Riccati Equation for Static Spaces and its Applications

In this paper, we derive a Riccati-type equation applicable to (sub-)static Einstein spaces and examine its various applications. Specifically, within the framework of conformally compactifiable manifolds, we prove a splitting theorem for the Riemannian universal covering. Furthermore, we demonstrate two distinct methods by which the Riccati equation can establish the connectivity of the conformal boundary under the static Einstein equation. Additionally, for compact static triples possessing positive scalar curvature, we establish the compactness of the universal covering.