Clifford-like parallelisms

TLDR: This work characterises the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms and establishes necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

Abstract: Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space $$({\mathbb P},{\mathrel {\parallel _{\ell }}},{\mathrel {\parallel _{r}}})$$ over a quaternion skew field we characterise the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms $$\mathrel {\parallel _{\ell }}$$ and $$\mathrel {\parallel _{r}}$$ , in a geometric and an algebraic way. Finally, we establish necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.