Abstract
Betten and Riesinger (Aequ Math 81:227–250, 2011) constructed parallelisms of \({\mathrm{PG}(3,\mathbb {R})}\) with automorphism group \({\mathrm{SO}(3, \mathbb {R})}\) by applying the reducible \({\mathrm{SO}(3, \mathbb {R})}\)-action to a rotational Betten spread. This was generalized by Löwen (Rotational spreads and rotational parallelisms and oriented parallelisms of PG(3,\({\mathbb {R}}\)). arXiv:1804.07615, 2018) so as to include oriented parallelisms (i.e., parallelisms of oriented lines). In this way, a much larger class of examples was produced. Here we show that, apart from Clifford parallelism, these are the only topological parallelisms admitting an automorphism group of dimension 3 or larger. In particular, we show that a topological parallelism admitting the irreducible action of \({\mathrm{SO}(3, \mathbb {R})}\) is Clifford.
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Löwen, R. Parallelisms of \(\mathrm{PG}(3,\mathbb R)\) admitting a 3-dimensional group. Beitr Algebra Geom 60, 333–337 (2019). https://doi.org/10.1007/s13366-018-0418-2
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DOI: https://doi.org/10.1007/s13366-018-0418-2