Homogeneous surfaces in Lie sphere geometry
type: Item
instance of: scholarly article
title: Homogeneous surfaces in Lie sphere geometry
zbMATH Open document ID: 1231.53014
author: Tongzhu Li
published in: Geometriae Dedicata
publication date: 2010-11-23T00:00:00Z
review text: This paper deals with certain classes of surfaces in the 3-sphere \(S^3 \subset {\mathbb R}^4\). It is shown that any flat canal surface in \(S^3\) is Lie equivalent to the stereographic projection of either a generic cylinder or a generic cone or a surface of revolution in \({\mathbb R}^3\). The non-flat canal surfaces in \(S^3\) are classified up to Lie equivalence. Moreover, a classification of Lie homogeneous surfaces is given. If \(US^3 := \{(x, \xi) \in S^3\times S^3\mid x\cdot \xi = 0\}\), then an immersion \(\lambda = (x, \xi): M \longrightarrow US^3\) is called a Lie geometric surface if \(dx\cdot \xi = 0\) holds. The Lie geometric surfaces which allow a Lie sphere transformation for any prescribed preimage-image pair of points chosen on them are called homogeneous Lie surfaces. The author gives a complete classification of such surfaces with respect to the Lie transformation group of \(S^3\). It is shown that there exist 19 types of such surfaces.
reviewed by: Anton Gfrerrer
Mathematics Subject Classification ID: 53A40
zbMATH DE Number: 5818355
zbMATH Keywords: Lie sphere geometry
zbMATH Keywords: canal surface
zbMATH Keywords: Lie homogeneous surface
zbMATH Keywords: 3-sphere
zbMATH Keywords: stereographic projection
MaRDI profile type: Publication
full work available at URL: s10711-010-9460-4
OpenAlex ID: W2034952953
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DOI: 10.1007/S10711-010-9460-4
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description: scientific article; zbMATH DE number 5818355