Calculus of Variations and Geometric Measure Theory
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E. Le Donne

Lipschitz and path isometric embeddings of metric spaces

created by ledonne on 17 May 2010



Inserted: 17 may 2010

Year: 2010


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 C1 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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