A comment on the integrability conditions of the conformal Killing equation
type: Item
instance of: scholarly article
title: A comment on the integrability conditions of the conformal Killing equation
zbMATH Open document ID: 0675.53025
DOI: 10.1007/BF00769870
author: C. D. Collinson
published in: General Relativity and Gravitation
publication date: 1989-01-01T00:00:00Z
review text: Let M be a differentiable manifold of dimension \(n\geq 4\) with metric \(g_{ab}\), Weyl tensor \(C^ a_{bcd}\), Ricci tensor \(R_{ab}\), Ricci scalar R and a tensor \(C_{abc}\) defined by \[ C_{abc}=\frac{2}{n- 2}[R_{c[b;a]}-\frac{1}{2(n-1)}R_{;[a}g_{b]c}] \] where square brackets denote skew-symmetrisation and a semicolon denotes a covariant derivative. \textit{K. Yano}, in his book ``The theory of Lie derivatives and its applications'' (1957; Zbl 0077.158) shows that the integrability conditions for the conformal Killing equation \({\mathcal L}_ Xg_{ab}=2\phi g_{ab}\) (where X is a vector field on M, \({\mathcal L}\) denotes a Lie derivative and \(\phi\) is a function on M) can be generated from the two conditions \[ {\mathcal L}_ XC^ a_{bcd}=0\quad and\quad {\mathcal L}_ XC_{abc}=-C^ d_{cab}\phi_{;d}. \] In this paper the author shows how the second of these conditions follows from the first and is, in this sense, redundant.
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 83C40
Mathematics Subject Classification ID: 53B50
zbMATH DE Number: 4105768
zbMATH Keywords: Weyl tensor
zbMATH Keywords: integrability conditions
zbMATH Keywords: conformal Killing equation
reviewed by: G. S. Hall
MaRDI profile type: Publication
Wikidata QID: Q125118342
cites work: Q3235035
description: scientific article; zbMATH DE number 4105768