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 $\mathfrak{F}_{t}$, which depends on the Ricci curvature $Ric$ and the scalar curvature $R$, and that satisfy a pinching condition of the form $Sec > \epsilon R$, where $\epsilon$ 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 $Sec > \frac{1}{48} R$ are Einstein and, by a known result, are isometric to $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$.