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.

As part of a programme to classify quasi-Einstein metrics (M, g, X) on closed manifolds and
near-horizon geometries of extreme black holes, we study such spaces when the vector field
X is divergence-free but not identically zero. This condition is satisfied by left-invariant quasi-
Einstein metrics on compact homogeneous spaces (including the near-horizon geometry of
an extreme Myers–Perry black hole with equal angular momenta in two distinct planes) and
on certain bundles over Kähler–Einstein manifolds. We find that these spaces exhibit a mild …