  We study the Schr\"odinger-Poisson system on the unit sphere $\SS^2$ of
$\RR^3$, modeling the quantum transport of charged particles confined on a
sphere by an external potential. Our first results concern the Cauchy problem
for this system. We prove that this problem is regularly well-posed on every
$H^s(\SS ^2)$ with $s>0$, and not uniformly well-posed on $L^2(\SS ^2)$. The
proof of well-posedness relies on multilinear Strichartz estimates, the proof
of ill-posedness relies on the construction of a counterexample which
concentrates exponentially on a closed geodesic. In a second part of the paper,
we prove that this model can be obtained as the limit of the three dimensional
Schr\"odinger-Poisson system, singularly perturbed by an external potential
that confines the particles in the vicinity of the sphere.
