Critical point metrics of the total scalar curvature (Q2890358)

Let \((M,g)\) be a compact Riemannian manifold. Einstein and Hilbert showed that the critical points of the total scalar curvature functional are Einstein metrics. The Yamabe problem [\textit{H. Yamabe}, Osaka Math. J. 12, 21--37 (1960; Zbl 0096.37201)] consists in showing that a Riemannian manifold \((M,g)\) of dimension \(n\geq 3\) has a metric \(g'\), conformal to \(g\), with constant scalar curvature. The solution to the Yamabe problem led \textit{A. L. Besse} [Einstein manifolds. Berlin etc.: Springer-Verlag (1987; Zbl 0613.53001)] to conjecture that the result of Einstein and Hilbert holds true when one restricts the total scalar curvature functional to the subset of Riemannian metrics with constant scalar curvature. The Euler-Lagrange equations for this restricted variational problem lead to a critical point equation (CPE) involving the traceless Ricci tensor of the metric. Hence, Besse's conjecture consists in showing that if the CPE has a (nontrivial) solution, the corresponding metric is Einstein.NEWLINENEWLINEThe upshot of this paper is a proof of Besse's conjecture when \((M,g)\) is a Riemannian manifold with parallel Ricci tensor, such that the CPE has a nontrivial solution. Indeed, the authors show that \((M,g)\) is Einstein and then, by a result of \textit{M. Obata} [J. Math. Soc. Japan 14, 333--340 (1962; Zbl 0115.39302)], isometric to a standard sphere. In order to obtain this result, they prove that, when the CPE has a nontrivial solution, the manifold can not be a warped product. Applications to \(3\)-manifolds are given.