Domain structure of black hole space-times

We introduce the domain structure for stationary black hole space-times. The domain structure lives on the submanifold of fixed points of the Killing vector fields. Depending on which Killing vector field has fixed points the submanifold is naturally divided into domains. The domain structure provides invariants of the space-time, both topological and continuous. It is defined for any space-time dimension and any number of Killing vector fields. We examine the domain structure for asymptotically flat space-times and find a canonical form for the metric of such space-times. The domain structure generalizes the rod structure introduced for space-times with D-2 commuting Killing vector fields. We analyze in detail the domain structure for Minkowski space, the Schwarzschild-Tangherlini black hole and the Myers-Perry black hole in six and seven dimensions. Finally, we consider the possible domain structures for asymptotically flat black holes in six and seven dimensions.