Representation of two-dimensional stable planes by Riemannian metrics and affine connections
by
Gerhard Gerlich
Technische Universität Braunschweig (Germany)

The classical examples of plane topological geometries, the real projective, affine, and hyperbolic planes, have one thing in common: Their lines are geodesics of a Riemannian metric. This leads to the question whether the lines of other (non-classical) topological geometries can also be described in this manner, or whether an affine connection exists such that the lines are geodesics of this affine connection. We consider two-dimensional compact projective planes in general and the three families of two-dimensional stable affine planes having a large (i. e. at least three-dimensional) collineation group, namely the Moulton planes M( s ) , the skew parabola planes E_{c, d} and the planes over Cartesian fields P_{\alpha, \beta, c}. It turns out that only the classical, desarguesian real projective plane has the property that its lines are generated by a Riemannian metric. Moreover, among the three families of non-classical affine planes with large collineation groups, only the Moulton planes admit affine connections that generate their line system. All of these connections can be specified explicitly. Among them is a Riemannian connection that is singular on one entire line and yields a Riemannian metric for two half-planes only.

Date received: February 5, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafv-09.