The general Kerr-de Sitter metrics in all dimensions

We give the general Kerr-de Sitter metric in arbitrary space-time dimension D≥4, with the maximal number [(D−1)/2] of independent rotation parameters. We obtain the metric in Kerr-Schild form, where it is written as the sum of a de Sitter metric plus the square of a null-geodesic vector, and in generalised Boyer-Lindquist coordinates. The Kerr-Schild form is simpler for verifying that the Einstein equations are satisfied, and we have explicitly checked our results for all dimensions D≤11. We discuss the global structure of the metrics, and obtain formulae for the surface gravities and areas of the event horizons. We also obtain the Euclidean-signature solutions, and we construct complete non-singular compact Einstein spaces on associated SD−2 bundles over S2, infinitely many for each odd D≥5.