A metric is said to have harmonic curvature if the exterior derivative of the Ricci tensor vanishes. It is known that there exists manifolds with dRic=0 that are not Einstein. My question is whether or not there are examples of manifolds that admit a metric with harmonic curvature but don't admit an Einstein metric.



harmonic curvature versus Einstein metric
There are many $3$-manifolds that admit a metric with harmonic curvature that don't admit an Einstein metric because in dimension $3$, 'Einstein' is equivalent to constant sectional curvature, while in dimension $3$, harmonic curvature is the same as conformally flat with constant scalar curvature. 


Since one can take connected sums of conformally flat manifolds and get conformally flat manifolds and since, by Schoen's solution of the Yamabe problem, every compact $3$-manifold has a conformal change to one with constant scalar curvature, one sees that there are many more compact $3$-manifolds that admit a metric with harmonic curvature than there are compact $3$-manifolds that have an Einstein metric.


