Abstract
Static perfect fluid spaces are of great interest in metric theories of gravitation, they being used in building realistic models of some compact objects, like neutron stars and white dwarfs. In this article, we show that a compact static perfect fluid space with non-trivial solution f and Ricci curvature in the direction of the vector field gradf bounded below by a certain lower bound, is necessarily a standard sphere and the converse statement also holds. Moreover, we establish two simple characterizations of the Euclidean spaces through connected static perfect fluid spaces.