SubRiemannian geodesics for Carnot groups of step 3

TL;DR

This paper proves that all subRiemannian geodesics in Carnot groups of step 3 are normal, using a reduction argument and the Goh condition related to the group's graded structure.

Contribution

It establishes that in step 3 Carnot groups, all subRiemannian geodesics are normal, extending understanding of geodesic regularity in these geometric structures.

Findings

All subRiemannian geodesics are normal in step 3 Carnot groups

The Goh condition is derived from the end-point mapping reformulation

The proof relies on the graded structure of Carnot groups

Abstract

In Carnot groups of step 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.