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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 29, 2015

Isometric Embeddings of Pro-Euclidean Spaces

  • Barry Minemyer


In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].


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Received: 2015-5-27
Accepted: 2015-10-9
Published Online: 2015-10-29

© 2015 Barry Minemyer

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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