Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A local characterization of smooth projective planes

Authors: Stefan Immervoll, Rainer Löwen and Ioachim Pupeza
Journal: Proc. Amer. Math. Soc. 138 (2010), 323-332
MSC (2000): Primary 51H25; Secondary 51H10
Published electronically: September 1, 2009
MathSciNet review: 2550198
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 2000, Bödi and Immervoll considered compact, connected smooth incidence geometries with mutually transversal point rows and mutually transversal line pencils. They made the very natural assumptions that the flag space is a $ 3l$-dimensional closed smooth submanifold of the product of the point space and the line space (both of which are $ 2l$-manifolds) and that both associated projections are submersions. They showed that then the number of joining lines of two distinct points and the number of intersection points of two distinct lines are constant. Here we prove that both constants are equal to one. Thus, smooth projective planes are characterized using only compactness and connectedness plus the purely local (in fact, infinitesimal) conditions stated above.

References [Enhancements On Off] (What's this?)

  • 1. N. Bourbaki, General Topology, Part 2. Hermann, Paris, 1966.MR 0205211 (34:5044b)
  • 2. R. Bödi, Smooth stable and projective planes, Habilitation Thesis, University of Tübingen, 1996.
  • 3. R. Bödi, $ 16$-dimensional smooth projective planes with large collineation groups, Geom. Dedic. 72, pp. 283-298, 1998. MR 1647708 (99m:51025)
  • 4. R. Bödi, Smooth Hughes planes are classical, Arch. Math. 73, pp. 73-80, 1999. MR 1696544 (2000i:51038)
  • 5. R. Bödi, S. Immervoll, Implicit characterizations of smooth incidence geometries, Geom. Dedic. 83, pp. 63-76, 2000. MR 1800011 (2001j:51018)
  • 6. T. Bröcker, K. Jänich, Introduction to Differential Topology, Cambridge University Press, Cambridge, 1987. MR 674117 (83i:58001)
  • 7. M. Greenberg, Lectures on Algebraic Topology, Benjamin, New York-Amsterdam, 1967. MR 0215295 (35:6137)
  • 8. S. Immervoll, Glatte affine Ebenen mit großer Automorphismengruppe und ihr projektiver Abschluß, Diploma Thesis, University of Tübingen, 1998.
  • 9. S. Immervoll, Smooth generalized quadrangles and isoparametric hypersurfaces of Clifford type, Forum Math. 14, pp. 877-899, 2002. MR 1932524 (2003i:53083)
  • 10. S. Immervoll, Isoparametric hypersurfaces and smooth generalized quadrangles, J. Reine Angew. Math. 554, pp. 1-17, 2003. MR 1952166 (2003m:51018)
  • 11. S. Immervoll, Real analytic projective planes with large automorphism groups, Adv. Geom. 3, pp. 163-176, 2003. MR 1967997 (2004b:51013)
  • 12. N. Knarr, R. Löwen, Four-dimensional compact projective planes admitting an affine Hughes group, Result. Math. 38, pp. 270-306, 2000. MR 1799719 (2001h:51022)
  • 13. L. Kramer, S. Stolz, A diffeomorphism classification of manifolds which are like projective planes, J. Differ. Geom. 77, pp. 177-188, 2007. MR 2355782 (2008g:57032)
  • 14. R. Löwen, Topological pseudo-ovals, elation Laguerre planes, and elation generalized quadrangles, Math. Z. 216, pp. 347-369, 1994. MR 1283074 (95h:51025)
  • 15. W.S. Massey, Algebraic Topology: An Introduction, Graduate Texts in Math., vol. 56, Springer-Verlag, New York-Heidelberg, 5th printing, 1977. MR 0448331 (56:6638)
  • 16. B. McKay, Dual curves and pseudoholomorphic curves, Selecta Math. 9, pp. 251-311, 2003. MR 1993485 (2004g:53099)
  • 17. B. McKay, Smooth projective planes, Geom. Dedic. 116, pp. 157-202, 2005. MR 2195446 (2007g:53085)
  • 18. B. McKay, Almost complex rigidity of the complex projective plane, Proc. Amer. Math. Soc. 135, pp. 597-603, 2007. MR 2255307 (2007f:53119)
  • 19. I. Pupeza, A local characterization of two-dimensional smooth projective planes, Diploma Thesis, Technische Universität Braunschweig, 2007.
  • 20. J. Otte, Differenzierbare Ebenen, Dissertation, Universität Kiel, 1992.
  • 21. J. Otte, Smooth projective translation planes, Geom. Dedic. 58, pp. 203-212, 1995. MR 1358234 (96h:51011)
  • 22. H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel, Compact Projective Planes, de Gruyter, Berlin, 1995. MR 1384300 (97b:51009)
  • 23. E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 51H25, 51H10

Retrieve articles in all journals with MSC (2000): 51H25, 51H10

Additional Information

Stefan Immervoll
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Rainer Löwen
Affiliation: Institut für Analysis und Algebra, Technische Universität Braunschweig, Pockelsstraße 14, D-38106 Braunschweig, Germany

Ioachim Pupeza
Affiliation: Institut für Analysis und Algebra, Technische Universität Braunschweig, Pockelsstraße 14, D-38106 Braunschweig, Germany
Address at time of publication: Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

Keywords: Smooth projective plane
Received by editor(s): September 14, 2008
Published electronically: September 1, 2009
Communicated by: Ted Chinburg
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society