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

Abstract

Abstract Let Q be an elliptic quadric of the real projective 3-space PG(3, ℝ) and denote by Q ¬i the set of non-interior points with respect to Q. A simple covering of Q ¬i by 2-secants of Q is called generalized line star with respect to Q. In [D. Betten, R. Riesinger, Topological parallelisms of the real projective 3-space. Results Math. 47 (2005), 226–241. MR2153495 (2006b:51009) Zbl 1088.51005] the authors give a construction P such that is a parallelism of PG(3, ℝ); cf. Theorem 1 below. In the present article, we are mainly interested in the plane analogues of gl-stars: the gl-pencils with respect to a conic; cf. Definition 3. If a gl-star is generated by rotating a gl-pencil about an axis , then we call a latitudinal gl-star and a latitudinal parallelism. We present a general construction process for gl-pencils by giving generating functions. Along this way we prove the existence of non-Clifford latitudinal parallelisms in PG(3, ℝ); moreover, we show that each latitudinal parallelism is topological.