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 $B_3$.'