Regular $3$-dimensional parallelisms of $\mbox{{\rm PG}}(3,\mathbb R)$

Abstract

In [8] the collineation groups of some known 5-, 4- and 3-dimensional topological regular parallelisms of $PG(3,\mathbb R)$ were determined. In the present article we concentrate on 3-dimensional regular parallelisms and prove: the 3-dimensional regular parallelisms are exactly those which can be constructed from generalized line stars, see [3]. We determine the collineation groups of 3-dimensional regular parallelisms and show that only group dimension 1 or 2 is possible. If the collineation group is 2-dimensional, then the parallelism is rotational which means that there is a rotation group $SO_2(\mathbb R)$ about some axis leaving the parallelism invariant. We give a construction method for the generalized line stars which induce these parallelisms.