On locally conformally flat critical metrics for quadratic functionals (Q2014336)

Let \((M^n,g)\) be a connected closed smooth manifold of dimension \(n\geqslant 3\) and let \(\mathcal M_1\) denote the space of equivalence classes of smooth Riemannian metrics of volume one on \(M^n\). In the reviewed paper the authors consider the quadratic functional \[ \mathcal F_{t,s}(g)=\int_M{|\mathrm{Ric}_g|^2dV_g}+t\int_M{R_g^2dV_g}+s\int_M{|Rm_g|^2dV_g}\,, \] defined for some real constants \(t\) and \(s\), where \(R_g\), \(\mathrm{Ric}_g\), \(Rm_g\) and \(dV_g\) denote, respectively, the scalar curvature, the Ricci curvature tensor, the Riemann curvature tensor of \(g\) and the volume form of \(M^n\). It is known that space form metrics are critical point for \(\mathcal F_{t,s}\). The authors investigate the converse problem and they find conditions for a critical metric to be a space form metric.