Insufficiency of the Ricci and Bianchi identities for characterising curvature
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 with an example of a tensor K, as above, on the manifold \({\mathbb{R}}^ 2\) which is not the curvature tensor of any metric on \({\mathbb{R}}^ 2\) but which agrees to infinite order at each \(p\in {\mathbb{R}}^ 2\) with the curvature tensor at p arising from some metric defined in a neighbourhood of p.
scientific article; zbMATH DE number 4115437
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