Parallelisms of \(\text{PG}(3, {\mathbb R})\) composed of non-regular spreads
In a series of articles the authors constructed topological regular parallelisms of the real projective 3-space PG\((3,\mathbb{R})\), that is, families \textbf{P} of regular spreads such that each line of PG\((3,\mathbb{R})\) is contained in exactly one spread of \textbf{P} and such that the operation of drawing a line \textbf{P}-parallel to a given line through a given point is continuous with respect to the natural topologies on the point set and lines set of PG\((3,{\mathbb R})\). In the paper under review the authors present two new, very different and beautiful constructions of parallelisms that are not regular. The first method is based on the spreads obtained in [Arch. Math. 21, 100--102 (1970; Zbl 0204.53501)] by the first author. These spreads \(\mathcal{S}(\mathfrak{B}_F)\) are determined by a homeomorphism \(F\) of \({\mathbb R}^{\geq 0}\). The second author [Geom. Dedicata 40, 145--163 (1991; Zbl 0742.51014)] has shown that, in affine coordinates with respect to a suitable plane at infinity, each such spread is composed of reguli contained in certain coaxial, concentric, rotational one-sheeted hyperboloids. If SO\(_3({\mathbb R})\) is the group of all rotations about the common center, then it is shown that the family \(\{g(\mathcal{S}(\mathfrak{B}_F ))\mid g \in \mathrm{SO}_3({\mathbb R})\}\) of spreads is a topological parallelism, which is irregular unless \(F\) is linear. The authors furthermore investigate rotational spreads. These are spreads \(\mathcal{S}\) of PG\((3,\mathbb{R})\), considered as the projective closure of a real Euclidean 3-space \(E\), for which there exists an (affine) line \(A\) such that \(\mathcal{S}\) is invariant under each rotation about \(A\). It is shown that \(\mathcal{S}\) must contain \(A\) and the line \(U\) at infinity of the planes orthogonal to \(A\), and that \(\mathcal{S}\) is composed of \(A\), \(U\), and a system of reguli carried by pairwise disjoint coaxial one-sheeted rotational hyperboloids in \(E\). The authors then prove that Betten's spreads \(\mathcal{S}(\mathfrak{B}_F)\) are precisely the rotational topological spreads having coaxial concentric carrier hyperboloids. Moreover, an example is given of a rotational spread having coaxial acentric carrier hyperboloids. For the second method of constructing parallelisms the authors examine the piecewise regular spreads obtained in [Math. Z. 128, 129--151 (1972; Zbl 0231.50011)] by the first author. These spreads \(\mathcal{S}(\mathfrak{B}_{72,w})\) depend on a real number \(w>1\) (\(w=1\) yields the regular spread) and are so-called PR2-spreads, that is, piecewise regular spreads that are composed of two segments of regular spreads joined together along a common regulus. Under the Klein correspondence between lines of projective 3-space and points of the Klein quadric in projective 5-space each regular segment of the PR2-spread gives rise to a segment on a line; furthermore, the underlying lines are parallel. From this correspondence a geometric interpretation of the parameter \(w\) is given as a cross-ratio of certain lines in projective 5-space. Starting from a suitable family \(\mathbf{S}\) of pairs of parallel linear segments a flock of a fixed elliptic quadric \(Q\) is obtained for each element of \(\mathbf{S}\). Embedding \(Q\) and with it each of these flocks into the Klein quadric and applying the Thas-Walker construction yields a spread for each member of \(\mathbf{S}\). The parallelism \(\mathbf{P}_{72}\) obtained in this way is composed of proper piecewise regular spreads, except for one member which is a regular spread. It is shown that \(\mathbf{P}_{72}\) is not topological and that the characteristic cross-ratios for each spread in \(\mathbf{P}_{72}\) cover precisely the closed interval \([1,3]\).
scientific article; zbMATH DE number 5909091
scientific article