The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized m-Bakry–Émery Ricci curvature, in which m is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound, as well as control on the Bakry–Émery vector field, the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized m-Bakry–Émery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson’s finiteness of fundamental group isomorphism types, volume comparison, the Abresch–Gromoll inequality, and a Cheng–Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature.

Cheeger-Colding的几乎分裂定理是在几乎非负的广义m -Bakry-ÉmeryRicci曲率的背景下建立的，其中m为正，并且相关矢量场不一定必须是函数的梯度。在这种情况下，表明具有直径上限和体积下限，以及对Bakry-Émery矢量场的控制，此类流形的基本组几乎是阿贝尔的。此外，针对广义m给出了有关Ricci曲率下界的已知结果的扩展-Bakry–Émery Ricci曲率。其中包括：Gromov和Gallot的第一个Betti数界，基本群同构类型的Anderson有限性，体积比较，Abresch-Gromoll不等式和Cheng-Yau梯度估计。最后，此分析应用于更高尺寸的固定真空黑洞，以发现低温层位必须具有有限的拓扑结构，类似于零温度层位（极端）所表现出的限制。