Hyperflock determining line sets and totally regular parallelisms of PG $${(3,\,{\mathbb{R}})}$$

Abstract

By a totally regular parallelism of the real projective 3-space \({\Pi_3:={{\rm PG}}(3, \mathbb {R})}\) we mean a family T of regular spreads such that each line of Π3 is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π5 associated with the Klein quadric H5 (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H5, i.e., a family H of elliptic subquadrics of H5 such that each point of H5 is on exactly one subquadric of H. Moreover, \({\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}}\) is a hyperflock determining line set, i.e., a set \({\mathcal {Z}}\) of 0-secants of H5 such that each tangential hyperplane of H5 contains exactly one line of \({\mathcal {Z}}\) . We say that \({{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}}\) is the dimension of T and that T is a dT- parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If \({\mathcal{G}}\) is a hyperflock determining line set, then \({\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}}\) is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.