Regular \(3\)-dimensional parallelisms of \(\mathrm{PG}(3,\mathbb{R})\)
type: Item
instance of: scholarly article
title: Regular \(3\)-dimensional parallelisms of \(\mathrm{PG}(3,\mathbb{R})\)
zbMATH Open document ID: 1336.51008
author: Dieter Betten
author: Rolf Riesinger
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2016-01-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1450389250
review text: 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 lar…
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A15
Mathematics Subject Classification ID: 51M30
zbMATH DE Number: 6535840
zbMATH Keywords: topological parallelism
zbMATH Keywords: regular parallelism
zbMATH Keywords: dimension of a regular parallelism
zbMATH Keywords: Clifford parallelism
zbMATH Keywords: rotational parallelism
zbMATH Keywords: hyperflock
zbMATH Keywords: generalized line star
zbMATH Keywords: generalized line pencil
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6535840