On mappings of a spatial domain which preserve angles and dilatations along a system of rays
type: Item
instance of: scholarly article
title: On mappings of a spatial domain which preserve angles and dilatations along a system of rays
zbMATH Open document ID: 0942.30002
DOI: 10.1007/BF02674619
author: M. T. Brodovich
published in: Siberian Mathematical Journal
publication date: 2000-04-13T00:00:00Z
review text: In 1933, D. E. Men'shov has proven that if a univalent continuous mapping acting from a planar domain into the plane preserves angles or dilations along a system of rays at every point of the domain then this mapping is holomorphic [Math. Ann. 109, 101-159 (1933; Zbl 0007.41502)]. A number of mathematicians strived for weakening the assumptions of this Men'shov's theorem. It turns out that the claim of the theorem is true even if the mapping is neither univalent nor continuous (see, for example, the author's article [Sib. Mat. Zh. 38, No. 5, 1005-1007 (1997; Zbl 0880.30004)] for more detail). Less is known about mappings in the 3-space. In his article [Sov. Math., Dokl. 7, 1321-1323 (1966; Zbl 0181.08801)], \textit{A. P. Kopylov} announced the following theorem: If \(D\) is a domain in \(\mathbb R^3\) and \(f\: D\to\mathbb R^3\) is a homeomorphism preserving angles between rays at every point of~\(D\), then \(f\) is conformal. In the article under review, the author proves the following theorem: Let \(Q\) be an open cube in \(\mathbb R^3\). For every point \(x\) of \(Q\), there is an everywhere dense countable system of rays \(t_i(x)\) \((i=1,2,\dots)\) in \(\mathbb R^3\) and a discontinuous function \(f\: Q\to \mathbb R^3\) such that \(f\) preserves dilatations and angles along the system of rays \(t_i(x)\) \((i=1,2,\dots)\) at every point \(x\) of \(Q\).
Mathematics Subject Classification ID: 30A05
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30A99
zbMATH DE Number: 1431635
zbMATH Keywords: monogeneity condition
MaRDI profile type: Publication
cites work: Generalized Douglas algebras and the corona theorem
cites work: Q5571085
cites work: Q5558032
description: scientific article; zbMATH DE number 1431635