Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
-
Rainer Löwen
Abstract
Using the Klein correspondence, regular parallelisms of PG(3, ℝ) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3, ℝ). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious.
Communicated by: T. Grundhöfer
References
[1] D. Betten, R. Riesinger, Topological parallelisms of the real projective 3-space. Results Math. 47 (2005), 226–241. MR2153495 Zbl 1088.51005Search in Google Scholar
[2] D. Betten, R. Riesinger, Generalized line stars and topological parallelisms of the real projective 3-space. J. Geom. 91 (2009), 1–20. MR2471992 Zbl 1168.51003Search in Google Scholar
[3] D. Betten, R. Riesinger, Hyperflock determining line sets and totally regular parallelisms of PG(3, ℝ). Monatsh. Math. 161 (2010), 43–58. MR2670230 Zbl 1208.51009Search in Google Scholar
[4] D. Betten, R. Riesinger, Automorphisms of some topological regular parallelisms of PG(3, ℝ). Results Math. 66 (2014), 291–326. MR3272630 Zbl 1316.51001Search in Google Scholar
[5] D. Betten, R. Riesinger, Regular 3-dimensional parallelisms of PG(3, ℝ). Bull. Belg. Math. Soc. Simon Stevin 22 (2015), 813–835. MR3435084 Zbl 1336.51008Search in Google Scholar
[6] T. Grundhöfer, R. Löwen, Linear topological geometries. In: Handbook of incidence geometry, 1255–1324, North-Holland 1995. MR1360738 Zbl 0824.51011Search in Google Scholar
[7] V. Guillemin, A. Pollack, Differential topology. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1974. MR0348781 Zbl 0361.57001Search in Google Scholar
[8] N. Knarr, Translation planes. Springer 1995. MR1439965 Zbl 0843.51004Search in Google Scholar
[9] R. Kühne, R. Löwen, Topological projective spaces. Abh. Math. Sem. Univ. Hamburg 62 (1992), 1–9. MR1182837 Zbl 0784.51012Search in Google Scholar
[10] R. Löwen, Projectivities and the geometric structure of topological planes. In: Geometry—von Staudt’s point of view Proc. Bad Windsheim, 1980), 339–372, Reidel, Dordrecht 1981. MR621322 Zbl 0458.51015Search in Google Scholar
[11] R. Löwen, Regular parallelisms from generalized line stars in P3ℝ: a direct proof. J. Geom. 107 (2016), 279–285. MR3519949 Zbl 1360.51007Search in Google Scholar
[12] R. Löwen, Rotational spreads and rotational parallelisms and oriented parallelisms of PG(3, ℝ). J. Geom. 110 (2019), Paper No. 11, 15 pages. MR3895741 Zbl 1410.51012Search in Google Scholar
[13] R. Löwen, G. F. Steinke, Regular parallelisms on PG(3, ℝ) admitting a 2-torus action. Bull. Belg. Math. Soc. Simon Stevin 28 (2021), 305–326. MR4355690 Zbl 1490.51001Search in Google Scholar
[14] K. Nagami, Cross sections in locally compact groups. J. Math. Soc. Japan 15 (1963), 301–303. MR158019 Zbl 0163.02801Search in Google Scholar
[15] H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel, Compact projective planes. De Gruyter 1995. MR1384300 Zbl 0851.51003Search in Google Scholar
[16] H. R. Salzmann, Topological planes. Advances in Math. 2 (1967), 1–60. MR220135 Zbl 0153.21601Search in Google Scholar
[17] A. A. Tuzhilin, Lectures on Hausdorff and Gromov–Hausdorff Distance Geometry. Lecture Notes, Peking University, 2019, arXiv:2012.00756v1 [math.MG]Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
Articles in the same Issue
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
