Lipschitz and path isometric embeddings of metric spaces

TL;DR

This paper demonstrates that sub-Riemannian manifolds can be embedded into Euclidean space while preserving curve lengths, extending Nash's theorem, and shows finite-dimensional metric spaces can be Lipschitz embedded, highlighting limitations for more general spaces.

Contribution

It extends Nash's $C^1$ embedding theorem to sub-Riemannian manifolds and establishes Lipschitz embeddings for finite packing dimension metric spaces.

Findings

Sub-Riemannian manifolds can be embedded preserving curve lengths.

Finite packing dimension metric spaces admit Lipschitz embeddings.

The result does not extend to all metric spaces, such as Finsler non-Riemannian manifolds.

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 packing dimension can be embedded in some Euclidean space via a Lipschitz map.