| The consistency condition on the Killing vectors is tested for the Kaluza-Klein reductions of type IIB supergravity and D = 11 supergravity on curved internal spaces. It is shown in general that for U(1) bundle spaces of arbitrary dimension, the Killing vectors associated with the isometries of the base manifold, when it is a product of two or more complex projective spaces, do not satisfy the consistency condition. A new way of constructing the Killing vectors of U(1) bundles over products of Einstein-Kahler spaces is given. Then, the non-linear Kaluza-Klein ansatz is constructed to describe the embeddings of the U(1)3, U(1)4 and U(1)2 truncations of D = 5, D = 4 and D = 7 gauged supergravities into the type IIB string and M-theory. These enable one to oxidise any associated lower dimensional solutions to D = 10 or D = 11. In particular, these general ansatz are used to embed the charged AdS5, AdS4 and AdS 7 black hole solutions in ten and eleven dimensions. The charges for the black holes with toroidal horizons may be interpreted as the angular momenta of D3-branes, M2-branes and M5-branes spinning in the transverse dimensions, in their near-horizon decoupling limits. The horizons of the black holes coincide with the worldvolumes of the branes. Finally, a rectangular three-dimensional lattice of intersecting domain walls, with arbitrary spacing, is shown to emerge naturally as a classical solution of M-theory. |
