Intrinsic flat distance

In mathematics, the intrinsic flat distance is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral currents lying in Euclidean space.

The Sormani–Wenger intrinsic flat (SWIF) distance is a distance between compact oriented Riemannian manifolds of the same dimension. More generally it defines the distance between two integral current spaces, (X,d,T), of the same dimension (see below). This class of spaces and this distance were first announced by mathematicians Sormani and Wenger at the Geometry Festival in 2009 and the detailed development of these notions appeared in the Journal of Differential Geometry in 2011. It has been described in Morgan's Huffington Post blog and has numerous applications.

The SWIF distance is an intrinsic notion based upon the (extrinsic) flat distance between submanifolds and integral currents in Euclidean space developed by Federer–Fleming. The definition imitates Gromov's definition of the Gromov–Hausdorff distance in that it involves taking an infima over all distance preserving maps of the given spaces into all possible ambient spaces Z. Once in a common space Z, the flat distance between the images is taken by viewing the images of the spaces as integral currents in the sense of Ambrosio–Kirchheim.

The rough idea in both intrinsic and extrinsic settings is to view the spaces as the boundary of a third space or region and to find the smallest weighted volume of this third space. In this way, spheres with many splines that contain increasingly small amounts of volume converge SWIF-ly to spheres.

Given two compact oriented Riemannian manifolds, M_i, possibly with boundary:

iff there is an orientation preserving isometry from M₁ to M₂. If M_i converge in the Gromov-Hausdorff sense to a metric space Y then a subsequence of the M_i converge SWIF-ly to an integral current space contained in Y but not necessarily equal to Y. For example, the GH limit of a sequence of spheres with a long thin neck pinch is a pair of spheres with a line segment running between them while the SWIF limit is just the pair of spheres. The GH limit of a sequence of thinner and thinner tori is a circle but the flat limit is the 0 space. In the setting with nonnegative Ricci curvature and a uniform lower bound on volume, the GH and SWIF limits agree. If a sequence of manifolds converge in the Lipschitz sense to a limit Lipschitz manifold then the SWIF limit exists and has the same limit.

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