A local characterization of smooth projective planes

In 2000, Bödi and Immervoll considered compact, connected smooth incidence geometries with mutually transversal point rows and tually transversal line pencils. They made the very natural assumptions the flag space is a 3/-dimensional closed smooth submanifold of the product of the point space and the line space (both of which are 2Z-manifolds) that both associated projections are submersions. They showed that then 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 compactness and connectedness plus the purely local (in fact, infinitesimal) conditions stated above.