Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

TL;DR

This paper explores the properties of geodesics in sub-Riemannian and almost-Riemannian geometries on SO(3) and S^2, providing explicit parameterizations, optimality conditions, and characterizations of periodic geodesics.

Contribution

It offers explicit parameterizations of geodesics on SO(3) and S^2 and analyzes their optimality and periodicity using symmetry and exponential map techniques.

Findings

Explicit parameterization of geodesics on SO(3)

Necessary optimality conditions for geodesics

Upper bounds on cut time and periodic geodesics on SO(3)

Abstract

In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on $S^2$. These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian geodesics on SO(3) and use it to get a parameterization of almost-Riemannian geodesics on $S^2$. We use symmetries of the exponential map to obtain some necessary optimality conditions. We present some upper bounds on the cut time in both cases and describe periodic geodesics on SO(3).