Is there a moduli space $\mathcal{M}$ of the closed, $n$-dimensional, simply connected manifolds with the following properties:
(i) Two abstract simply connected closed manifolds $M, N\in\mathcal{M}$ are identified if their embeddings in $\mathbb{R}^{\infty}$ (actually $\mathbb{R}^{\max(2\dim M,2\dim N)}$ suffices by the Whitney embedding theorem) are the same, i.e., if the images of their $C^k$-smooth embeddings are equal: $im(\text{Emb}(M,\mathbb{R}^{\infty}))=im(\text{Emb}(N,\mathbb{R}^{\infty}))$;
(ii) $\mathcal{M}$ is itself a manifold which carries a natural Hausdorff topology induced by the metric $d(M,N)=\sup_{(p,q)\in M\times N}d(\text{Emb}(B(p;r),\mathbb{R}^{\infty}),\text{Emb}(B(q;r),\mathbb{R}^{\infty}))$ where $B(p;r):=\{x\in M: d(p,x)<r\}$ ("the distance between the two embeddings").
For instance, could one use the affine Grassmannian $Graff(n,V)$? I have also seen the use of the moduli space of a manifold in this series, defined as $$ \Psi_d(\mathbb{R}^{n+1}) = \left\{ \omega\subseteq\mathbb{R}^n\ \middle\vert \begin{array}{l} \text{topologically closed smooth } d\text{-dimensional}\\ \text{submanifold, i.e. closed as a subset of } \mathbb{R}^n, \\ \partial\omega = \emptyset. \end{array}\right\}. $$How well are these objects known and have they been studied?
Thoughts: If we consider $Graff(n,\mathbb{R}^{\infty})$, then we can identify each point in it as the tangent bundle of one such manifold. Then we could possibly use the exponential map to give a description of this moduli space. I am curious if there is a known expression which exhausts all such manifolds.
Please note I am posting this here, as this was not received well on MO. Before you vote to close this, could you suggest a way to improve the question? I am not an expert, so it would be nice to receive constructive feedback instead of an immediate adverse response.
Thanks in advance! Any help would be appreciated.
