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

By a totally regular parallelism of the real projective 3-space P₃:=PG(3, mathbb R){Pi_3:={{rm PG}}(3, mathbb {R})} we mean a family T of regular spreads such that each line of Π ₃ 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 π ₅ associated with the Klein quadric H ₅ (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H ₅, i.e., a family H of elliptic subquadrics of H ₅ such that each point of H ₅ is on exactly one subquadric of H. Moreover, {p₅(span  l(X))|X ? T}=:H_T{{pi_5({{rm span}} ,lambda(mathcal {X}))vertmathcal {X}inbf{T}}=:mathcal {H}_{bf{T}}} is a hyperflock determining line set, i.e., a set Z{mathcal {Z}} of 0-secants of H ₅ such that each tangential hyperplane of H ₅ contains exactly one line of Z{mathcal {Z}} . We say that dim(span H_T)=:d_T{{{rm dim}}({{rm span}},mathcal {H}_{bf{T}})=:d_{bf{T}}} is the dimension of T and that T is a d _T - 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 G{mathcal{G}} is a hyperflock determining line set, then {l^-1 (p₅(X) ?H₅) | X ? G}{{lambda^{-1},{rm (}pi_5(X){,cap H_5),|, Xinmathcal{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.