New Rigidity Results for Critical Metrics of Some Quadratic Curvature Functionals

Abstract: 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} $.