Vacuum static spaces with harmonic curvature

TL;DR

This paper classifies n-dimensional vacuum static spaces with harmonic curvature, revealing they are locally isometric to four specific geometric types, thus advancing understanding of their structure.

Contribution

It provides a comprehensive classification of vacuum static spaces with harmonic curvature, including both complete and incomplete cases, identifying four fundamental geometric types.

Findings

Vacuum static spaces with harmonic curvature are classified into four types.

Complete vacuum static spaces with harmonic curvature fall into these four categories.

The classification includes local isometries to Einstein and product manifolds.

Abstract

In this article we make a thorough classification of (not necessarily complete) $n$-dimensional vacuum static spaces $(M,g,f)$ with harmonic curvature and, as a corollary, obtain a classification of complete vacuum static spaces with harmonic curvature. Indeed, we showed that $(M,g,f)$ is locally isometric to one of the following four types; (i) the Riemannian product of an Einstein manifold and a vacuum static space, (ii) the warped product over an interval with an Einstein manifold as fiber, (iii) an Einstein manifold, (iv) the Riemannian product of two Einstein manifolds.