The intrinsic flat distance between riemannian manifolds and other integral current spaces

Abstract

Inspired by the Gromov-Hausdorff distance, we define a new notion called the intrinsic flat distance between oriented m di- mensional 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 iso-metric 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 ori-ented 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 includ- ing the fact that they are countably Hm rectifiable spaces and present numerous examples. © 2011 Journal of Differential Geometry. © 2011 Applied Probability Trust.