Home Mathematics Modified classical flat Minkowski planes
Article
Licensed
Unlicensed Requires Authentication

Modified classical flat Minkowski planes

  • Günter F. Steinke EMAIL logo
Published/Copyright: July 22, 2017
Become an author with De Gruyter Brill
Author Information Explore this Subject
Advances in Geometry
From the journal Advances in Geometry Volume 17 Issue 3

Abstract

We construct anew family of flat Minkowski planes that admit the simple group PSL2(ℝ) as a group of automorphisms. These planes are obtained from the classical flat Minkowski plane by ‘bending’ circles at a distinguished circle.

Key words: Classical real Minkowski plane; flat Minkowski plane; automorphism group; group dimension; Klein–Kroll type
MSC 2010: 51H15

R. Löwen

References

[1] R. Artzy, H. Groh, Laguerre and Minkowski planes produced by dilatations. J. Geom. 26 (1986), 1-20. MR837764 Zbl 0598.5100410.1007/BF01221003Search in Google Scholar

[2] D. Betten, Die Projektivitätengruppe der Moulton-Ebenen. J. Geom. 13 (1979), 197-209. MR560441 Zbl 0426.5100910.1007/BF01919755Search in Google Scholar

[3] E. Hartmann, Beispiele nicht einbettbarer reeller Minkowski-Ebenen. Geom. Dedicata10 (1981), 155-159. MR608137 Zbl 0454.5100410.1007/BF01447419Search in Google Scholar

[4] J. Jakóbowski, H.-J. Kroll, A. Matraś, Minkowski planes admitting automorphism groups of small type. J. Geom. 71 (2001), 78-84. MR1848314 Zbl 1003.5100410.1007/s00022-001-8554-4Search in Google Scholar

[5] M. Klein, A classification of Minkowski planes. PhD Thesis, University of Haifa, 1985.10.1007/BF01231025Search in Google Scholar

[6] M. Klein, Classification of Minkowski planes by transitive groups of homotheties. J. Geom. 43 (1992), 116-128. MR1148261 Zbl 0746.5100910.1007/BF01245947Search in Google Scholar

[7] M. Klein, H.-J. Kroll, A classification of Minkowski planes. J. Geom. 36 (1989), 99-109. MR1022339 Zbl 0694.5100510.1007/BF01231025Search in Google Scholar

[8] H.-J. Kroll, Eine Kennzeichnung der miquelschen Minkowski-Ebenen durch Transitivitätseigenschaften. Geom. Dedicata58 (1995), 75-77. MR1353301 Zbl 0838.5100210.1007/BF01263477Search in Google Scholar

[9] H.-J. Kroll, A. Matraś, Minkowski planes with Miquelian pairs. Beiträge Algebra Geom. 38 (1997), 99-109. MR1447989 Zbl 0885.51006Search in Google Scholar

[10] R. Löwen, Projectivities and the geometric structure of topological planes. In: Geometry— von Staudt’s point of view(Proc. NATO Adv. Study Inst., Bad Windsheim, 1980), volume 70 of NATO Adv. Study Inst. Ser., Ser. C:Math. Phys. Sci., 339-372, Reidel 1981. MR621322 Zbl 0458.5101510.1007/978-94-009-8489-9_13Search in Google Scholar

[11] R. Löwen, A local “fundamental theorem” for classical topological projective spaces. Arch. Math. (Basel) 38 (1982), 286-288. MR656196 Zbl 0486.5101410.1007/BF01304789Search in Google Scholar

[12] W. A. Pierce, Moulton planes. Canad. J. Math. 13 (1961), 427-436. MR0123960 Zbl 0103.1340410.4153/CJM-1961-035-6Search in Google Scholar

[13] W. A. Pierce, Collineations of affine Moulton planes. Canad. J. Math. 16 (1964), 46-62. MR0157264 Zbl 0135.3930310.4153/CJM-1964-005-9Search in Google Scholar

[14] W. A. Pierce, Collineations of projective Moulton planes. Canad. J. Math. 16 (1964), 637-656. MR0166677 Zbl 0136.1530210.4153/CJM-1964-064-4Search in Google Scholar

[15] B. Polster, The piecewise projective group as group of projectivities. Results Math. 30 (1996), 122-135. MR1402430 Zbl 0871.5100810.1007/BF03322185Search in Google Scholar

[16] B. Polster, Toroidal circle planes that are not Minkowski planes. J. Geom. 63 (1998), 154-167. MR1651572 Zbl 0928.5101110.1007/BF01221246Search in Google Scholar

[17] B. Polster, G. Steinke, Geometries on surfaces, volume 84 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2001. MR1889925 Zbl 0995.5100410.1017/CBO9780511549656Search in Google Scholar

[18] B. Polster, G. F. Steinke, Criteria for two-dimensional circle planes. Beiträge Algebra Geom. 35 (1994), 181-191. MR1312665 Zbl 0821.51013Search in Google Scholar

[19] H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel, Compact projective planes. De Gruyter 1995. MR1384300 Zbl 0851.5100310.1515/9783110876833Search in Google Scholar

[20] A. Schenkel, Topologische Minkowski-Ebenen. PhD thesis, Erlangen-Nürnberg, 1980.Search in Google Scholar

[21] G. Steinke, An extension property for classical topological Benz planes. Arch. Math. (Basel) 41 (1983), 190-192. MR719424 Zbl 0522.5100710.1007/BF01196877Search in Google Scholar

[22] G. F. Steinke, Some Minkowski planes with 3-dimensional automorphism group. J. Geom. 25 (1985), 88-100. MR815010 Zbl 0589.5102710.1007/BF01222947Search in Google Scholar

[23] G. F. Steinke, Topological affine planes composed of two Desarguesian half planes and projective planes with trivial collineation group. Arch. Math. (Basel) 44 (1985), 472-480. MR792373 Zbl 0564.5100610.1007/BF01229332Search in Google Scholar

[24] G. F. Steinke, A family of 2-dimensional Minkowski planes with small automorphism groups. Results Math. 26 (1994), 131-142. MR1290685 Zbl 0812.5100910.1007/BF03322292Search in Google Scholar

[25] G. F. Steinke, A family of flat Minkowski planes admitting 3-dimensional simple groups of automorphisms. Adv. Geom. 4 (2004), 319-340. MR2071809 Zbl 1067.5100810.1515/advg.2004.019Search in Google Scholar

[26] G. F. Steinke, On the Klein-Kroll types of flat Minkowski planes. J. Geom. 87 (2007), 160-178. MR2372525 Zbl 1155.5100910.1007/s00022-007-1913-zSearch in Google Scholar

[27] G. F. Steinke, Anote on Minkowski planes admitting groups of automorphisms of some large types with respect to homotheties. J. Geom. 103 (2012), 531-540. MR3017061 Zbl 1278.5100410.1007/s00022-013-0143-9Search in Google Scholar

Received: 2015-9-18
Published Online: 2017-7-22
Published in Print: 2017-7-26

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  3. Classification of 3-dimensional left-invariant almost paracontact metric structures
  5. On the topology of the spaces of curvature constrained plane curves
  7. On deformations of parallel G2 structures and almost contact metric structures
  9. A flag representation of projection functions
  11. A fundamental theorem for submanifolds of multiproducts of real space forms
  13. Intriguing sets of quadrics in PG(5, q)
  15. Successive radii and ball operators in generalized Minkowski spaces
  17. Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms
  19. The Archimedean projection property
  21. Kernels of numerical pushforwards
  23. Modified classical flat Minkowski planes
https://doi.org/10.1515/advgeom-2017-0026
Keywords for this article
Classical real Minkowski plane; flat Minkowski plane; automorphism group; group dimension; Klein–Kroll type

Articles in the same Issue

  1. Frontmatter
  3. Classification of 3-dimensional left-invariant almost paracontact metric structures
  5. On the topology of the spaces of curvature constrained plane curves
  7. On deformations of parallel G2 structures and almost contact metric structures
  9. A flag representation of projection functions
  11. A fundamental theorem for submanifolds of multiproducts of real space forms
  13. Intriguing sets of quadrics in PG(5, q)
  15. Successive radii and ball operators in generalized Minkowski spaces
  17. Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms
  19. The Archimedean projection property
  21. Kernels of numerical pushforwards
  23. Modified classical flat Minkowski planes
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/advgeom-2017-0026/pdf
Scroll to top button