I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a few options in mind). (Since $M$ is not necessarily simply-connected, I guess that no nice topology could make $\mathcal{C}$ connected.) However, I know nothing about infinite-dimensional manifolds, hence my question: could anybody please point me to some monograph or rich survey article on the subject?

I've read a brief presentation by Andrew Stacey from which I've understood that infinite-dimensional Riemann manifolds can be strong or weak, the weak ones can in particular be weak co-Riemannian, and that there might also be problems regarding the local-triviality of the tangent bundle. I guess that the results that I'm looking for are the ones that best apply to spaces of smooth curves with fixed endpoints (which rules out the theory of loop spaces).

(One of the reasons I'm trying to do this is in order for me to compactify this manifold and still get "something with a distance". Maybe having this end in mind could narrow down the focus area of your answers.)


Firstly, I would suggest taking a look at the work of Peter Michor, who is actually a frequent contributor to MO. Regarding Riemannian metrics on spaces of curves, you might be interested in this nice recent paper. Also, for more on infinite-dimensional Riemannian geometry including some metric aspects, take a look at the work of Brian Clarke, Boris Khesin, Jonatan Lenells, Gerard Misiolek, Stephen Preston and collaborators, such as this and this paper.

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