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

2017

Acta Math. Hungar.

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

Section: Main Results and Examplesmentioning

confidence: 79%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pencilled regular parallelisms

Havlicek

2017

Acta Math. Hungar.

“…In a series of articles [3][4][5][6] the authors constructed regular parallelisms. By contrast we exhibit in the present paper examples of irregular parallelisms.…”

Section: Introductionmentioning

confidence: 99%

Parallelisms of $${{\rm PG}(3, {\mathbb R})}$$ composed of non-regular spreads

Betten

2011

Aequat. Math.

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Any continuous strictly monotonic function F : R ≥0 → R with F (0) = 0 and F (t) → ∞ for t → ∞ gives rise to a topological rotational spread of PG (3, R); this spread is non-regular, if F is not linear. The action of the group SO 3 (R) on this spread yields a topological parallelism of PG (3, R). The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P 72 of PG (3, R) which is composed of piecewise regular spreads each consisting of two segments which are tacked together along a common regulus. Using Klein's correspondence of line geometry and the Thas-Walker construction we represent every parallel class of P 72 via two parallel half-lines being noninterior to a given sphere in R 3 . The parallelism P 72 contains exactly one regular spread, all other members of P 72 are piecewise regular spreads with two segments. However, P 72 is not topological.Mathematics Subject Classification (2010). 51H10, 51A15, 51A40, 51M30.

“…Definition 2.1). In [2][3][4][5][6] we exhibit many examples of topological non-Clifford parallelisms.…”

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confidence: 99%

Clifford parallelism: old and new definitions, and their use

Betten

2012

J. Geom.

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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 PG(3, 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 PG(3, R) onto those lines of PG(5, R) having no common point with the Klein quadric; a regular parallelism P of PG(3, 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 PG(3, 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 PG(3, 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 Aute(P C ) of all automorphic collineations and dualities of the Clifford parallelism P C and show Aute(P C ) ∼ = (SO3R × SO3R) Z2.Mathematics Subject Classification (2010). 51A15, 51M30, 51H10.Keywords. Topological parallelism, complexification, focal line of a regular spread, imaginary quadric, Klein's correspondence of line geometry, elliptic displacement.