Under what circumstances do we know whether a given tensor of 4th rank could be the curvature tensor of a manifold? For instance, if I specify some arbitrary functions R_{ijkl} (with the necessary symmetries under interchange i<->j, k<->l, and ij<->kl), is it necessarily the case that there is a manifold for which R_{ijkl} is the curvature? I guess that satisfying the Bianchi identities is a necessary precondition.(adsbygoogle = window.adsbygoogle || []).push({});

And if so, is there any way to construct the metric explicitly?

Thanks,

Dave

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# How can we tell if a given tensor is a curvature tensor?

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