We generalise the standard, flat p-brane solutions sourced by a dilaton and a form field, by taking the worldvolume to be a curved Einstein space, such as (anti-)de Sitter space. Our method is based on reducing the p-branes to domain walls and then a
llowing these domain walls to be curved. For de Sitter worldvolumes this extends some recently constructed warped de Sitter non-compactifications. We restrict our analysis to solutions that possess scaling behavior and demonstrate that these scaling solutions are near-horizon limits of a more general solution. Finally, our framework can equally be used for spacelike branes and the uplift of the domain wall/cosmology correspondence becomes in this context a more general timelike/spacelike brane correspondence.
In this note we compare the geodesic formalism for spherically symmetric black hole solutions with the black hole effective potential approach. The geodesic formalism is beneficial for symmetric supergravity theories since the symmetries of the large
r target space leads to a complete set of commuting constants of motion that establish the integrability of the geodesic equations of motion, as shown in arXiv:1007.3209. We point out that the integrability lifts straightforwardly to the integrability of the equations of motion with a black hole potential. This construction turns out to be a generalisation of the connection between Toda molecule equations and geodesic motion on symmetric spaces known in the mathematics literature. We describe in some detail how this generalisation of the Toda molecule equations arises.
