In this talk we deal with embedding results for geodesic metric spaces that are homeomorphic to manifolds.

I will give an introduction to the subRiemannian Heisenberg structure on R^3. Such a metric space is not biLipschitz embeddable into any Euclidean space. However, it can be embedded into R^4 preserving the length of all the curves.

If time permits, I will discuss more general metrics.