# Relationship between Carnot-Caratheory Distance and Levi-Civita Connection

Suppose that $$G\cong H\times K$$ is a nilpotent Lie group, where $$H,K$$ are Lie-Subgroups for which $$H$$ and $$K$$ are Lie-Subgroups such that

• $$H$$ is Commutative
• $$K$$ is Compact

Then $$G$$ admits a bi-invariant metric then the (Levi-Civita) Riemannian exponential and Lie Group exponential maps coincide. Since the Riemannian structure induces a natural distance structure, then the metric can be expressed using the Lie group structure also.

However, since $$G$$ is nilpotent then the Carnot-Caratheory distance also is well-defined.

Question My question, is how are these two structures related? Since they both seem to be unique and induced by $$G$$'s nice Lie Group geometry.