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.
Highlights
It is a widely used strategy in mathematics to define a new structure by modifying a given one
The starting point is a projective double space (P, r ), that is, a projective space P together with a left parallelism and a right parallelism r on its line set such that the so-called double space axiom (DS) is satisfied
In case (A), H is a quaternion skew field with centre F, the left and right parallelisms do not coincide and, in general, there are Clifford-like parallelisms of P(HF ), r different from and r
Summary
It is a widely used strategy in mathematics to define a new structure by modifying a given one. Theorem 4.3 establishes that a Clifford-like parallelism of P(HF ), , r is Clifford precisely when it admits an automorphism that fixes all its parallel classes and acts non-trivially on the point set of the projective space P(HF ). In our proof of Theorem 4.3 we could use the fact that the existence of a proper non-trivial dilatation (namely a nonidentical collineation with a fixed point and the property that all parallel classes remain invariant) is possible only in the commutative case, i.e. in our case (B) (see Marchi and Perelli Cippo 1980, Teorema 2 or Karzel and Maxson 1984, (II.10)). In contrast to our considerations, in this beautiful result only the “size” of an automorphism group is taken into account and not its action on the parallel classes
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