Sub-Riemannian geodesics on the 3-D sphere

TL;DR

This paper investigates sub-Riemannian geodesics on the 3-dimensional sphere, modeled as the Lie group SU(2), using Hamiltonian methods to analyze the geometric structure and geodesic equations.

Contribution

It provides a detailed analysis of sub-Riemannian geodesics on the sphere viewed as a Lie group, employing Hamiltonian formalism to solve the geodesic equations.

Findings

Explicit solutions for sub-Riemannian geodesics on S^3

Characterization of geodesic optimality and cut loci

Application of Hamiltonian formalism to non-commutative Lie groups

Abstract

The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.