An integral inequality for compact Bach-flat $\mathcal{A}_{2}$-manifolds

We establish a Catino-type integral inequality for closed Bach-flat $\mathcal{A}_2$-manifolds, namely Riemannian manifolds with nonnegative scalar curvature and constant nonnegative second Schouten curvature. In the equality case, we derive rigidity results showing that the manifold is either Einstein or isometrically covered by $\mathbb{S}^{1}\times\mathbb{S}^{n-1}(\kappa)$ endowed with the product metric.