Bach-flat critical metrics for quadratic curvature functionals (Q1994109)

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.