Rigidity of quasi-Einstein metrics

We call a metric quasi-Einstein if the m-Bakry–Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, it contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kähler quasi-Einstein metrics.