# Vanishing conditions on Weyl tensor for Einstein-type manifolds

## TL;DR

This paper establishes conditions under which Einstein-type manifolds with specific Weyl tensor properties are locally warped products, leading to implications for the nonexistence of multiple black holes in static spacetimes.

## Contribution

It proves that Einstein-type manifolds with divergence-free Weyl tensor and zero radial Weyl curvature are locally warped products, extending geometric understanding and black hole uniqueness results.

## Key findings

- Manifolds are locally warped products under given conditions
- Zero radial Weyl curvature implies specific geometric structure
- Results imply nonexistence of multiple black holes in certain spacetimes

## Abstract

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional Einstein fibers, provided that the potential function is proper. As a consequence, we prove a result about the nonexistence of multiple black holes in static spacetimes.