New rigidity results for critical metrics of some quadratic curvature functionals

We prove a new rigidity result for metrics defined on closed smooth n-manifolds that are critical for the quadratic functional Ft, which depends on the Ricci curvature Ric and the scalar curvature R, and that satisfy a pinching condition of the form Sec > ε R, where ε is a function of t and n, while Sec denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying Sec > 148 R are Einstein and, by a known result, are isometric to S4, RP4 or CP2.