Parallelisms of $${{\rm PG}(3, {\mathbb R})}$$ composed of non-regular spreads

Abstract

Any continuous strictly monotonic function $${F : {{\mathbb R}^{{\geq}0} \to {\mathbb R}}}$$ with F(0) = 0 and F(t) → ∞ for t → ∞ gives rise to a topological rotational spread of $${{\rm PG}\,(3,{\mathbb R})}$$ ; this spread is non-regular, if F is not linear. The action of the group $${SO_3({\mathbb R})}$$ on this spread yields a topological parallelism of $${{\rm PG}\,(3,{\mathbb R})}$$ . The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P 72 of $${{\rm PG}\,(3,{\mathbb R})}$$ which is composed of piecewise regular spreads each consisting of two segments which are tacked together along a common regulus. Using Klein’s correspondence of line geometry and the Thas–Walker construction we represent every parallel class of P 72 via two parallel half-lines being non-interior to a given sphere in $${{\mathbb R}^3}$$ . The parallelism P 72 contains exactly one regular spread, all other members of P 72 are piecewise regular spreads with two segments. However, P 72 is not topological.