Topological affine planes with affine connections

Abstract

Abstract We consider the question whether the system of lines of a two-dimensional topological plane can be described as the system of geodesics of a Riemannian metric or an affine connection. In [4] we have shown that non-classical compact projective planes do not admit Riemannian metrics. Here we use different methods that are suited to two-dimensional planes in general. We apply them to three families of topological affine planes with large collineation groups. It turns out that for the standard models of the skew-parabola planes and the planes over cartesian fields no affine connection, and hence no Riemannian metric exist. However, for the Moulton planes affine connections do exist and we determine all of them. Among them is at least one Riemannian connection. We also give an example where no Riemannian metric exists. Moreover, we derive a characterization of the classical affine plane in the case that ℝ2 acts by vector space translations as a subgroup of the collineation group.