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I'm reading Gromov's notes http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf

and at page 7 they say that there is a unique second order differential operator $S$ from the space of Riemannian metrics on a fixed manifold $M$ such that

1) $S$ is $Diffeo(M)$-equivariant

2) $S$ is linear wrt the second derivatives of its argument, a Riemannian metric $g$

Can you point out a reference or an idea for a proof of this statement? In the notes it appears to be none.

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  • $\begingroup$ I don't have a proof, but it seems to me that more is true. Namely, any scalar function defined using a Riemannian metric and its derivatives up to second order that is equivariant under diffeomorphisms has to be a function of the scalar curvature. $\endgroup$
    – Deane Yang
    Commented Oct 27, 2014 at 19:36
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    $\begingroup$ @DeaneYang: Well, that isn't true. For example, the functional that gives the square norm of the Riemann curvature tensor satisfies that condition as well, and it's not a function of the scalar curvature $\endgroup$
    – Robert Bryant
    Commented Oct 27, 2014 at 20:13
  • $\begingroup$ Robert, oops. You're right. I once knew stuff like that. $\endgroup$
    – Deane Yang
    Commented Oct 27, 2014 at 20:17
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    $\begingroup$ If by a Riemannian invariant one means a polynomial in the derivatives of the metric tensor $g$ and $(det g)^{-1}$ that maintains its form under changes of coordinate, then it follows from work of H. Weyl that any Riemannian invariant is a linear combination of complete contractions of covariant derivatives of the Riemannian curvature tensor. For precise statements see the introduction of the book of Fefferman and Graham on ambient metrics. The characterization of the scalar curvature should follow because there is only one such contraction linear in the second derivatives of $g$. $\endgroup$
    – Dan Fox
    Commented Oct 28, 2014 at 11:54

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