The Carnot-Carathéodory distance on $2$-step groups

Abstract

Combining Varadhan's formula, Loewner's theorem with the method of stationary phase, we study the exact formula of the Carnot-Carathéodory distance on $2$-step groups. The method is also adapted to determine all normal geodesics from the identity element to any given point (up to a set of measure zero). As an application, we characterize the squared sub-Riemannian distance as well as the cut locus on generalized Heisenberg-type groups and on star graphs respectively. Furthermore, the long-standing open problem of Gaveau-Brockett is completely solved in the case of $N_{3,2}$, the free Carnot group of step two and 3 generators.

52 pages, 3 figures. The main results of this paper, combined with the main results in arXiv:2102.09860 and recent progress, should be included in a single forthcoming paper