Twistors and Cartan connections
type: Item
instance of: scholarly article
title: Twistors and Cartan connections
zbMATH Open document ID: 0997.53038
DOI: 10.1023/A:1006762718907
published in: Annals of Global Analysis and Geometry
publication date: 2002-11-10T00:00:00Z
review text: The main result of the paper gives a general approach for constructing a twistor space with a CR-structure for a manifold with a geometric structure. It states the following: Let \(G\) be a closed subgroup of a Lie group \(L\) and let \(W\) be a closed subgroup of the complexification \(L^{{\mathbb C}}\). Set \(K=W\cap G\). Let \(\pi:P\to M\) be a principal \(G\)-bundle over a manifold \(M\) and \({ k}\) a Cartan connection of type \(L\) on \(P\). If \(K=W\cap L\), then there exists a natural CR-structure on the manifold \(Z=P/K\). This CR-structure is integrable if and only if \(\Omega({ k}^{-1}({m}), { k}^{-1}({ m}))\subset { m}\) where \(\Omega\) is the curvature form of the Cartan connection \({ k}\) and \({ m}=\operatorname {Lie} W\) is the Lie algebra of \(W\). The integrability condition is further discussed in the case when the Lie algebra \(\operatorname {Lie} L\) is graded and \(\operatorname {Lie} G\), \(\operatorname {Lie} W\) are graded subalgebras. Finally, the authors present two applications of their construction -- for manifolds with a \(G\)-structure of the so-called CR-type and for the manifold of the infinite \(1\)-coframes of a given manifold.
reviewed by: Johann Davidov
Mathematics Subject Classification ID: 53C28
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 57N16
zbMATH DE Number: 1530054
zbMATH Keywords: twistor spaces
zbMATH Keywords: CR-structures
zbMATH Keywords: Cartan connections
author: Dmitri Alekseevsky
author: M. M. Graev
MaRDI profile type: Publication
full work available at URL: a:1006762718907
OpenAlex ID: W1583022007
description: scientific article; zbMATH DE number 1530054