Lipschitz and Path Isometric Embeddings of Metric Spaces

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 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. A map f : X → Y between two metric spaces X and Y is called a path isometry (probably a better name is a length preserving map) if, for all curves γ in X, one has LY (f ◦ γ) = LX(γ). Here LX and LY denote the lengths of the parametrized curves with respect to the distances of X and of Y , respectively. From the definition, a path isometry is not necessarily injective. The first aim of the following paper is to show that any sub-Riemannian manifold can be mapped into some Euclidean space via a path isometric embedding, i.e., a topological embedding that is also a path isometry. Sub-Riemannian manifolds are metric spaces when endowed with the Carnot-Carathéodory distance dCC associated to the fixed sub-bundle and Riemannian structure. For an introduction to sub-Riemannian geometry see [Bel96, Gro99, BBI01, Mon02, Bul02, LD10]. An equivalent statement of our first result is the following. Denote by E the k-dimensional Euclidean space. Our result says that, for every sub-Riemannian manifold (M,dCC), there exists a path connected subset Σ ⊂ E, for some k ∈ N, such that, when Σ is endowed with the path distance dΣ induced by the Euclidean length, then the metric space (Σ, dΣ) is isometric to (M,dCC). After such a fact one should wonder which are the length metric spaces obtained as subsets of E with induced length structure. We show that any distance on R that is coming from a norm but not from a scalar product cannot be obtained in such a way. We conclude the paper by showing another positive result for general metric spaces: every metric space of finite Hausdorff dimension has a Lipschitz embedding into some E. Date: May 17, 2010. 1