Parallelisms of \(\text{PG}(3, {\mathbb R})\) composed of non-regular spreads
type: Item
instance of: scholarly article
title: Parallelisms of \(\text{PG}(3, {\mathbb R})\) composed of non-regular spreads
zbMATH Open document ID: 1222.51009
author: Dieter Betten
author: Rolf Riesinger
published in: Aequationes Mathematicae
publication date: 2011-06-17T00:00:00Z
review text: In a series of articles the authors constructed topological regular parallelisms of the real projective 3-space PG\((3,\mathbb{R})\), that is, families \textbf{P} of regular spreads such that each line of PG\((3,\mathbb{R})\) is contained in exactly one spread of \textbf{P} and such that the operation of drawing a line \textbf{P}-parallel to a given line through a given point is continuous with respect to the natural topologies on the point set and lines set of PG\((3,{\mathbb R})\). In the paper under review the authors present two new, very different and beautiful constructions of parallelisms that are not regular. The first method is based on the spreads obtained in [Arch. Math. 21, 100--102 (1970; Zbl 0204.53501)] by the first author. These spreads \(\mathcal{S}(\mathfrak{B}_F)\) are determined by a homeomorphism \(F\) of \({\mathbb R}^{\geq 0}\). The second author [Geom. Dedicata 40, 145--163 (1991; Zbl 0742.51014)] has shown that, in affine coordinates with respect to a suitable plane at infinity, each such spread is composed of reguli contained in certain coaxial, concentric, rotational one-sheeted hyperboloids. If SO\(_3({\mathbb R})\) is the group of all rotations about the common center, then it is shown that the family \(\{g(\mathcal{S}(\mathfrak{B}_F ))\mid g \in \mathrm{SO}_3({\mathbb R})\}\) of spreads is a topological parallelism, which is irregular unless \(F\) is linear. The authors furthermore investigate rotational spreads. These are spreads \(\mathcal{S}\) of PG\((3,\mathbb{R})\), considered as the projective closure of a real Euclidean 3-space \(E\), for which there exists an (affine) line \(A\) such that \(\mathcal{S}\) is invariant under each rotation about \(A\). It is shown that \(\mathcal{S}\) must contain \(A\) and the line \(U\) at infinity of the planes orthogonal to \(A\), and that \(\mathcal{S}\) is composed of \(A\), \(U\), and a system of reguli carried by pairwise disjoint coaxial one-sheeted rotational hyperboloids in \(E\). The au…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A15
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51M30
zbMATH DE Number: 5909091
zbMATH Keywords: projective 3-space
zbMATH Keywords: parallelism
zbMATH Keywords: topological parallelism
zbMATH Keywords: Clifford parallelism
zbMATH Keywords: rotational spread
zbMATH Keywords: piecewise regular spread
zbMATH Keywords: Klein quadric
zbMATH Keywords: flock
zbMATH Keywords: Thas-Walker construction
MaRDI profile type: Publication
full work available at URL: s00010-011-0081-2
OpenAlex ID: W2069082369
cites work: Nicht-desarguessche 4-dimensionale Ebenen
cites work: Topological parallelisms of the real projective 3-space
cites work: Constructing topological parallelisms of PG(3, ℝ) via rotation of generalized line pencils
cites work: Generalized line stars and topological parallelisms of the real projective 3-space
cites work: Hyperflock determining line sets and totally regular parallelisms of \(\text{PG}(3,\mathbb{R})\)
cites work: Q4104481
cites work: Coset switching in parallelisms
cites work: Q4764159
cites work: Q3694152
cites work: Q5699076
cites work: Parallelism-inducing groups
cites work: Minimal parallelism-inducing groups
cites work: Q3579790
cites work: Computational line geometry
cites work: Examples of rigid, topological spreads of the real projective 3- space.
cites work: Spreads admitting net generating regulizations
cites work: Piecewise regular spreads
cites work: Q4104072
DOI: 10.1007/S00010-011-0081-2
description: scientific article; zbMATH DE number 5909091