Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

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.

Electrostatic system with divergence-free Bach tensor and non-null cosmological constant

“…There are some well-known classification results of some important geometric structures like static vacuum manifolds and Ricci solitons carrying a metric such that the Bach tensor is free from divergence (cf. [3,5,11,13,15]). Any threedimensional a Riemannian manifold is locally conformally flat if, and only if, its Cotton tensor C is identically zero.…”

Vacuum Static Spaces with Positive Isotropic Curvature

“…In [15] (cf. [14], [33]), we derived some useful relations of the Bach tensor B and the Cotton tensor C to the tensor T and its divergence δT . In view of (1.7) and(1.8), one can see that thses relations are still valid in vacuum static spaces if we just replace the function 1 + f in CPE case by f .…”

“…If T = 0, then up to a finite quotient and appropriate scaling, M is either isometric to a sphere S n , or a warped product, Proof. It is proved in [14] that a vacuum static space (M n , g, f ) with T = 0 is Bach-flat. The conclusion follows from a result due to Qing and Yuan [27].…”

“…Here z ○ z is defined by 14]) Suppose that ω = 0. Then the functions α and z 2 are both constant on each level set f −1 (c).…”

Besse conjecture with positive isotropic curvature

“…This structural tensor T has deep relations to the critical point equation (1.1) and the Cotton tensor (cf. [9]).…”