Isotropic quasi-Einstein manifolds

Abstract

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.