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In a series of papers (1, 2, 3) P. Gilkey et al. discuss certain identities satisfied by the curvature tensor of a (pseudo)-Riemannian metric.

Contrary to the Bianchi or Ricci identities, these ones only hold in certain dimensions. The simplest of them is well-known: on any Riemannian manifold of dimension $2$, the Einstein tensor vanishes, i.e. $$ Ricc = \frac{r}{2} \, g \ , $$ where $Ricc$ and $r$ denote the Ricci tensor and the scalar curvature of $g$, respectively.

More generally, for any Riemannian manifold of even dimension $n = 2k$, the authors consider $k$ identities satisfied by the curvature, and characterize them as the only identities satisfying certain homogeneity condition.

Nevertheless, all these identities are second-order, i.e., only involve second-derivatives of the metric $g$. My question is:

  • Does anybody know any "higher-order" dimensional curvature identity?

As far as I can follow the reasoning of the paper, I guess there should be plenty of them, but I cannot come up with any particular expression.

NOTE: For the curious reader, the identities studied by Gilkey et al. are related to the Chern-Gauss-Bonnet theorem: in even dimension, the integral of the Pfaffian is a topological constant, hence independent of the Riemannian metric; therefore, the Euler-Lagrange equations of this variational problem are identically zero, thus yielding a curvature identity.

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Maybe you could ask Peter Gilkey by email? – Deane Yang Aug 3 '13 at 4:01
Yeah, good idea, thanks Deane – Capitaine Haddock Aug 3 '13 at 16:26

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