Riemannian and sub-Riemannian structures on a cotangent bundle of Heisenberg group

Abstract

In this paper we give a classification of left invariant sub-Riemannian structures on cotangent bundle of $2n+1$ dimensional Heisenberg group ${\rm T}^*H_{2n+1}$. We show that the sub Riemannian metric is tamed by the corresponding Riemannian metric on ${\rm T}^*H_{2n+1}$. We also describe Riemannian and sub-Riemannian geodesics on ${\rm T}^*H_{2n+1}$.