Some days ago, I listened a report about the width of Riemann manifold. I am interesting in the space of closed curves in Riemann manifold. It seemly has good topology and different construction.For example, let $(M,g)$ be a Riemann manifold, $Z_1(M)$ is the space of all closed curves in $M$, let $\sigma:[0,1]\rightarrow Z_1(M)$ be a famliy of curves, then we can compute $\frac{d\sigma}{ds}$ , then the minimum point is the geodesic ,and there are many other interesting results .But I forget to ask the reporter which paper tell the details about the space of closed curves.
I want to know the detail about the space of closed curves, what I should read ?
The literature is massive. Some keywords are morse theory, calculus of variations, and global analysis.
Topologically the space of continuous curves is given the compact-open topology. This is bad for doing analysis. Better is to look at spaces with some Sobolev regularity. You can look at the books of Klingenberg for a reference. Another option is to look at the space of smooth curves. This has no Hilbert manifold structure, but a Frechet one. I've heard the book of Kriegl and Michor as a reference, but it goes deep into the functional analysis. Also have a look at books of Struwe and of Jost.