Isometric Immersion of $S^1\to M$ $M$ be any Riemannian manifold, and $S^1$ is a circle.
We can give Manifold structure to $C^\infty(S^1, M)$ modeled on nuclear frechet space.
Take $Imm(S^1, M):\{f\in C^\infty(S^1,M): f \text{ is an immersion} \}$.  This is open subset of $C^\infty(S^1,M)$ hence it is also a nuclear frechet manifold.
Consider set of all isomoetric immersion of $S^1\to M$, can we give some differentiable structure here.  Please provide the reference where people have already studied the isometric immersed loops over a manifold.
Edit:   Can we see  Set of isometric immersion as a manifold modeled over some Locally convex space.
 A: The case where $\dim M = 3$ is considered in Brylinski's book Loop spaces, characteristic classes and geometric quantization (I don't know where it originates, presumably there are references in the book if it doesn't originate there - I don't have a copy to hand).
The method given there ought to generalise.  Look at an isometric immersion, say $\alpha \colon S^1 \to M$.  A chart at $\alpha$ for the full loop space, $L M = C^\infty(S^1, M)$, has domain $\Gamma_{S^1}(\alpha^* T M)$.  So long as the chart map is chosen carefully (and I'm pretty sure it can be done so), the following should work.  As $\alpha$ is an immersion, we have a non-zero section $\alpha' \in \Gamma_{S^1}(\alpha^* T M)$.  The fibrewise orthogonal complement (in the induced metric from $T M$) of this defines a subbundle, say $E_\alpha$, of $\alpha^* T M$.  We take sections of this, $\Gamma_{S^1}(E_\alpha)$, and restrict the chart map to this.  The idea being that if $\beta$ is the image of a section $X$ then $\|\beta'\|^2 = \|\alpha'\|^2 + 2\langle \alpha', X \rangle + \| X\|^2$ and to first order, as $\langle \alpha', X \rangle = 0$ then this is $\|\alpha'\|^2 = 1$ (as $\alpha$ is an immersion).  To make this precise one would need to choose the original chart map very carefully, but as I said I don't think that would be difficult and the details should generalise from the 3-manifold case as given in Brylinski's book.
A: Let $(M, g)$ be a smooth Riemannian manifold, and $f \colon S^1 \to M$ a smooth immersion. The metric $f^* g$ on $S^1$ is flat, since $S^1$ is one-dimensional. Since the flat metric is unique up to diffeomorphism then, given any fixed metric $h$ on $S^1$, there exists a diffeomorphism $\varphi \colon S^1 \to S^1$ such that the pull-back $f^* g$ under $\varphi$ is $h$. In particular, given any immersion $f$, there exists a $\varphi$ such that $f \circ \varphi$ is an isometry. As such, the space of isometric immersions may be identified with the quotient of the space of smooth immersions by isometric diffeomorphisms (i.e. rotations) of $S^1$. 
