Maybe this is a vague question. In Besse's book Einstein manifolds, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference between $\delta W^{\pm}=0$ and Einstein?
Are there examples of metrics with $\delta W^+=\delta W^-=0$ but not Einstein? What if $\delta W^+=\delta W^-=0$ and constant scalar curvature, is it equivalent to Einstein? Thank you very much.
A very simple example of a non-Einstein manifold with harmonic Weyl tensor is the product $\mathbb{S}^2\times \mathbb{R}^2$ (with the usual metrics). In Besse's book there are more examples. The example above has also constant scalar curvature. So the assumptions $\delta W=0$ and constant scalar curvature are not sufficient to say that the manifold is Einstein.
Since any Einstein manifold satisfies $\delta W=0$ we can consider it as a generalization of the Einstein condition. However, as said before, they are not equivalent. One can characterize the condition $\delta W=0$ in terms of the Schouten tensor $S=\frac{1}{n-2}\left(Ric-\frac{s}{2(n-1)}g\right).$ Indeed, $\delta W=0$ if and only if $S$ is a Codazzi tensor, that is, $(\nabla_XS)(Y,Z)=(\nabla_YS)(X,Z).$
In the particular case of dimension four we can decompose the Weyl tensor $W=W^++W^-.$ A further generalization of $\delta W=0$ is $\delta W^+=0$ or $\delta W^-=0.$
If you are interested into the study of this manifolds the following papers could be useful:
The geometry of weakly self-dual Kahler surfaces, by V. Apostolov, D. M. J. Calderbank and P. Gauduchon:
http://arxiv.org/pdf/math/0104233.pdf
Compact Kahler surfaces with harmonic anti-self-dual Weyl tensor by W. Jelonek:
http://www.sciencedirect.com/science/article/pii/S0926224502000761
