Static uniform-density stars must be spherical in general relativity

In this paper the uniqueness of the static solutions of Einstein's equation that represent isolated uniform-density perfect-fluid stellar models is demonstrated: any static asymptotically flat space-time containing only a uniform-density perfect fluid confined to a spatially compact world tube is necessarily spherically symmetric. This result generalizes to relativistic uniform-density models the well known Newtonian theorem of Carleman and Lichtenstein