Generalized Affine Connections Associated with the Space of Centered Planes

On the metric dimension of affine planes, biaffine planes and generalized quadrangles

In this paper the metric dimension of (the incidence graphs of) particular partial linear spaces is considered. We prove that the metric dimension of an affine plane of order q ≥ 13 is 3q − 4 and describe all resolving sets of that size if q ≥ 23. The metric dimension of a biaffine plane (also called a flag-type elliptic semiplane) of order q ≥ 4 is shown to fall between 2q−2 and 3q−6, while fo...

Properties of Generalized Berwald Connections

Affine Connections with W = 0

If ∇ is a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl’s result that W vanishes if and only if (M,∇) is (locally) diffeomorphic to RP with ∇, when transported to RP, in the projective class of ∇LC , the Levi-Civita connection of the Fubini–Study metric on...

Affine and projective planes

Locally polar geometries with affine planes

The study of geometries on the absolute points of polarities in projective spaces has been started by Veldkamp [21], who was the first to give a synthetic characterization of these geometries, which he called polar spaces. As part of his work on spherical buildings [19], Tits extended Veldkamp’s results to a somewhat larger class of geometries related to pseudo-quadratic forms. In 1974, Buekenh...