Koranyi Metric On Heisenberg Group, vs Carnot Caratheodory Distance 

Construction of the sub-Riemannian distance on the Heisenberg group was involved and required notions from differential geometry. Fortunately, there is simpler way to define a bi-Lipschitz equivalent distance that only uses the group multiplication. In particular it does not require any knowledge of paths, Riemannian metric, or anything else about manifolds.