I want to write this in mathematical notation: "Let us represent a ball, $B_3$, with a metric $g$ as a point on manifold. Let $M$ be the (infinite dimensional) manifold formed from every ball with all possible smooth metrics." In such a way that smoothly going from one point on $M$ to another smoothly varies the metric of the ball.
Does this "manifold of manifolds" have a name? (This would be a topological manifold unless one defined some kind of `meta metric' on it.)
Edit: As Michael pointed out this is more precisely described as 'a space such that every point corresponds to a Riemannian metric on $B_3$.'
Here are some general comments.
If $M$ is a smooth manifold, the collection of Riemannian metrics on $M$ is an open subset of the infinite-dimensional vector space $\Gamma(M, S^2(TM)^*)$ where $S^2$ denotes the second symmetric power.
If $M$ is compact, $\Gamma(M, S^2(TM)^*)$ is a Fréchet space, and hence the collection of Riemannian metrics on $M$ is an infinite-dimensional Fréchet manifold.