Regular Parallelisms Research Articles

Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case

Abstract Using the Klein correspondence, regular parallelisms of PG(3, ℝ) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3, ℝ). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious.

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

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.

New Uniform Subregular Parallelisms of PG(3, 4) Invariant under an Automorphism of Order 2

Abstract A spread in PG(n, q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A parallelism is uniform if all its spreads are isomorphic. Up to isomorphism, there are three spreads of PG(3, 4) – regular, subregular and aregular. Therefore, three types of uniform parallelisms are possible. In this work, we consider uniform parallelisms of PG(3, 4) which possess an automorphism of order 2. We establish that there are no regular parallelisms, and that there are 8253 nonisomorphic subregular parallelisms. Together with the parallelisms known before this work, this yields a total of 8623 known subregular parallelisms of PG(3, 4).

Pencilled regular parallelisms

Over any field \({\mathbb{K}}\), there is a bijection between regular spreads of the projective space \({{\rm PG}(3,\mathbb{K})}\) and 0-secant lines of the Klein quadric in \({{\rm PG}(5,\mathbb{K})}\). Under this bijection, regular parallelisms of \({{\rm PG}(3,\mathbb{K})}\) correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be pencilled if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being pencilled. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.

New Regular Parallelisms of PG(3, 5)

AbstractWe construct all regular parallelisms of with automorphisms of order 3. Their number is 8. The two cyclic parallelisms found by Prince are among them. The other six ones are the first examples of noncyclic regular parallelisms and the first examples of regular parallelisms that do not belong to the infinite families of Penttila and Williams.

Regular $3$-dimensional parallelisms of $\mbox{{\rm PG}}(3,\mathbb R)$

In [8] the collineation groups of some known 5-, 4- and 3-dimensional topological regular parallelisms of $PG(3,\mathbb R)$ were determined. In the present article we concentrate on 3-dimensional regular parallelisms and prove: the 3-dimensional regular parallelisms are exactly those which can be constructed from generalized line stars, see [3]. We determine the collineation groups of 3-dimensional regular parallelisms and show that only group dimension 1 or 2 is possible. If the collineation group is 2-dimensional, then the parallelism is rotational which means that there is a rotation group $SO_2(\mathbb R)$ about some axis leaving the parallelism invariant. We give a construction method for the generalized line stars which induce these parallelisms.

Regular parallelisms from generalized line stars in $${P_{3} \mathbb{R}}$$ P 3 R : a direct proof

A generalized line star with respect to an elliptic quadric contained in the Klein quadric gives rise to a regular parallelism in real projective 3-space. This was shown by Betten and Riesinger (Results Math 47:226–241, 2005) using the Thas–Walker construction. They remark that the resulting description of the parallelism is equivalent to a much simpler one, but again the proof is hard. In practice, they always work with that simpler construction. We show that the Thas–Walker approach is not needed here. In fact, one can derive all relevant properties of the parallelism from the simple construction. We also give coordinate-free, short proofs of the topological properties of regular parallelisms in general and of those obtained from generalized line stars in particular.

Automorphisms of Some Topological Regular Parallelisms of $${{\rm PG}(3, \mathbb{R})}$$ PG ( 3 , R )

In a precedent article we constructed various topological regular parallelisms of the real projective 3-space \({{\rm PG}(3, \mathbb{R})}\) via hyperflock determining line sets of \({{\rm PG}(5, \mathbb{R})}\) (see Betten and Riesinger in Mh Math 161:43–58, 2010). In the present paper we discuss for some of these parallelisms their automorphism groups consisting of all automorphic collineations and all automorphic dualities, especially we compute their group dimension. Thus we are able to present: (1) topological regular 5-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0, (2) topological regular 4-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0 or 1, (3) topological regular 3-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 1.

On transitive parallelisms of $$PG(3,4)$$ P G ( 3 , 4 )

A parallelism in $$PG(n,q)$$ is transitive if it has an automorphism group which is transitive on the spreads. A parallelism is regular if all its spreads are regular. In $$PG(3,4)$$ no examples of transitive and no regular parallelisms are known. Transitive parallelisms in $$PG(3,4)$$ must have automorphisms of order 7. That is why we construct all 482 parallelisms with automorphisms of order 7 and establish that there are neither transitive, nor regular ones among them. We conclude that there are no transitive parallelisms in $$PG(3,4)$$ . The investigation is computer-aided. We use GAP (Groups, Algorithms, Programming—a System for Computational Discrete Algebra) to find a subgroup of order 7 and its normalizer in the automorphism group of $$PG(3,4)$$ . For all the other constructions and tests we use our own software written in C++.

Clifford parallelism: old and new definitions, and their use

Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space $${{\rm PG}(3,\mathbb{R})}$$ whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein’s definition). Our new access to the topic ‘Clifford parallelism’ is free of complexification and involves Klein’s correspondence λ of line geometry together with a bijective map γ from all regular spreads of $${{\rm PG}(3,\mathbb{R})}$$ onto those lines of $${{\rm PG}(5,\mathbb{R})}$$ having no common point with the Klein quadric; a regular parallelism P of $${{\rm PG}(3,\mathbb{R})}$$ is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with γ is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of $${{\rm PG}(3,\mathbb{R})}$$ are Clifford (D3 = dimensionality definition). Submission of (D2) to λ−1 yields a complexification free definition of a Clifford parallelism which uses only elements of $${{\rm PG}(3,\mathbb{R})}$$ : A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group $${Aut_e({\bf P}_{\bf C})}$$ of all automorphic collineations and dualities of the Clifford parallelism P C and show $${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$$ .

Regular parallelisms in kinematic spaces

Here we propose a definition of regular parallelism in a linear space not necessarily embedded onto a projective space and we investigate its properties in the particular case of kinematic spaces. We prove that the kinematic parallelisms are always regular in that sense and we deduce some results on the group of translations acting transitively on the pointset.

Hyperflock determining line sets and totally regular parallelisms of PG $${(3,\,{\mathbb{R}})}$$

By a totally regular parallelism of the real projective 3-space \({\Pi_3:={{\rm PG}}(3, \mathbb {R})}\) we mean a family T of regular spreads such that each line of Π3 is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π5 associated with the Klein quadric H5 (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H5, i.e., a family H of elliptic subquadrics of H5 such that each point of H5 is on exactly one subquadric of H. Moreover, \({\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}}\) is a hyperflock determining line set, i.e., a set \({\mathcal {Z}}\) of 0-secants of H5 such that each tangential hyperplane of H5 contains exactly one line of \({\mathcal {Z}}\) . We say that \({{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}}\) is the dimension of T and that T is a dT- parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If \({\mathcal{G}}\) is a hyperflock determining line set, then \({\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}}\) is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.

The Doubly Transitive t-Parallelisms

Johnson showed that the only doubly transitive parallelisms of PG(3, q) are exactly the two regular parallelisms in PG(3, 2). This article completely generalizes this result to doubly transitive t-parallelisms of arbitrary finite projective spaces.

Two-Transitive Parallelisms

In this article, the parallelisms which admit doubly transitive groups are completely classified as the two regular parallelisms in PG(3,2).

Transitive Parallelisms

In this article, the groups acting on parallelisms in PG(3, q), q = pr, which are generated by collineations of order a p-primitive divisor of q3 − 1 are completely determined. In particular, when the group generated is non-solvable only the groups PSL(2,7) or A7 are possible. If the parallelism is transitive then either it is one of the two regular parallelisms in PG(3,2) or the group is solvable and is contained in гL(1,q3)/Z, where Z denotes the scalar group of order q − 1.

The Cyclic Parallelisms of PG(3,5)

We determine, by a computer search, all the cyclic parallelisms of PG(3,5). There are 45 of them, up to projective equivalence. In particular, there are two cyclic regular parallelisms of PG(3,5). Previously, no example was known of a regular parallelism in PG(3,q), forqodd.

Regular parallelisms from translation planes

Let π be a translation plane of order q 4 admitting a collineation group G isomorphic to SL(2, q), p r = q, in the translation complement. It is shown that if the p-elements of G are elations and L is an elation axis then the kernel of π is GF( q) and G acts 1 2 - transitive on l ∞− N∩l ∞ (where N denotes the elation net) if and only if there is a regular parallelism of L (considering L as PG(3, q)). With no assumption on the p-elements, it is shown that the existence SL(2, q)× z 1+ q+ q 2 as a collineation group implies that the p-elements are elations, the kernel is GF( q) and any elation axis L admits a regular parallelism as PG(3, q).

On regular parallelisms in PG(3, q)

On regular parallelisms in PG(3, q)