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.
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. If the collineation group is 2-dimensional, then the parallelism is rotational which means that there is a rotation group SO 2 (R) about some axis leaving the parallelism invariant. We give a construction method for the generalized line stars which induce these parallelisms.
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