In the process of calculation of a problem in tensor analysis, I have encountered with an expression given by $$C^{ij}R_{ij}=0.$$

This is for a Riemannian manifold of dimension 3. Assuming the manifold is compact, what can be said about the manifold? Here $C^{ij}$ and $R_{ij}$ are Cotton-york tensor and Ricci tensor respectively.

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    $\begingroup$ Doesn't the Cotton tensor have three indices? In general, I get the impression that it's easier to produce and classify curvature scalars than it is to find any application or interpretation of a particular scalar. $\endgroup$
    – user21349
    Commented May 21, 2018 at 13:48
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    $\begingroup$ This is a variant of cotton tensor called cotton-york tensor. It has two indices. You can also look at the paper titled " the cotton flow" $\endgroup$ Commented May 21, 2018 at 14:08
  • $\begingroup$ The manifold is Riemannian and dimension is three. $\endgroup$ Commented May 21, 2018 at 14:08
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    $\begingroup$ @BenCrowell: If the Cotton tensor is viewed as a one-form valued two-form, then on a $3$-manifold it can be identified with a tensor with two-indices via the metric duality of $1$ and $2$ forms, and this is often done. $\endgroup$
    – Dan Fox
    Commented May 21, 2018 at 15:12


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