Regular parallelisms from generalized line stars in ${P_{3} \mathbb{R}}$: a direct proof

Abstract

A generalized line star with respect to an elliptic quadric contained in the Klein quadric gives rise to a regular parallelism in real projective 3-space. This was shown by Betten and Riesinger (Results Math 47:226–241, 2005) using the Thas–Walker construction. They remark that the resulting description of the parallelism is equivalent to a much simpler one, but again the proof is hard. In practice, they always work with that simpler construction. We show that the Thas–Walker approach is not needed here. In fact, one can derive all relevant properties of the parallelism from the simple construction. We also give coordinate-free, short proofs of the topological properties of regular parallelisms in general and of those obtained from generalized line stars in particular.