  The BPS Skyrme model has many exact analytic solutions in flat space. We
generalize the model to a curved space or spacetime and find that the solutions
can only be BPS for a constant time-time component of the metric tensor. We
find exact solutions on the curved spaces: a 3-sphere and a 3-hyperboloid; and
we further find an analytic gravitating Skyrmion on the 3-sphere. For the case
of a nontrivial time-time component of the metric, we suggest a potential for
which we find analytic solutions on anti-de Sitter and de Sitter spacetimes in
the limit of no gravitational backreaction. We take the gravitational coupling
into account in numerical solutions and show that they are well approximated by
the analytic solutions for weak gravitational coupling.
