However, Hamilton's theorem on 3-manifolds with $Ric>0$ and Brendle-Schoen differentiable sphere theorem would be interesting examples maybe. In these case, Ricci Flow (without surgery) creates a deformation between your initial metric (with $Ric>0$ or strictly quarter pinched) and metric of constant curvature $1$, since only quotients of the sphere bear such metrics, the initial manifold was a quotient of a sphere.