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Proceedings of the American Mathematical Society
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
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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 $ 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.


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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: http://dx.doi.org/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
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.