Collapsing manifold

For the concept in homotopy, see collapse (topology).

In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics g_n, such that as n goes to infinity the manifold is close to a k-dimensional space, where k < n, in the Gromov–Hausdorff distance sense. Generally there are some restrictions on the sectional curvatures of (M, g_n). The simplest example is a flat manifold, whose metric can be rescaled by 1/n, so that the manifold is close to a point, but its curvature remains 0 for all n.

Examples

Generally speaking there are two types of collapsing:

(1) The first type is a collapse while keeping the curvature uniformly bounded, say $|\sec(M_i)|\le 1$.

Let M_i be a sequence of n dimensional Riemannian manifolds, where sec(M_i) denotes 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 ε(n) such that if $|\sec(M_i)|\le 1$ and Inj(M_i) < ε(n), then M_i admits an injectivity radius of the manifold M. Roughly speaking the N-structure is a locally action of a nilmanifold, which is a generalization of an F-structure, introduced by Cheeger and Gromov. This theorem generalized previous theorems of Cheeger-Gromov and Fukaya where they only deal with the torus action and bounded diameter cases respectively.

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

This is closely related to the so-called almost nonnegatively curved manifold case which generalizes non-negatively curved manifolds as well as almost flat manifolds. A manifold is said to be almost nonnegatively curved if it admits a sequence of metrics g_i, such that $\sec(M,g_i)\ge -1/n$ and ${\rm diam}(M,g_i)\le 1/n$. The role that an almost nonnegatively curved manifold plays in this collapsing case when curvature is bounded below is the same as an almost flat manifold plays in the curvature bounded case.

When curvature is only bounded below, the limit space is an Alexandrov space. A theorem says that on the regular part of the limit space, there is a fibration structure when i is sufficiently large, the fiber is an almost nonnegatively curved manifold.^{[citation needed]} Here the regular means the (δ,n)-strainer radius is uniformly bounded from below by some number.

What happens at a 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 that there exists a ε(n) and δ(n) such that if a 3-dimensional manifold M satisfies Vol(M) < ε(n) then 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.

External links

• Examples of collapsing manifolds (PDF)