Bijections of \({\mathbb{R}}^ n\) onto itself
type: Item
instance of: scholarly article
title: Bijections of \({\mathbb{R}}^ n\) onto itself
zbMATH Open document ID: 0654.51022
DOI: 10.1007/BF00183024
author: R. Daniel Mauldin
author: Richard J. Gardner
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: Affine and continuous maps are characterized within the class of bijections of \({\mathbb{R}}^ n\) by the property that a certain set of geometric or topological figures is mapped into itself or another set of this type. A basic tool for the authors' characterization of affine maps is the fundamental theorem of projective geometry [\textit{G. Darboux}, Math. Ann. 17, 55-62 (1880)]. For example, affine maps are characterized by: each circle is mapped into the boundary of a strictly convex body or onto the boundary of a convex body (of any dimension). Also the existence of bijections of \({\mathbb{R}}^ n\) mapping circles into unions of two line segments, into quadrilaterals or onto unions of countably many line segments is shown. Continuity is implied by the following properties: i) circles are mapped onto simple closed curves \((n=2)\); ii) simple closed curves are mapped onto compact sets. The first result does not extend to \(n=3\). Finally in the infinite-dimensional case the preservation of circles is shown to characterize similarities.
Mathematics Subject Classification ID: 51N10
Mathematics Subject Classification ID: 51N20
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 4067740
zbMATH Keywords: bijections of \({\mathbb{R}}^ n\) onto itself
zbMATH Keywords: continuous maps
zbMATH Keywords: affine maps
reviewed by: Bernd Wegner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00183024
OpenAlex ID: W1993143589
description: scientific article; zbMATH DE number 4067740