  We consider the motion of n point particles of positive masses that interact
gravitationally on the 2-dimensional hyperbolic sphere, which has negative
constant Gaussian curvature. Using the stereographic projection, we derive the
equations of motion of this curved n-body problem in the Poincar\'e disk, where
we study the elliptic relative equilibria. Then we obtain the equations of
motion in the Poincar\'e upper half plane in order to analyze the hyperbolic
and parabolic relative equilibria. Using techniques of Riemannian geometry, we
characterize each of the above classes of periodic orbits. For n=2 and n=3 we
recover some previously known results and find new qualitative results about
relative equilibria that were not apparent in an extrinsic setting.
