Constructing topological parallelisms of PG(3, ℝ) via rotation of generalized line pencils

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.

Section: Regulus Spread Parallelismmentioning

confidence: 99%

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

mentioning

confidence: 99%

Clifford parallelism: old and new definitions, and their use

2012

J. Geom.

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“…Denote by Q ¬ i ⊂ span Q the set of non-interior points with respect to Q and by S (2) Q the set of all 2-secants of Q. A generalized line star S with respect to 2 Q determines the mapping κ S : Q ¬ i ⊂ span Q → S (2) Q ; x → κ S (x) (1.4) where κ S (x) denotes the unique line of S being incident with the point x ∈ Q ¬ i ; we speak of the "S-construction κ S ". As Q ¬ i carries the topology of the ambient projective 3-space span Q and as S (2) Q is part of the Grassmann manifold G Q…”

Section: Vol 91 (2008)mentioning

confidence: 99%

“…All examples from [1,Section 5], all examples from [2], and all examples from Section 3 of the present paper are topological axial generalized line stars. In [1,Section 6] we exhibit non-topological axial examples and also non-topological nonaxial examples.…”

Section: Definition 17 a Generalized Line Starmentioning

confidence: 99%

Generalized Line Stars and Topological Parallelisms of the Real Projective 3-Space

2008

J. geom.

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Let Q be an elliptic quadric of the real projective 3-space PG(3, R) =: Π3 and denote by Q ¬ i the set of non-interior points with respect to Q. A simple covering S of Q ¬ i by 2-secants of Q is called generalized line star with respect to Q. The generalized line star S is called continuous, if the determination of the unique line of S through a given point of Q ¬ i is continuous. A parallelism is a family P of spreads such that each line of Π3 is contained in exactly one spread of P; two lines of Π3 are P-parallel if, and only if, they are members of the same spread of P. The parallelism P is called topological, if the operation of drawing a line P-parallel to a given line through a given point is continuous. In [1] the authors give a construction P such that P(S) is a parallelism of Π3; cf. Theorem 1.4 below.In the present paper the authors prove: If S is continuous, then P(S) is a topological parallelism of Π3. The authors construct continuous generalized line stars by composing so-called 3-pencils (cf. Definition 3.3 below). If A is a generalized line star of [1, Section 5] or of [2] or from Section 3 of the present paper, then there exists a line A such that each line of A has non-empty intersection with A; we speak of an axial generalized line star. In Section 4 examples of non-axial continuous generalized line stars are constructed.Mathematics Subject Classification (2000). 51H10; 51A40, 51M30.

“…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

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.