Lipschitz and path isometric embeddings of metric spaces
Metric Geometry
2016-02-17 v2 Differential Geometry
Abstract
We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite packing dimension can be embedded in some Euclidean space via a Lipschitz map.
Keywords
Cite
@article{arxiv.1005.1623,
title = {Lipschitz and path isometric embeddings of metric spaces},
author = {Enrico Le Donne},
journal= {arXiv preprint arXiv:1005.1623},
year = {2016}
}
Comments
19 pages, in Theorem 2.9 the packing dimension replaces the Hausdorff dimension, final version