Vlasov's equation describes the time evolution of the distribution function for a collisionless physical system of identical particles, such as plasma or galaxies. Together with Poisson's equation, which yields the potential, it forms the Vlasov-Poisson system. In Euclidean space this system has been extensively studied in the past century. It has been recently shown that the Valsov-Poisson system exhibits an interesting, counter-intuitive phenomenon called Landau damping. Our universe, however, may not be at on a large scale, so it is important to introduce and study a natural extension of the Vlasov-Poisson systems to spaces of constant curvature. Our starting point is the unit sphere S2, but we further restrict our study to one of its great circles. We show that, even for this reduced model, the potential function has more singularities than in the classical case. Our main result is to derive a Penrose stability criterion for the linear Landau damping phenomenon. / Graduate / 0405 / shengyis@uvic.ca