## Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial

### A characterization of Clifford parallelism by automorphisms

Rainer Löwen

#### Abstract

Betten and Riesinger have shown that Clifford parallelism on real projective space is the only topological parallelism that is left invariant by a group of dimension at least 5. We improve the bound to 4. Examples of different parallelisms admitting a group of dimension $≤ 3$ are known, so 3 is the “critical dimension”.

#### Article information

Source
Innov. Incidence Geom. Algebr. Topol. Comb., Volume 17, Number 1 (2019), 43-46.

Dates
Accepted: 23 March 2017
First available in Project Euclid: 26 February 2019

https://projecteuclid.org/euclid.iig/1551206774

Digital Object Identifier
doi:10.2140/iig.2019.17.43

Mathematical Reviews number (MathSciNet)
MR3986546

Zentralblatt MATH identifier
06983417

#### Citation

Löwen, Rainer. A characterization of Clifford parallelism by automorphisms. Innov. Incidence Geom. Algebr. Topol. Comb. 17 (2019), no. 1, 43--46. doi:10.2140/iig.2019.17.43. https://projecteuclid.org/euclid.iig/1551206774

#### References

• M. Berger, Geometry II, Springer-Verlag, Berlin, 1987.
• D. Betten and R. Löwen, “Compactness of the automorphism group of a topological parallelism on real projective 3-space”, Results Math. 72:1-2 (2017), 1021–1030.
• D. Betten and R. Riesinger, “Generalized line stars and topological parallelisms of the real projective 3-space”, J. Geom. 91:1-2 (2009), 1–20.
• D. Betten and R. Riesinger, “Parallelisms of ${\rm PG}(3,\mathbb R)$ composed of non-regular spreads”, Aequationes Math. 81:3 (2011), 227–250.
• D. Betten and R. Riesinger, “Clifford parallelism: old and new definitions, and their use”, J. Geom. 103:1 (2012), 31–73.
• D. Betten and R. Riesinger, “Automorphisms of some topological regular parallelisms of ${\rm PG}(3,\mathbb{R})$”, Results Math. 66:3-4 (2014), 291–326.
• D. Betten and R. Riesinger, “Collineation groups of topological parallelisms”, Adv. Geom. 14:1 (2014), 175–189.
• W. Klingenberg, Lineare Algebra und Geometrie, Springer, 1984.
• H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, and M. Stroppel, Compact projective planes, De Gruyter Expositions in Mathematics 21, Walter de Gruyter & Co., Berlin, 1995.
• J. A. Tyrrell and J. G. Semple, Generalized Clifford parallelism, Cambridge Tracts in Mathematics and Mathematical Physics 61, Cambridge University Press, 1971.