Isotropic quasi-Einstein manifolds

We investigate the local structure of four-dimensional Lorentzian quasi-Einstein manifolds under conditions on the Weyl tensor. We show that if the Weyl tensor is harmonic and the potential function preserves this harmonicity then, in the isotropic case, the manifold is necessarily a pp-wave. Using the quasi-Einstein equation, further conclusions are obtained for pp-waves. In particular, we show that a four-dimensional pp-wave is conformally Einstein if and only if it is locally conformally flat or has harmonic Weyl tensor.