Riccati Equation for Static Spaces and its Applications

TL;DR

This paper derives a Riccati-type equation for static Einstein spaces, explores its applications in conformally compactifiable manifolds, and proves results on boundary connectivity and universal covering compactness.

Contribution

It introduces a Riccati equation tailored for static Einstein spaces and applies it to prove new geometric and topological properties.

Findings

Splitting theorem for the Riemannian universal covering

Connectivity of the conformal boundary established

Universal covering compactness for positive scalar curvature cases

Abstract

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.