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 ${\mathbb{CP}}^2$.

17 pages