Vanishing conditions on Weyl tensor for Einstein-type manifolds

Benedito Leandro

Vol. 314 (2021), No. 1, 99–113

DOI: 10.2140/pjm.2021.314.99

Abstract

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 $\left(n-1\right)$-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.

Keywords

Einstein-type manifolds, Weyl tensor, harmonic Weyl curvature.

Mathematical Subject Classification 2010

Primary: 53C20, 53C21, 53C25

Milestones

Received: 11 June 2019

Revised: 1 July 2020

Accepted: 24 July 2021

Published: 15 October 2021