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

摘要

Let Q be an elliptic quadric of the real projective 3-space PG(3, mathbbR{mathbb{R}}) =: Õ₃prod_{3} and denote by Q¬i the set of non-interior points with respect to Q. A simple covering mathfrakGmathfrak{G} of Q¬i by 2-secants of Q is called generalized line star with respect to Q. The generalized line star mathfrakGmathfrak{G} is called continuous, if the determination of the unique line of mathfrakGmathfrak{G} through a given point of Q¬i is continuous. A parallelismis a family P of spreads such that each line of Õ₃prod_{3} is contained in exactly one spread of P; two lines of Õ₃prod_{3} 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 mathbbP{mathbb{P}} such that mathbbP(mathfrakG){mathbb{P}}(mathfrak{G}) is a parallelism of Õ₃prod_{3} ; cf. Theorem 1.4 below. In the present paper the authors prove: If mathfrakGmathfrak{G} is continuous, then mathbbP(mathfrakG){mathbb{P}}(mathfrak{G}) is a topological parallelism of Õ₃prod_{3}. The authors construct continuous generalized line stars by composing so-called 3-pencils (cf. Definition 3.3 below). If 프 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 프 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.