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A local characterization of smooth projective planes
Author(s):
Stefan
Immervoll;
Rainer
Löwen;
Ioachim
Pupeza
Journal:
Proc. Amer. Math. Soc.
138
(2010),
323-332.
MSC (2000):
Primary 51H25;
Secondary 51H10
Posted:
September 1, 2009
MathSciNet review:
2550198
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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  -dimensional closed smooth submanifold of the product of the point space and the line space (both of which are  -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:
-
- 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,
 -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)
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Additional Information:
Stefan
Immervoll
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email:
stim@fa.uni-tuebingen.de
Rainer
Löwen
Affiliation:
Institut für Analysis und Algebra, Technische Universität Braunschweig, Pockelsstraße 14, D-38106 Braunschweig, Germany
Email:
r.loewen@tu-bs.de
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
Email:
ioachim.pupeza@mpq.mpg.de
DOI:
10.1090/S0002-9939-09-10100-4
PII:
S 0002-9939(09)10100-4
Keywords:
Smooth projective plane
Received by editor(s):
September 14, 2008
Posted:
September 1, 2009
Communicated by:
Ted Chinburg
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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