Regular parallelisms on PG(3,R) from generalized line stars: The oriented case

TL;DR

This paper extends the theory of regular parallelisms in PG(3,R) to the oriented case, revealing more possibilities and complexities through refined dual object constructions and detailed orientation analysis.

Contribution

It introduces a refined approach to characterize oriented regular parallelisms using dual objects, expanding understanding beyond the non-oriented case.

Findings

More oriented parallelisms than non-oriented ones.

Complexity in classifying oriented regular spreads.

Refined dual object constructions for oriented cases.

Abstract

Using the Klein correspondence, regular parallelisms of PG(3,R) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3,R). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious.