  In this paper, we focus on the geometry of compact conformally flat manifolds
$(M^n,g)$ with positive scalar curvature. Schoen-Yau proved that its universal
cover $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$
such that $M^n$ is a Kleinian manifold. Moreover, the limit set of the Kleinian
group has Hausdorff dimension $<\frac{n-2}{2}$. If additionally we assume that
the non-local curvature $Q_{2\gamma}\geq 0$ for some $1<\gamma<2$, the
Hausdorff dimension of the limit set is less than or equal to
$\frac{n-2\gamma}{2}$. If $Q_{2\gamma}>0$, then the above inequality is strict.
Moreover, the above upper bound is sharp. As applications, we obtain some
topological rigidity and classification theorems in lower dimensions.
