  We generalize the Newtonian n-body problem to spaces of curvature k=constant,
and study the motion in the 2-dimensional case. For k>0, the equations of
motion encounter non-collision singularities, which occur when two bodies are
antipodal. This phenomenon leads, on one hand, to hybrid solution singularities
for as few as 3 bodies, whose corresponding orbits end up in a
collision-antipodal configuration in finite time; on the other hand, it
produces non-singularity collisions, characterized by finite velocities and
forces at the collision instant. We also point out the existence of several
classes of relative equilibria, including the hyperbolic rotations for k<0. In
the end, we prove Saari's conjecture when the bodies are on a geodesic that
rotates elliptically or hyperbolically. We also emphasize that fixed points are
specific to the case k>0, hyperbolic relative equilibria to k<0, and Lagrangian
orbits of arbitrary masses to k=0--results that provide new criteria towards
understanding the large-scale geometry of the physical space.
