Locally classical stable planes
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Rainer Löwen
Abstract
We show that a simply connected stable plane with connected lines is isomorphic to an open subplane of a classical projective plane (i.e., a plane over the real or complex numbers, the quaternions or the octonions) if it has that property locally. We also give several examples indicating that our hypotheses cannot be relaxed much further.
Communicated by: T. Grundhöfer
References
[1] D. Betten, Topologische Geometrien auf dem Möbiusband. Math. Z. 107 (1968), 363–379. MR238331 Zbl 0167.4900210.1007/BF01110068Suche in Google Scholar
[2] H. Busemann, The geometry of geodesics. Academic Press 1955. MR75623 Zbl 0112.37002Suche in Google Scholar
[3] U. Felgner, Hilberts “Grundlagen der Geometrie” und ihre Stellung in der Geschichte der Grundlagendiskussion. Jahresber. Dtsch. Math.-Ver. 115 (2014), 185–206. MR3158103 Zbl 1305.0102810.1365/s13291-013-0071-5Suche in Google Scholar
[4] C. D. Feustel, Homotopic arcs are isotopic. Proc. Amer. Math. Soc. 17 (1966), 891–896. MR196724 Zbl 0152.4090410.1090/S0002-9939-1966-0196724-4Suche in Google Scholar
[5] T. Grundhöfer, R. Löwen, Linear topological geometries. In: Handbook of incidence geometry, 1255–1324, North-Holland 1995. MR1360738 Zbl 0824.5101110.1016/B978-044488355-1/50025-6Suche in Google Scholar
[6] P. Leymann, Continuous deformation of differentiable planes. Doctoral Dissertation, Universität Würzburg, 2023.Suche in Google Scholar
[7] R. Löwen, Vierdimensionale stabile Ebenen. Geometriae Dedicata 5 (1976), 239–294. MR428187 Zbl 0344.5000310.1007/BF00145961Suche in Google Scholar
[8] R. Löwen, A local “fundamental theorem” for classical topological projective spaces. Arch. Math. (Basel) 38 (1982), 286–288. MR656196 Zbl 0486.5101410.1007/BF01304789Suche in Google Scholar
[9] R. Löwen, Topology and dimension of stable planes: on a conjecture of H. Freudenthal. J. Reine Angew. Math. 343 (1983), 108–122. MR705880 Zbl 0524.5701110.1515/crll.1983.343.108Suche in Google Scholar
[10] R. Löwen, Ends of surface geometries, revisited. Geom. Dedicata 58 (1995), 175–183. MR1358231 Zbl 0845.5101210.1007/BF01265636Suche in Google Scholar
[11] R. Löwen, E. Soytürk, G. F. Steinke, Blowing up points and embedding flat stable planes in the nonorientable compact surface of genus one. Topology Appl. 155 (2008), 1041–1055. MR2419283 Zbl 1146.5101010.1016/j.topol.2008.01.012Suche in Google Scholar
[12] R. Löwen, G. F. Steinke, Actions of
[13] J. Martin, D. Rolfsen, Homotopic arcs are isotopic. Proc. Amer. Math. Soc. 19 (1968), 1290–1292. MR232394 Zbl 0179.2810210.1090/S0002-9939-1968-0232394-6Suche in Google Scholar
[14] F. R. Moulton, A simple non-Desarguesian plane geometry. Trans. Amer. Math. Soc. 3 (1902), 192–195. MR1500595 JFM 33.0497.0410.1090/S0002-9947-1902-1500595-3Suche in Google Scholar
[15] C. Polley, Lokal desarguessche Salzmann-Ebenen. Arch. Math. (Basel) 19 (1968), 553–557. MR238155 Zbl 0181.4820310.1007/BF01898780Suche in Google Scholar
[16] C. Polley, Lokal desarguessche Geometrien auf dem Möbiusband. Arch. Math. (Basel) 23 (1972), 346–347. MR315574 Zbl 0248.5003010.1007/BF01304893Suche in Google Scholar
[17] C. Polley, Lokales Verhalten zweidimensionaler topologischer Geometrien. Doctoral Dissertation, Universität Tübingen, 1972.Suche in Google Scholar
[18] H. Salzmann, Kollineationsgruppen ebener Geometrien. Math. Z. 99 (1967), 1–15. MR212671 Zbl 0146.4160410.1007/BF01118683Suche in Google Scholar
[19] H. Salzmann, Geometries on surfaces. Pacific J. Math. 29 (1969), 397–402. MR248855 Zbl 0181.2330110.2140/pjm.1969.29.397Suche in Google Scholar
[20] H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel, Compact projective planes, volume 21 of De Gruyter Expositions in Mathematics. de Gruyter 1995. MR1384300 Zbl 0851.5100310.1515/9783110876833Suche in Google Scholar
[21] M. Stroppel, Endomorphisms of stable planes. Sem. Sophus Lie 2 (1992), 75–81. MR1188636 Zbl 0789.51018Suche in Google Scholar
[22] M. Stroppel, A note on Hilbert and Beltrami systems. Results Math. 24 (1993), 342–347. MR1244287 Zbl 0791.5101110.1007/BF03322342Suche in Google Scholar
[23] C. Thomassen, The Jordan–Schönflies theorem and the classification of surfaces. Amer. Math. Monthly 99 (1992), 116–130. MR1144352 Zbl 0773.5700110.1080/00029890.1992.11995820Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
