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.

作为对极端黑洞的闭合流形和近地平线几何结构上的准爱因斯坦度量（ M，  g，  X ）进行分类的程序的一部分，我们研究当矢量场X无散度但不全为零时的此类空间。这个条件可以通过紧致均匀空间（包括两个不同平面中具有相等角动量的极端迈尔斯-佩里黑洞的近地平线几何）和凯勒-爱因斯坦流形上的某些束上的左不变准爱因斯坦度量来满足。我们发现这些空间表现出一种温和的刚性形式：它们总是允许由X生成的单参数等距组。还获得了进一步的几何和拓扑限制。