Isometric Embeddings of Pro-Euclidean Spaces
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...
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| Format: | Article |
| Language: | English |
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De Gruyter
2015-10-01
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| Series: | Analysis and Geometry in Metric Spaces |
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| Online Access: | https://doi.org/10.1515/agms-2015-0019 |
| Summary: | 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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| ISSN: | 2299-3274 |