A local characterization of smooth projective planes
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- by Stefan Immervoll, Rainer Löwen and Ioachim Pupeza PDF
- Proc. Amer. Math. Soc. 138 (2010), 323-332 Request permission
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
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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
- Received by editor(s): September 14, 2008
- Published electronically: September 1, 2009
- Communicated by: Ted Chinburg
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 323-332
- MSC (2000): Primary 51H25; Secondary 51H10
- DOI: https://doi.org/10.1090/S0002-9939-09-10100-4
- MathSciNet review: 2550198