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 > 1/48 R are Einstein and, by a known result, are isometric to S⁴, RP⁴ or CP².