  We establish an explicit correspondence of Einstein gravity on the squashed
spheres that are the $U(1)$ bundles over $\mathbb{CP}^m$ to the Kaluza-Klein
AdS gravity on the tori. This allows us to map the Ricci-flat Kerr metrics in
odd dimensions with all equal angular momenta to charged Kaluza-Klein AdS black
holes that can be lifted to become singly rotating M-branes and D3-branes.
Furthermore, we find maps between Ricci-flat gravitational instantons to the
AdS domain walls. In particular the supersymmetric bolt instantons correspond
to domain walls that can be interpreted as distributed M-branes and D3-branes,
whilst the non-supersymmetric Taub-NUT solutions yield new domain walls that
can be lifted to become solutions in M-theory or type IIB supergravity. The
correspondence also inspires us to obtain a new superpotential in the
Kaluza-Klein AdS gravity in four dimensions.
