Collapsing manifold

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In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M which admits a sequence of metric gn, such that the as n goes to infinity the manifold is closed to a k(< n)-dimensional space in the Gromov-Hausdorff distance sense. Generally there are some restrictions on the sectional curvatures of (M, gn). The simple example is the flat manifold which you can rescale the metric by 1/n, so that the manifold is closed a point, but it's curvature remains 0 for all n.

Generally speaking there are two types collapsing:

(1) The first type is the collapsing while keeps curvature uniformly bouned, say |\sec(M_i)|\le 1.

Let Mi be a sequence of n dimensional Riemannian manifolds, we use sec(Mi) to denote the sectional curvature of the ith manifold. There is a theorem proved by Jeff Cheeger, Kenji Fukaya and Mikhail Gromov, which states that: There exists a constant \varepsilon(n) such that if |\sec(M_i)|\le 1 and {\rm Inj}(M_i)<\varepsilon(n), then Mi admits a N-structure, where Inj(M) means injectivity radius of manifold M. Roughly speaking the N-structure is a locally action of a nilmanifold, which is a generlization of F-structure which introduced by Cheeger and Gromov. This theorem generlized the previous theorm of Cheeger-Gromov and Fukaya where the only deal with the torus action and bounded diameter case respectively.

(2) The second type is the collapsing while only keeps the lower bound of curvature, say \sec(M_i)\ge -1.

This is closed related to the so called almost nonnegatively curved manifold which generlized the nonnegatively curved manifold as well as almost flat manifold. A manifold is said almost nonegatively curved if it admits a sequence of metric gi, such that \sec(M,g_i)\ge -1/n and {\rm diam}(M,g_i)\le 1/n. The role that almost nonnegatively curved manifold plays in this collapsing case when curvature bounded blow is the same as almost flat manifold plays in curvature bounded case.

When curvature only bounded below, the limit space is Alexandrov space. A theorem of Yamaguchi said that on the regular part of the limit space, there is a fibration structure when i sufficiently large, the fiber is almost nonnegatively curved manifold. Here the regular means the (δ,n)-strainer radius is uniformly bounded by below by some number.

What happens at singular point? There is no answer to this question at this time. But on dimension 3, Yamaguchi gives a full classification of this type collapsed manifold. He proved the following that there exists a \varepsilon(n) and δ(n) such that if a 3-dimensional manifold M satisfies {\rm Volum}(M)<\varepsilon(n) the one of the following is true (i) M is a graph manifold or (ii) M has diameter less than δ(n) and has finite fundamental group.