4.4 Orbit space structure

The structure theorem [65Jump To The Next Citation Point] applies to stationary toroidal Kaluza–Klein black holes and provides the following product structure
⟨⟨Mext⟩⟩ ≈ ℝ × Σ,
with the stationary vector being tangent to the ℝ factor and where Σ, endowed with the induced metric, is an n-dimensional Riemannian manifold admitting a 𝕋n−2 action by isometries. A careful analysis of the topological properties of such toroidal actions [169Jump To The Next Citation Point], based on deep results from [260Jump To The Next Citation Point, 261], allows one to show that the orbit space n− 2 ⟨⟨Mext⟩⟩∕(ℝ × 𝕋 ) is homeomorphic to a half plane with boundary composed of segments and corners; the segments being the projection of either a component of the event horizon or an axis of rotation (the set of zeros of a linear combination of axial vectors), and the corners being the projections of the intersections of two axes. Moreover, the interiors are in fact diffeomorphic. To establish this last fundamental result it is necessary to exclude the existence of exceptional orbits of the toroidal action; this was done by Hollands and Yazadjiev in [169Jump To The Next Citation Point] by extending the results in [260] to the KK-black hole setting. In particular one obtains the following decomposition
n−2 + ⟨⟨Mext ⟩⟩ ∖ (∪𝒜i ) ≈ ℝ × 𝕋 × ℝ × ℝ ,
where ∪𝒜i is the union of all axes; we note that such product structure is necessary to the construction of Weyl coordinates [76Jump To The Next Citation Point, 65Jump To The Next Citation Point] and, consequently, indispensable to perform the desired reduction of the vacuum equations.

As already discussed, basic properties of black rings show that a classification of KK-black holes in terms of mass, angular momenta and horizon topology is not possible. But, as argued by Hollands and Yazadjiev [169Jump To The Next Citation Point], the angular momenta and the structure of the orbit space characterize such black holes if one further assumes non-degeneracy of the event horizon. This orbit space structure is in turn determined by the interval structure of the boundary of the quotient manifold, a concept related to Harmark’s rod structure developed in [137] (see also [101, Section]). Note that the interval structure codifies the horizon topology.

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