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 $B_3$.'

  • $\begingroup$ 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. $\endgroup$ – onurcanbkts May 30 '19 at 15:37
  • $\begingroup$ "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? $\endgroup$ – Thomas Andrews May 30 '19 at 15:51
  • $\begingroup$ It seems like you are describing a space such that every point corresponds to a Riemannian metric on $B^3$. Is that right? $\endgroup$ – Michael Albanese May 30 '19 at 15:55
  • $\begingroup$ @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 $S_1$. $\endgroup$ – zooby May 30 '19 at 18:01
  • $\begingroup$ @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. $\endgroup$ – Michael Albanese May 30 '19 at 18:10

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.

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  • $\begingroup$ 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? $\endgroup$ – zooby May 30 '19 at 21:26

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