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 C 1 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 Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.
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Le Donne, E. Lipschitz and path isometric embeddings of metric spaces. Geom Dedicata 166, 47–66 (2013). https://doi.org/10.1007/s10711-012-9785-2
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DOI: https://doi.org/10.1007/s10711-012-9785-2