In this work we are mostly interested in uniqueness results for four-dimensional black holes. This leads us naturally to consider those vacuum Kaluza–Klein spacetimes with enough symmetries to lead to four-dimensional spacetimes after dimensional reduction, providing henceforth four-dimensional black holes. It is convenient to start with a very short overview of the subject; the reader is referred to [101, 172] and references therein for more information. Standard examples of Kaluza–Klein black holes are provided by the Schwarzschild metric multiplied by any spatially flat homogeneous space (e.g., a torus). Non-trivial examples can be found in [272, 211]; see also [200, 172] and reference therein.

4.1 Black holes in higher dimensions

4.2 Stationary toroidal Kaluza–Klein black holes

4.3 Topology of the event horizon

4.4 Orbit space structure

4.5 KK topological censorship

4.6 Classification theorems for KK-black holes

4.2 Stationary toroidal Kaluza–Klein black holes

4.3 Topology of the event horizon

4.4 Orbit space structure

4.5 KK topological censorship

4.6 Classification theorems for KK-black holes

Living Rev. Relativity 15, (2012), 7
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