# Vacuum static spaces with vanishing of complete divergence of Bach tensor and Weyl tensor

## TL;DR

This paper investigates vacuum static spaces where the complete divergence of the Bach and Weyl tensors vanish, revealing implications for harmonicity, black hole non-existence, and the Besse conjecture.

## Contribution

It establishes new links between divergence conditions of curvature tensors and geometric properties, including harmonicity and Bach-flatness, and proves the Besse conjecture under weaker assumptions.

## Key findings

- Vanishing divergence implies harmonicity of the metric.
- Non-existence of multiple black holes under these conditions.
- Proof of the Besse conjecture with weaker assumptions.

## Abstract

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Bach tensor and Weyl tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. On the other hand, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.