The purpose of this article is to investigate the geometry of static perfect fluid space-time metrics on compact manifolds with boundary. In the first part, we provide a boundary estimate for static perfect fluid space-time. In the second part, we establish a unified Böchner type formula for a large class of spaces that include the static perfect fluid space-time, critical metrics of the volume functional, static spaces and CPE metrics. Moreover, as a consequence of such a formula we obtain a gap result for a compact static perfect fluid space-time.

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We consider the static vacuum Einstein space-time when the spatial factor (or, base) is conformal to a pseudo-Euclidean space, which is invariant under the action of a translation group. We characterize all such solitons. Moreover, we give examples of static vacuum Einstein solutions for Einstein’s field equation. Applications provide an explicit example of a complete static vacuum Einstein space-time.

In this paper we provide a method capable of producing an infinite number of solutions for Einstein's equation on static spacetimes with perfect fluid as a matter field. All spacetimes of this type which are symmetric with respect to a given group of translations and whose spatial factor is conformally flat, are characterized. We use this method to give some exact solutions of the referred equation.