Insufficiency of the Ricci and Bianchi identities for characterising curvature
type: Item
instance of: scholarly article
title: Insufficiency of the Ricci and Bianchi identities for characterising curvature
zbMATH Open document ID: 0681.53007
DOI: 10.1016/0393-0440(89)90006-5
published in: Journal of Geometry and Physics
publication date: 1989-01-01T00:00:00Z
review text: The author discusses the following problem: Let M be a manifold with global metric g (of any signature) and corresponding curvature tensor R with local components \(R^ a_{bcd}\). Then R satisfies the Bianchi and Ricci identities with respect to the covariant derivative associated with g, which are differential identities on R. In what sense are these identities integrability conditions for a curvature tensor? More precisely, if \(K^ a_{bcd}\) is a tensor on M with all the algebraic symmetries of a curvature tensor and if there exists a metric \(\gamma\) on M such that K satisfies the Bianchi and Ricci identities with respect to the covariant derivative arising from \(\gamma\) then is K the curvature tensor of some metric on M ? The answer is in the negative as follows by taking (M,g) to be a manifold of constant curvature (\(\neq 0)\) with curvature tensor R and taking \(K=\alpha R\) where \(\alpha\) is a constant different from 0 or 1. The proof is based on the theorems concerning the extent to which a metric curvature tensor determines the metric from which it arises [the reviewer and \textit{C. B. G. McIntosh}, Int. J. Theor. Phys. 22, 469-476 (1983; Zbl 0523.53037); the reviewer, Gen. Relativ. Gravitation 15, 581-589 (1983; Zbl 0514.53018)]. A similar negative answer is given regarding when the tensor K is the curvature of some connection on M (in the context of gauge theories). The author continues with a discussion of the equivalence problem [in the notation of \textit{S. Sternberg}'s book ``Lectures on differential geometry'' (Prentice Hall 1964; Zbl 0129.131) and \textit{J. Ehlers} in E. B. Christoffel: the influence of his work on mathematics and the physical science, Int. Symp., Aachen 1979, 526-542 (1981; Zbl 0674.53001)]. He shows that the ``shape'' function used by these authors may, for a tensor like K above, be identical to that of a curvature tensor R arising from a metric g on M without K being the curvature of any metric on M. The paper concludes w…
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 83C99
zbMATH DE Number: 4115437
zbMATH Keywords: Bianchi identities
zbMATH Keywords: Ricci identities
zbMATH Keywords: integrability conditions
zbMATH Keywords: curvature tensor
zbMATH Keywords: equivalence problem
author: Alan D. Rendall
reviewed by: G. S. Hall
MaRDI profile type: Publication
full work available at URL: 0393-0440(89)90006-5
OpenAlex ID: W2024803319
cites work: Q3935814
cites work: Q3928594
cites work: Algebraic determination of the metric from the curvature in general relativity
cites work: Does a generic connection depend continuously on its curvature?
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description: scientific article; zbMATH DE number 4115437