Rigidity of critical metrics for quadratic curvature functions on closed Riemannian manifolds

We study rigidity of critical metrics for quadratic curvature functions $\mathcal {F}_{t,s}(g)$ involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor. In particular, when $s=0$, we give new characterizations by pointwise inequalities involving the Weyl curvature and the traceless Ricci tensor for critical metrics with divergence-free Cotton tensor. We also provide a few rigidity results for locally conformally flat critical metrics.