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.