Parallelisms of $$\mathrm{PG}(3,\mathbb R)$$ PG ( 3 , R ) admitting a 3-dimensional group

Abstract

Betten and Riesinger [1] constructed Parallelisms of PG(3, R) with automorphism group SO 3 R by applying the reducible SO 3 R-action to a rotational Betten spread. This was generalized in [8] 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 SO 3 R is Clifford.MSC 2010: 51H10, 51A15, 51M30