In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of possible fibers, so on). On the other hand, a lot of information is also known about warped product metrics of harmonic Weyl curvature (as seen here).

I'm looking for results about warped product metrics satisfying a condition (purely) on the Weyl tensor which generalizes conformal flatness. By purely I mean I want conditions only on the Weyl tensor, so that, for instance, imposing harmonic Weyl curvature + constant scalar curvature (which implies harmonic Riemannian curvature) is not what I'm looking for. For instance, one could ask what is known about warped product metrics with parallel Weyl tensor (i.e $\nabla W = 0$ instead of $W = 0$) - this is obviously a stronger condition than harmonic Weyl curvature but I haven't found any papers dealing with it specifically. I'm not that interested in parallel Weyl curvature, this is just an example of what I'm looking for. Ideally, a good condition would be weaker than conformal flatness, stronger than harmonic Weyl curvature but simultaneously weaker than parallel Weyl tensor (half harmonic Weyl curvature would fit in nicely, but I haven't found any results along that line either).

Any help is appreciated! Thanks in advance.

In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of possible fibers, so on). On the other hand, a lot of information is also known about warped product metrics of harmonic Weyl curvature (as seen here).

I'm looking for results about warped product metrics satisfying a condition (purely) on the Weyl tensor which generalizes conformal flatness. By purely I mean I want conditions only on the Weyl tensor, so that, for instance, imposing harmonic Weyl curvature + constant scalar curvature (which implies harmonic Riemannian curvature) is not what I'm looking for. For instance, one could ask what is known about warped product metrics with parallel Weyl tensor (i.e $\nabla W = 0$ instead of $W = 0$) - this is obviously a stronger condition than harmonic Weyl curvature but I haven't found any papers dealing with it specifically. I'm not that interested in parallel Weyl curvature, this is just an example of what I'm looking for. Ideally, a good condition would be weaker than conformal flatness, stronger than harmonic Weyl curvature but simultaneously weaker than parallel Weyl tensor (although interesting, half harmonic Weyl curvature would not fit in these criteria, but I haven't found any results along that line either).

Any help is appreciated! Thanks in advance.