A reduced parallel transport equation on Lie Groups with a left-invariant metric

TL;DR

This paper derives a parallel transport equation on Lie groups with left-invariant metrics, demonstrating its application on SE(3) for stable, efficient numerical transport, implemented via the geomstats package.

Contribution

It introduces a new derivation of the parallel transport equation on Lie groups with left-invariant metrics and demonstrates its practical application and implementation.

Findings

Stable and efficient parallel transport implementation on SE(3)

Comparison shows advantages over pole ladder algorithm

Open-source implementation available online

Abstract

This paper presents a derivation of the parallel transport equation expressed in the Lie algebra of a Lie group endowed with a left-invariant metric.The use of this equation is exemplified on the group of rigid body motions SE(3), using basic numerical integration schemes, and compared to the pole ladder algorithm. This results in a stable and efficient implementation of parallel transport. The implementation leverages the python package geomstats and is available online.