  The classification of the possible equilibrium shapes that a self-gravitating
fluid can take in a Riemannian manifold is a classical problem in mathematical
physics. In this paper it is proved that the equilibrium shapes are
isoparametric submanifolds. Some geometric properties of the equilibrium shapes
are also obtained, specifically the relationship with the isoperimetric problem
and the group of isometries of the manifold. This work follows the new
geometric approach to the problem developed by the author (see math-ph/0305038)
