# What is the way to write a manifold of manifolds?

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 $$B_3$$.'

• This is basically the definition of Riemanian manifold if I understand your definition correctly. Note that, in a manifold, the metric is not defined on the manifold itself, but rather on the tangent space of each point on that manifold. – onurcanbkts May 30 '19 at 15:37
• "Every ball with all possible smooth metrics" is a bit of a vague concept. Once you define this clearly, how does this set form an infinite dimensional manifold? When are two balls close to each other? – Thomas Andrews May 30 '19 at 15:51
• It seems like you are describing a space such that every point corresponds to a Riemannian metric on $B^3$. Is that right? – Michael Albanese May 30 '19 at 15:55
• @Michael yes. that's right. Space' or 'manifold' seem somewhat interchangeable to me. e.g is $\mathbb{R}^3$ a space or an infinite flat manifold? I guess a manifold is an unlabelled space? e.g. the space of all 2D rotations is a circle $S_1$. – zooby May 30 '19 at 18:01
• @zooby: A space usually refers to a topological space, while a manifold is a very specific type of topological space; the two terms are not interchangeable. Every manifold is a (topological) space, but not conversely. – Michael Albanese May 30 '19 at 18:10

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.
• If we take all metrics on $B_3$ that all share a common metric on the boundary $S_3$, then I wonder would this a be closed Frechet Manifold? – zooby May 30 '19 at 21:26