Sub-Riemannian geodesics on nested principal bundles

TL;DR

This paper explores the geometric properties of sub-Riemannian geodesics on nested principal bundles, utilizing Hamiltonian formalism and providing examples like the Gromoll-Meyer sphere and twistor space.

Contribution

It introduces a Hamiltonian approach to analyze geodesics in non-holonomic systems related by Lie group actions, with new geometric insights and examples.

Findings

Hamiltonian formalism effectively describes geodesics

Identifies geometric structures on exotic spheres and twistor space

Provides explicit parametric forms of geodesics

Abstract

We study the interplay between geodesics on two non-holono\-mic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.