Geodesics and shortest arcs of some sub-Riemannian metrics on the Lie groups $SU(2)\times\mathbb{R}$ and $SO(3)\times\mathbb{R}$ with three-dimensional generating distributions

TL;DR

This paper characterizes geodesics, shortest paths, and conjugate loci in specific sub-Riemannian geometries on certain Lie groups, providing explicit formulas and geometric insights.

Contribution

It explicitly determines geodesics, cut loci, and distances for left-invariant sub-Riemannian metrics on $SU(2)\times\mathbb{R}$ and $SO(3)\times\mathbb{R}$, advancing understanding of these structures.

Findings

Explicit formulas for geodesics and shortest arcs.

Descriptions of cut and conjugate loci.

Distances between arbitrary elements computed.

Abstract

We find geodesics, shortest arcs, cut loci, first conjugate loci, distances between arbitrary elements for some left-invariant sub-Riemannian metrics on the Lie groups $SU(2)\times\mathbb{R}$ and $SO(3)\times\mathbb{R}$.