# Isotropic quasi-Einstein manifolds

## TL;DR

This paper characterizes four-dimensional Lorentzian quasi-Einstein manifolds with harmonic Weyl tensor, showing that in the isotropic case they are necessarily $pp$-waves, and explores properties of these $pp$-waves.

## Contribution

It provides a classification of isotropic quasi-Einstein manifolds with harmonic Weyl tensor as $pp$-waves and analyzes their properties.

## Key findings

- Isotropic quasi-Einstein manifolds with harmonic Weyl tensor are $pp$-waves.
- Further properties of $pp$-waves satisfying the quasi-Einstein equation are derived.
- The potential function preserves harmonicity in these 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.