Rigidity of quasi-Einstein metrics: the incompressible case

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 form of rigidity: they always admit a one-parameter group of isometries generated by X. Further geometrical and topological restrictions are also obtained.