A local characterization of smooth projective planes

Immervoll, Stefan; Löwen, Rainer; Pupeza, Ioachim

doi:10.1090/S0002-9939-09-10100-4

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.

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