Collapsing Manifold

Collapsing Manifold

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.