  We derive a map between Einstein spaces of positive and negative curvature,
including scalar matter. Starting from a space of positive curvature with some
dimensions compactified on a sphere and analytically continuing the number of
compact dimensions, we obtain a space of negative curvature with a compact
hyperbolic subspace, and vice versa. Prime examples of such spaces are de
Sitter (dS) and anti-de Sitter (AdS) space, as well as black hole spacetimes
with (A)dS asymptotics and perturbed versions thereof, which play an important
role in holography. This map extends work done by Caldarelli et al., who map
asymptotically AdS spaces to Ricci-flat ones. A remarkable result is that the
boundary of asymptotically AdS spaces is mapped to a brane in the bulk of de
Sitter, and perturbations near the AdS boundary are sourced by a stress tensor
confined to this brane. We also calculate the Brown-York stress tensor for the
perturbed AdS metric, which turns out to be the negative of the stress tensor
on the de Sitter brane. The map can also be used as a solution generator, and
we obtain a Kerr/AdS solution with hyperbolic horizon from a known Kerr/dS one.
