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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.
==Intrinsic flat distance==
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.〔"Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces" by Sormani and Wenger, ''Journal of Differential Geometry'', Vol 87, 2011, 117–199〕 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. 〔
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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