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Journal of Geometry

, 110:11 | Cite as

Rotational spreads and rotational parallelisms and oriented parallelisms of \({\mathrm{PG}(3,{\mathbb {R}})}\)

  • Rainer LöwenEmail author
Article

Abstract

We introduce topological parallelisms of oriented lines (briefly called oriented parallelisms). Every topological parallelism (of lines) on \({\mathrm{PG}(3,{\mathbb {R}})}\) gives rise to a parallelism of oriented lines, but we show that even the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way. In fact we determine all parallelisms of both types that admit a reducible \({\mathrm{SO}_3 {\mathbb {R}}}\)-action (only the Clifford parallelism admits a larger group (Löwen in Innov Incid Geom. arXiv:1702.03328), and it turns out surprisingly that there are far more oriented parallelisms of this kind than ordinary parallelisms. More specifically, Betten and Riesinger (Aequ Math 81:227–250, 2011) construct ordinary parallelisms by applying \({\mathrm{SO}_3 {\mathbb {R}}}\) to rotational Betten spreads. We show that these are the only ordinary parallelisms compatible with this group action, but also the ‘acentric’ rotational spreads considered by them yield oriented parallelisms. The automorphism group of the resulting (oriented or non-oriented) parallelisms is always \({\mathrm{SO}_3 {\mathbb {R}}}\), no matter how large the automorphism group of the non-regular spread is. The isomorphism type of the parallelism depends not only on the isomorphism type of the spread used, but also on the rotation group applied to it. We also study the rotational Betten spreads used in this construction and their automorphisms.

Keywords

Rotational Betten spread topological parallelism reducible \({\mathrm{SO}_3 {\mathbb {R}}}\)-action 

Mathematics Subject Classification

51H10 51A15 51M30 

Notes

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

References

  1. 1.
    Betten, D.: Nicht-desarguessche 4-dimensionale Ebenen. Arch. Math. 21, 100–102 (1970)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Betten, D.: 4-dimensionale Translationsebenen mit genau einer Fixrichtung. Geom. Dedic. 3, 405–440 (1975)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Betten, D.: 4-dimensionale Translationsebenen mit 7-dimensionaler Kollineationsgruppe. J. Reine Angew. Math. 285, 126–148 (1976)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Betten, D.: 4-dimensionale Translationsebenen mit kommutativer Standgruppe. Math. Z. 154, 125–141 (1977)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Betten, D., Riesinger, R.: Parallelisms of \({\rm PG}(3; {\mathbb{R}})\) composed of non-regular spreads. Aequ. Math. 81, 227–250 (2011)Google Scholar
  6. 6.
    Betten, D., Riesinger, R.: Clifford parallelism: old and new definitions, and their use. J. Geom. 102, 31–73 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Betten, D., Riesinger, R.: Collineation groups of topological parallelisms. Adv. Geom. 14, 175–189 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Betten, D., Riesinger, R.: Regular 3-dimensional parallelisms of PG(3, R). Bull. Belg. Math. Soc. Simon Stevin 22, 1–23 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Betten, D., Löwen, R.: Compactness of the automorphism group of a topological parallelism on real projective 3-space. Res. Math. 72, 1021–1030 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Klingenberg, W.: Lineare Algebra und Geometrie, 1st edn. Springer, Berlin (1984)CrossRefGoogle Scholar
  11. 11.
    Knarr, N.: Translation Planes. Springer Lecture Notes in Mathematics. Springer, New York (1995)Google Scholar
  12. 12.
    Löwen, R.: A characterization of Clifford parallelism by automorphisms. Innov. Incid. Geom. 17(1), 43–46 (2019)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Löwen, R.: Compactness of the automorphism group of a topological parallelism on real projective 3-space: the disconnected case. Bull. Belg. Math. Soc. Simon Stevin. 25(5) (2018). arXiv:1710.05558
  14. 14.
    Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R., Stroppel, M.: Compact Projective Planes. de Gruyter, Berlin (1995)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Analysis und AlgebraTechnische Universität BraunschweigBrunswickGermany

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