On conformally flat manifolds with constant positive scalar curvature

We classify compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either ${\mathbb{S}}^n$ with the round metric, ${\mathbb{S}}^1\times {\mathbb{S}}^{n-1}$ with the product metric or ${\mathbb{S}}^1\times {\mathbb{S}}^{n-1}$ with a rotationally symmetric Derdzi'nski metric.