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

Abstract

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.

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

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In [8] the collineation groups of some known 5-, 4-and 3-dimensional topological regular parallelisms of PG(3, 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.

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“…2 Modified Pl ücker map and the group PO 6 (R, 3) We recall the notions of Pl ücker map, Pl ücker coordinates and Klein quadric, see [8, 2.2] and [13, p.363-367]. Let PG(3, R) = (P 3 (R), L 3 ) be the 3-dimensional projective point-line geometry, then we define for each line L ∈ L 3 a point of the projective space P 5 (R) in the following way: Choose two different points (s 0 , s 1 , s 2 , s 3 )R and (t 0 , t 1 , t 2 , t 3 )R on L and set (s 0 , s 1 , s 2 , s 3 )R ∨ (t 0 , t 1 , t 2 , t 3 )R = L → (p 0 , p 1 , p 2 , p 3 , p 4 , p 5 )R with…”

Regular parallelisms on PG(3,R) admitting a 2-torus action

“…In the 3-dimensional case, the hfd line set H defining a regular parallelism can be replaced by a simpler object, namely, by a compact so-called generalized line star or gl star S. The gl star is obtained by applying to the set H of lines the polarity π 3 induced by π 5 on the 3-space P 3 generated by H, S = π 3 π 5 (Π). This is due to Betten and Riesinger [2]; see [12] for a simpler approach and proof. The defining property of a gl star, which is equivalent to the fact that the gl star corresponds to a 3-dimensional parallelism, is this: S is a set of lines in a 3-dimensional subspace P 3 of PG(5, R), and Q = P 3 ∩ K is an elliptic quadric such that every line in S meets Q in two points, and every point of P 3 not in the interior of Q is incident with precisely one line from S. Here, a point is called exterior if it is incident with an exterior line, that is, a 0-secant of Q.…”

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

“…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.…”