Criteria for two-dimensional circle planes
type: Item
instance of: scholarly article
title: Criteria for two-dimensional circle planes
zbMATH Open document ID: 0821.51013
author: Günter F. Steinke
author: Burkard Polster
published in: Beiträge zur Algebra und Geometrie
publication date: 1995-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/231374
full work available at URL: no.2
review text: The paper is concerned with the following problem. Suppose you are given a circle plane, i.e., a Möbius plane, a Laguerre plane or a Minkowski plane. Suppose the point space of the circle plane can be equipped with a topology such that it is homeomorphic to the point space of the corresponding classical circle plane over the real numbers. Is there a topology on the circle space, such that the circle plane becomes a topological plane, i.e., such that geometric operations are continuous. If the answer is positive, then there is only one such topology on the circle space, namely the topology induced by the Hausdorff metric. The authors present a way to solve this problem by looking at the derived structure. To obtain the derived structure at a point \(s\) one removes all points parallel to \(s\). Circles through \(s\) and parallel classes not containing \(s\) become lines in the derived structure. This yields an affine plane. In the first part of the paper the authors investigate what circles not incident with \(s\) have to become in the derived plane such that the circle plane may become a topological circle plane. These results are then used to show that if for a circle plane as above at every point the derived affine plane is a topological affine plane, then the circle plane can be made a topological circle plane. If a Laguerre plane or a Minkowski plane is given, one only needs that for every point of a parallel class the derived affine plane is a topological affine plane to obtain the desired result. The proof is based on the fact that if a circle plane is given such that the point space is a topological space homeomorphic to the point space of the corresponding classical real circle plane and such that circles are simply closed curves, then the Hausdorff metric induces a topology on the circle set such that the circle plane becomes a topological circle plane. Unfortunately, it is not known whether a similar result holds for 4- dimensional circle planes. To people…
reviewed by: Andreas E. Schroth
Mathematics Subject Classification ID: 51H15
Mathematics Subject Classification ID: 51B10
Mathematics Subject Classification ID: 51B15
Mathematics Subject Classification ID: 51B20
zbMATH DE Number: 767396
zbMATH Keywords: topological geometry
zbMATH Keywords: topological oval
zbMATH Keywords: parabolic curve
zbMATH Keywords: hyperbolic curve
zbMATH Keywords: dimension
zbMATH Keywords: circle plane
zbMATH Keywords: Möbius plane
zbMATH Keywords: Laguerre plane
zbMATH Keywords: Minkowski plane
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 767396