$M$ 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.



Isometric Immersion of $S^1\to M$
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.


