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

“…We thereby generalise and unify Theorems 5.1, 5.5, and 5.6 in [6]. These theorems are more explicit than our result, but tailored to real projective spaces; see also [8,Rem. 8.1].…”

Section: Main Results and Examplesmentioning

confidence: 59%

“…Throughout this paper we stick as close as possible to the notions and the terminology in [8], even though we work over an arbitrary ground K field rather than over R. By λ : L 3 → H 5 we denote Klein's correspondence of line geometry, whose image is the Klein quadric H 5 in PG(5, K) = (P 5 , L 5 The polarity of PG(5, K) associated with H 5 is denoted by π 5 . A subquadric of the Klein quadric is the section of H 5 by an r-dimensional subspace of PG(5, K), r ∈ {−1, 0, .…”

Section: Preliminariesmentioning

confidence: 99%

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Pencilled regular parallelisms

Havlicek

Riesinger²

2017

Acta Math. Hungar.

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Over any field K, there is a bijection between regular spreads of the projective space PG(3, K) and 0-secant lines of the Klein quadric in PG(5, K). Under this bijection, regular parallelisms of PG(3, 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.Mathematics Subject Classification: 51A15 51M30

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“…Non-Clifford examples of regular ordinary parallelisms with a two-dimensional (torus) group have been given in [3], compare also [5] for a fresh view of the construction. Regular parallelisms with an even smaller group are described in [2]. (A topological parallelism is regular iff its elements are R. Löwen all isomorphic to the complex spread.…”

Section: Introductionmentioning

confidence: 99%

Parallelisms of $$\mathrm{PG}(3,\mathbb R)$$ PG ( 3 , R ) admitting a 3-dimensional group

Löwen

2018

Beitr Algebra Geom

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Betten and Riesinger [1] constructed Parallelisms of PG(3, R) with automorphism group SO 3 R by applying the reducible SO 3 R-action to a rotational Betten spread. This was generalized in [8] so as to include oriented parallelisms (i.e., parallelisms of oriented lines). In this way, a much larger class of examples was produced.Here we show that, apart from Clifford parallelism, these are the only topological parallelisms admitting an automorphism group of dimension 3 or larger. In particular, we show that a topological parallelism admitting the irreducible action of SO 3 R is Clifford.MSC 2010: 51H10, 51A15, 51M30

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

Cited by 10 publications

References 9 publications

Compactness of the Automorphism Group of a Topological Parallelism on Real Projective 3-Space

Compactness of the Automorphism Group of a Topological Parallelism on Real Projective 3-Space

Pencilled regular parallelisms

Parallelisms of $$\mathrm{PG}(3,\mathbb R)$$ PG ( 3 , R ) admitting a 3-dimensional group

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