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

Abstract

. We study rigidity of critical metrics for quadratic curvature functions 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 inequali-ties 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. Secondary 53C21.