*Preprint*

**Inserted:** 17 may 2010

**Year:** 2010

**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.

**Keywords:**
Heisenberg group, path isometry, Nash embedding

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