Conformally flat Lie groups
type: Item
instance of: scholarly article
title: Conformally flat Lie groups
zbMATH Open document ID: 0903.22002
DOI: 10.1007/PL00004600
author: Stephan Maier
published in: Mathematische Zeitschrift
publication date: 1998-12-16T00:00:00Z
review text: The aim of the present paper is on the one hand to prove that in dimension \(n \geq 3\) there is a one-to-one correspondence between left-invariant Möbius structures and left-invariant conformally flat structures on Lie groups and on the other hand to describe the complete list of Möbius and conformally flat Lie groups, respectively. Moreover, it is shown that any such Lie group is isometric, as a Riemannian manifold, to one of \(E^n\), \(H^n\), \(S^3\), \(E^1 \times S^3\), \(E^1 \times H^{n-1}\), or \(S^3 \times H^{n-3}\) endowed with their standard metrics. Let \(M(n)\), \(n \geq 2\), be the group of diffeomorphisms of the sphere \(S^n\) which map round \((n-1)\)-spheres to round \((n-1)\)-spheres. A manifold \(M\) is said to be modelled on \((S^n, M(n))\) if \(M\) has a maximal atlas of charts \((U, \phi)\), where \(\phi: U \rightarrow S^n\) is a homeomorphism on its image such that \(\phi_1 \phi_2^{-1}\) is a restriction of an element in \(M(n)\). Such a structure is called a Möbius structure and can also be interpreted as a conformally flat pseudogroup structure on \(M\), the pseudogroup being given by the restrictions of elements in \(M (n)\) to open subsets of the sphere [\textit{N. H. Kuiper}, Ann. Math., II. Ser. 50, No. 4, 916-924 (1949; Zbl 0041.09303)]. A Möbius structure on a Lie group \(G\) is called left invariant if the left action of \(G\) on itself is via Möbius transformations, i.e., diffeomorphisms of \(M\) which preserve the Möbius structure. In general a conformally flat structure on a manifold is a class of conformally equivalent metrics such that locally there exist conformal diffeomorphisms onto open subsets of Euclidean space. In dimension \(n \geq 3\), conformally flat structures induce Möbius structures and this sets up a one-to-one correspondence between Möbius structure and flat conformal structure [\textit{P. Kulkarni} and \textit{U. Pinkall}, Differential geometry, Lect. Notes Math. 1209, 190-209 (1986; Zbl 0612.57017)]. Then it is po…
reviewed by: Sergio Console
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 1185897
zbMATH Keywords: conformally flat structure
zbMATH Keywords: Möbius structure
zbMATH Keywords: conformally flat Lie groups
zbMATH Keywords: conformally flat pseudogroup structure
zbMATH Keywords: manifold
Wikidata QID: Q115390586
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1185897