Cartan geometries and dynamics
type: Item
instance of: scholarly article
title: Cartan geometries and dynamics
zbMATH Open document ID: 0978.53057
DOI: 10.1023/A:1005219805069
published in: Geometriae Dedicata
publication date: 2002-02-07T00:00:00Z
review text: The authors study Cartan connections in a \(J\)-principal bundle \(Q\) over a closed manifold \(M\). To such a connection associated is a Lie group \(L\), with Lie algebra \(l\), which admits \(J\) as a closed subgroup. The standard example of a Cartan connection arises if \(L\) admits a discrete subgroup \(\Gamma\) such that the double coset space \(\Gamma\setminus L/J\) is a closed manifold. In this case every subgroup \(G\) of the centralizer of \(J\) in \(L\) acts on \(\Gamma\setminus L\) as a group of principal bundle automorphisms preserving the Cartan connection. On the other hand, the existence of a sufficiently large group \(G\) of principal bundle automorphisms forces the Cartan connection \(\Theta\) to have constant curvature or to be standard. The authors derive several rigidity results of this kind. As an example, they show that \(\Theta\) is standard if \(G\) is a connected semisimple Lie group of rank \(\geq 2\) without compact factors, \(J\) is compact, the action of \(G\) is ergodic with respect to an invariant volume on the bundle \(Q\) and that there is a 1-parameter subgroup \(g_t\) of \(G\) which preserves the Cartan connection up to a diagonalizable 1-parameter subgroup of \(GL(l)\).
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 22F50
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 37C85
zbMATH DE Number: 1485573
zbMATH Keywords: principal bundle
zbMATH Keywords: Cartan connections
zbMATH Keywords: coset space
zbMATH Keywords: principal bundle automorphisms
zbMATH Keywords: constant curvature
zbMATH Keywords: rigidity
author: Renato Feres
author: Philipp Lampe
MaRDI profile type: Publication
full work available at URL: a:1005219805069
OpenAlex ID: W26509040
description: scientific article; zbMATH DE number 1485573