In the studies of active contours they describe the set of all simple smooth closed curves on $\mathbb{R}^2$ to be a Riemannian Manifold $M$. The tangent space at a curve $c$, $T\_cM$ is a set of vector fields defined on $c(t)$. The Riemannian metric is $\langle x,y \rangle\_c = \int\_c x(t)\cdot y(t)dt$. Is this in fact a manifold? If so how would you embed it in $\mathbb{R}^n$ i.e. what are the coordinate charts of this manifold?



Is the set of all smoothed closed simple curves on $\mathbb{R}^2$ a manifold?
As Ryan Budney explained it is not a finite dimensional manifold. First let us notice that the space of all smooths loops $L\mathbb{R}^2=C^{\infty}(S^1,\mathbb{R}^2)$ is a topological vector space, it is Fréchet and it is an inverse Hilbert limit vector space.
If you consider the subset $Emb(S^1,\mathbb{R}^2)\subset L\mathbb{R}^2$ of embeddings this is an open subset hence an infinite dimensional manifold of Fréchet type. 
Let us go further, so far we have considered parametrized curves, if we want to get rid of the parametrization we have to divide by the action of the infinite dimensional Lie group $Diff(S^1)$ and we get a smooth principal bunde
$$Emb(S^1,\mathbb{R}^2)\rightarrow Emb(S^1,\mathbb{R}^2)/Diff(S^1)$$
I suppose that this is the base of this bundle you are interested in.
Maybe a good reference for all this stuff is Kriegl-Michor "The Convenient Setting of Global Analysis." Mathematical Surveys and Monographs, Volume: 53 (Chapter IX, section 44).


If you are interested by a discussion of Riemannian metric on these spaces have a look at:
Michor, P. ; Mumford, D.
Riemannian geometries on spaces of plane curves. 
J. Eur. Math. Soc. 8 (2006), no. 1, 1–48.


