Conformal geometry of discrete groups and manifolds
type: Item
instance of: scholarly article
title: Conformal geometry of discrete groups and manifolds
zbMATH Open document ID: 0965.57001
author: Boris N. Apanasov
published in: De Gruyter Expositions in Mathematics
publication date: 2000-09-13T00:00:00Z
review text: A geometry in the sense of Felix Klein's Erlangen program is a pair \((X,G)\) where \(X\) is a manifold and \(G\) a Lie group acting transitively on \(X\). An \((X,G)\)-manifold is a manifold \(M\) composed of open subsets of \(X\) such that the transition functions are restrictions of elements of \(G\) to open subsets of \(X\). A conformal structure on a manifold is modelled on \((S^n, \text{Möb}(n))\) where \(\text{Möb}(n)\) denotes the group of Möbius (or conformal) transformations of the \(n\)-sphere \(S^n\). Equivalently, a conformal (or better, conformally flat) structure is a conformal class of Riemannian metrics, each locally conformally equivalent to the standard flat metric. In fact, in dimensions \(n \geq 3\), by a theorem of Liouville any conformal map between open connected subsets of \(S^n\) is the restriction of a Möbius transformation. In dimension 2, by the classical uniformization theorem for Riemann surfaces, each conformal structure on a surface is globally represented by a Riemannian metric of constant curvature, i.e. modelled on one of the three classical geometries: spherical, Euclidean or hyperbolic. In any dimension these three geometries are specializations of the conformal geometry \((S^n, \text{Möb}(n))\). In dimension 3, many of the 3-manifolds belonging to one of Thurston's eight 3-dimensional (locally homogeneous Riemannian) geometries admit conformal structures (but not the solvable and nilpotent manifolds). In dimensions \(n \geq 3\), conformal structures are in general less rigid than locally homogeneous Riemannian structures; moreover, the conformal approach unifies different geometric structures admitting conformal models. The present book is a systematic study of conformal and hyperbolic structures on manifolds, in dimensions \(n \geq 3\), and of discrete groups of Möbius transformations (Kleinian groups); a unifying aspect of the subject is that of the developing map \(d: \widetilde M \to S^n\), where \(\widetilde M\) denotes th…
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 30-02
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 51-02
Mathematics Subject Classification ID: 32G99
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 1505373
zbMATH Keywords: conformal and hyperbolic manifolds
zbMATH Keywords: Kleinian group
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1505373