The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces
C. Sormani, S. Wenger
备注：
(78 pages, 18 figures) Changes from the 2008 version: added examples appendix, added 3.1, fixed 3.2, fixed typos. Changes from v2: minor changes, finalized for submission. Changes from v3: Accepted by JDG, introduction revised, new citations, minor changes, reformatted for JDG, Changes now: minor typos
发表信息：
Journal of Differential Geometry, Vol 87, 117-199 (2011)
Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented $m$ dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim's extension of Federer-Fleming's notion of integral currents to arbitrary metric spaces. We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably $\mathcal{H}^m$ rectifiable spaces and present numerous examples.
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证明了任何局部共形平坦黎曼流形，只要其Jacobi算子沿每条测地线具有常特征值，则该流形具有常主Ricci曲率。本文给出了4，5，6，7，8维具有常Ricci特征值的局部共形平坦黎曼流形的局部分类（直到等距）.证明了任意具有常主Ricci曲率的n维$(4\leq n \leq 8)$局部共形平坦黎曼流形是黎曼局部对称空间.
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