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

Abstract

Any continuous strictly monotonic function F : R ≥0 → R with F (0) = 0 and F (t) → ∞ for t → ∞ gives rise to a topological rotational spread of PG (3, R); this spread is non-regular, if F is not linear. The action of the group SO 3 (R) on this spread yields a topological parallelism of PG (3, R). The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P 72 of PG (3, 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 noninterior to a given sphere in 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.Mathematics Subject Classification (2010). 51H10, 51A15, 51A40, 51M30.

Clifford parallelism: old and new definitions, and their use

“…8 a very illustrative generation of a Clifford parallelism in a projectively extended Euclidean 3-space E 3 is taken from [6]: If a rotational regular spread S of E 3 with center o is submitted to the group of all rotations about o, then a Clifford parallelism V is generated, in other words, V is the orbit of S under the group SO 3 R. For this assertion we give two proofs thus contraposing, at the one hand (Sect. 8.1), the argumentation via complexification to verify the Klein definition (cf.…”

“…Definition 2.1). In [2][3][4][5][6] we exhibit many examples of topological non-Clifford parallelisms.…”

“…(b) If F is linear, then P F is a Clifford parallelism, (c) otherwise P F is irregular.Part (a) and (c) of Theorem 8.1 are proved in[6], a proof of part (b) is missing. As we now have a clear idea of the concept 'Clifford parallelism', so we can give belated proofs of part (b) of Theorem 8.1.…”