Characterising Clifford parallelisms among Clifford-like parallelisms

Abstract

We recall the notions of Clifford and Clifford-like parallelisms in a 3-dimensional projective double space. In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it. In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones.

Projective Metric Geometry and Clifford Algebras

“…(a) The canonical filtration of Cl(V , Q); (b) for any homogeneous m ∈ Cl × (V , Q), the canonical action of the left translation λ m (right translation ρ m ) on the union of the projective spaces P Cl 0 (V , Q) and P Cl 1 (V , Q) ; (c) the canonical action of the reversal α on the union of the projective spaces P Cl 0 (V , Q) and P Cl 1 (V , Q) ; (19); (e) the point set M(V , Q) arising from the Lipschitz monoid Lip(V , Q) according to (21) and the group G(V , Q) ∼ = Lip × (V , Q)/F × as in (22); (f) the action of the group G(V , Q) on the projective space P(V , Q) as in (25).…”

“…Ultimately, one is lead to the following question: to which extent does the general theory of kinematic spaces (including the theory of Clifford parallelism) overlap with our findings as sketched in Remark 5.3. We refer, among others, to [5,22,33,47]. Going up one dimension (dim V = 4), one finds M 0 (V , Q) and M 1 (V , Q) as siblings of the classical Study quadric (see [52]) in a projective space of dimension 7; also here there are many results scattered over the literature; see [24, p. 463], [35, 3.4.2], [36, 6.2] and [51,Ch.…”

Projective metric geometry and Clifford algebras

“…the twisted adjoint representation of the quotient group Lip × (V, Q)/F × . By virtue of the inverse of ( 23), the twisted adjoint representation (24) and the canonical action of O ′ (V, Q) on P(V, Q), the group G(V, Q) as in (22) acts on the projective space P(V, Q). Explicitly, for all F p ∈ G(V, Q) and all X ∈ P(V, Q), this action of G(V, Q) takes the form…”

“…(e) the point set M(V, Q) arising from the Lipschitz monoid Lip(V, Q) according to (21) and the group G(V, Q) Lip × (V, Q)/F × as in (22);…”

“…Ultimately, one is lead to the following question: to which extent does the general theory of kinematic spaces (including the theory of Clifford parallelism) overlap with our findings as sketched in Remark 5.3. We refer, among others, to [5], [22], [33] and [47]. Going up one dimension (dim V = 4), one finds M 0 (V, Q)…”