Imagine that there is a class of Riemannian metrics $\mathcal{R}$ on 3-dimensional manifolds such that

- $\mathcal{R}$ is locally finite dimensional; i.e., there are finite number of real parameters which describe the metric locally at any point;
- Any closed 3-dimensional manifold admits a metric from $\mathcal{R}$;
- $\mathcal{R}$ is is invariant with respect to Ricci flow.

Likely *such $\mathcal{R}$ does not exist*.
Otherwise there is a good chance to simplify the Perelman's proof.

Can it be proved that such $\mathcal{R}$ does not exist?

UPDATE: You need to assume that metric depend continuously on the real parameters and $\mathcal{R}$ is closed in a reasonable topology.