Bach-flat critical metrics for quadratic curvature functionals
type: Item
instance of: scholarly article
title: Bach-flat critical metrics for quadratic curvature functionals
zbMATH Open document ID: 1403.58006
author: Weimin Sheng
author: Lisheng Wang
published in: Annals of Global Analysis and Geometry
publication date: 2018-11-01T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional compact smooth manifold and let \(\mathcal M\) denote the space of smooth Riemannian metrics on \(M\). In this paper, the authors study the critical metrics for quadratic curvature functionals \(\mathcal F_\tau= \int_M |\mathrm{Ric}|^2dV_g+\tau\int_MR^2dV_g\) involving the Ricci curvature and scalar curvature in the space \(\mathcal M_1\) of Riemannian metrics with unit volume. For these functionals, Einstein metrics are always critical metrics. However, a converse problem is not always true. The purpose of this paper is to show that, under the condition that the critical metrics are Bach-flat, a partial converse is true. We recall the definition of the Bach tensor. Let \(W\) be the Weyl conformal curvature tensor of \((M^n,g)\). The Bach tensor is defined by \(B_{ij}=\frac{1}{n-3}\nabla^k\nabla^l W_{ikjl} + \frac{1}{n-2}R^{kl} W_{ikjl}\) which was first introduced by R. Bach to investigate conformal relativity. In dimension 4, the Bach tensor arises as the Euler-Lagrange equations of the \(L^2\)-norm of the Weyl curvature tensor \(W\). In this paper the authors obtained a number of conditions on the metric \(g\in\mathcal M_1\) and the value \(\tau\), under which it turns out that if the metric \(g\) is critical point of functional \(\mathcal F_\tau\), then it will be Einstein.
Mathematics Subject Classification ID: 58E11
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 6970103
zbMATH Keywords: Bach-flat
zbMATH Keywords: quadratic functionals
zbMATH Keywords: Einstein metric
zbMATH Keywords: critical points
zbMATH Keywords: Riemannian functionals
zbMATH Keywords: space of Riemannian metrics
reviewed by: Nicolai K. Smolentsev
MaRDI profile type: Publication
full work available at URL: s10455-018-9606-4
OpenAlex ID: W2790711905
cites work: Q4398855
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Wikidata QID: Q126015457
DOI: 10.1007/S10455-018-9606-4
description: scientific article; zbMATH DE number 6970103