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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.