathrm{PG}(5,\mathbb{R})\) of the hyperflock determining line set associated with \(\mathrm{P}\), and the dimension \(n\) of the collineation group of \(\mathbf{P}\), which is a Lie group. The authors have previously shown that \(2\leq m\leq 5\) [Monatsh. Math. 161, No. 1, 43--58 (2010; Zbl 1208.51009)] and that \(n\leq 6\) [Adv. Geom. 14, No. 1, 175--189 (2014; Zbl 1291.51011)]; \(m=2\) or \(n=6\) characterizes the Clifford parallelism. In [Result. Math. 66, No. 3--4, 291--326 (2014; Zbl 1316.51001)], they provided examples of topological regular parallelisms with parameters \((m,n)=(5,0), (4,0), (4,1), (3,1)\). In the paper under review, the authors continue their study of regular topological parallelisms and exclusively deal with 3-dimensional regular parallelisms (i.e., \(m=3\)). It is shown that these parallelisms are exactly those regular parallelisms which can be constructed from non-ordinary generalized line stars, or those non-Clifford regular parallelisms whose hyperflock determining line sets are contained in the span of some elliptic quadric of the Klein quadric, or regular parallelisms whose latent line set is a regular spread. Examples with \((m,n)=(3,2)\) are presented and it is shown that \(m=n=3\) cannot occur. In order to prove the latter, it is shown that if \(\mathcal{A}\) is a topological generalized line star that induces \(\mathbf{P}\), then the connected component of the collineation group of \(\mathcal{A}\) is either the identity or it is the rotation group \(\mathrm{SO}_2(\mathbb{R})\) about some axis. Since the group dimension of \(\mathbf{P}\) turns out to be 1 plus the dimension of the collineation group of \(\mathcal{A}\), it follows that \(\mathbf{P}\) has group dimension \(n\leq 2\). In the last section, a construction for a large family of rotational 3-dimensional regular parallelisms \(\mathbf{P}\) is given, that is, there is a rotation group \(\mathrm{SO}_2(\mathbb{R})\) about some axis leaving the parallelism \(\mathbf{P}\) invariant.
scientific article; zbMATH DE number 6535840