  Systems of identical particles with equal charge are studied under a special
type of confinement. These classical particles are free to move inside some
convex region S and on the boundary of it $\Omega$ (the $S^{d-1}-$ sphere, in
our case). We shall show how particles arrange themselves under the sole action
of the Coulomb repulsion in many dimensions in the usual Euclidean space,
therefore generalizing the so called Thomson problem to many dimensions. Also,
we explore how the problem varies when non-Euclidean geometries are considered.
We shall see that optimal configurations in all cases possess a high degree of
symmetry, regardless of the concomitant dimension or geometry.
