  We obtain a natural extension of the Vlasov-Poisson system for stellar
dynamics to spaces of constant Gaussian curvature $\kappa\ne 0$: the unit
sphere $\mathbb S^2$, for $\kappa>0$, and the unit hyperbolic sphere $\mathbb
H^2$, for $\kappa<0$. These equations can be easily generalized to higher
dimensions. When the particles move on a geodesic, the system reduces to a
1-dimensional problem that is more singular than the classical analogue of the
Vlasov-Poisson system. In the analysis of this reduced model, we study the
well-posedness of the problem and derive Penrose-type conditions for linear
stability around homogeneous solutions in the sense of Landau damping.
