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Journal of Geometry

, 110:2 | Cite as

Clifford-like parallelisms

  • Hans HavlicekEmail author
  • Stefano Pasotti
  • Silvia Pianta
Open Access
Article
  • 114 Downloads
Part of the following topical collections:
  1. Karzel Anniversary Topical Collection

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.

Keywords

Blend of parallelisms Clifford parallelism projective double space quaternion skew field 

Mathematics Subject Classification

51A15 51J15 

Notes

Acknowledgements

Open access funding provided by TU Wien (TUW).

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische UniversitätViennaAustria
  2. 2.DICATAM-Sez. MatematicaUniversità degli Studi di BresciaBresciaItaly
  3. 3.Dipartimento di Matematica e FisicaUniversità Cattolica del Sacro CuoreBresciaItaly

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