Rotational spreads and rotational parallelisms and oriented parallelisms of $${\mathrm{PG}(3,{\mathbb {R}})}$$ PG ( 3 , R )

TLDR: In this paper, it was shown that the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way, and that there are far more oriented parallelism of this kind than ordinary parallelisms.

Abstract: We introduce topological parallelisms of oriented lines (briefly called oriented parallelisms). Every topological parallelism (of lines) on $${\mathrm{PG}(3,{\mathbb {R}})}$$ gives rise to a parallelism of oriented lines, but we show that even the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way. In fact we determine all parallelisms of both types that admit a reducible $${\mathrm{SO}_3 {\mathbb {R}}}$$ -action (only the Clifford parallelism admits a larger group (Lowen in Innov Incid Geom. arXiv:1702.03328 ), and it turns out surprisingly that there are far more oriented parallelisms of this kind than ordinary parallelisms. More specifically, Betten and Riesinger (Aequ Math 81:227–250, 2011) construct ordinary parallelisms by applying $${\mathrm{SO}_3 {\mathbb {R}}}$$ to rotational Betten spreads. We show that these are the only ordinary parallelisms compatible with this group action, but also the ‘acentric’ rotational spreads considered by them yield oriented parallelisms. The automorphism group of the resulting (oriented or non-oriented) parallelisms is always $${\mathrm{SO}_3 {\mathbb {R}}}$$ , no matter how large the automorphism group of the non-regular spread is. The isomorphism type of the parallelism depends not only on the isomorphism type of the spread used, but also on the rotation group applied to it. We also study the rotational Betten spreads used in this construction and their automorphisms.