Manifold of immersions of a manifold

Question

Let M be a Riemannian manifold, S a closed immersed submanifold of M, and consider the space of smooth perturbations, i.e. functions X : S → NS (the normal bundle of S in M), p → (p, X_p) for some X_p ∈ N_pS.

I am vaguely aware that this space comes equipped with an inner product, where X⋅Y is given by integrating X_p⋅Y_p over S. As I understand it, this inner product turns the space of immersed images of S into an infinite-dimensional Riemannian manifold, something like as follows (for now, let M = ℝⁿ):

Let H be the (Hilbert) space of all smooth maps S → M, I ⊂ H the proper open subset of maps which are immersions, and G the diffeomorphism group of S. Then G acts on I (on the right) by composition, and I/G inherits a Riemannian structure from H.

I have never seen this explained in detail. I have no infinite-dimensional geometry, relatively little Riemannian geometry and am very sketchy on how one would make the above rigorous. So my first question is: is the above correct, and if so, where - if anywhere - can I find a decent explanation, preferably one which doesn't assume too much background knowledge?

My second question is: might it be possible to generalize the above to maps which "aren't quite smooth"? Specifically, I'm interested in maps whose components can be written as the difference of convex functions (see my previous question here). These include C² and piecewise-linear maps in particular; the original motivation for all this was to apply the calculus of variations to immersed polyhedra in a rigorous manner (see some of my other previous questions for background), but I thought it would be nice to try to unify it with the existing theory.

Such maps give rise to a wider class of "not-quite-smooth" manifolds, along with corresponding notions of pushforward, immersion etc. I feel that if the above is true for smooth manifolds, it should generalize to these ones as well. But then, maybe not. It would really help if I could see a careful proof of the smooth case...

Answer 1

$Imm(S,M)/Diff(S)$ are called shape spaces. They are not manifolds in general, but are infinite dimensional orbifolds. You find a lot of information in the following papers. The last one is an overview article summarizing a lot of results. A surprising phenomenon is that the Riemannian metric on shape space induced by the $L^2$-metric on the space of immersions always has vanishing geodesic distance.

• Peter W. Michor; David Mumford: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48. (pdf)

• Peter W. Michor; David Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math. 10 (2005), 217--245. (pdf)

• Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics 3, 4 (2011), 389-438. (pdf)

• Martin Bauer, Martins Bruveris, Peter W. Michor: Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. 37 pages. arXiv:1305.1150.