A Global Coordinate-Free Approach to Invariant Contraction on Homogeneous Manifolds

TL;DR

This paper introduces a global, coordinate-free method for analyzing contraction on homogeneous manifolds using invariant metrics and Lie group structures, avoiding local charts and applying to control problems.

Contribution

It provides a novel global condition for contraction on homogeneous spaces using invariant metrics and matrix measures, with applications to control and geometry.

Findings

No great circle can be contained in a contraction region on a sphere.

The method characterizes contraction using matrix measures on Lie groups.

Applied to attitude control, it computes reachable sets effectively.

Abstract

In this work, we provide a global condition for contraction with respect to an invariant Riemannian metric on reductive homogeneous spaces. Using left-invariant frames, vector fields on the manifold are horizontally lifted to the ambient Lie group, where the Levi-Civita connection is globally characterized as a real matrix multiplication. By linearizing in these left-invariant frames, we characterize contraction using matrix measures on real square matrices, avoiding the use of local charts. Applying this global condition, we provide a necessary condition for a prescribed subset of the manifold to possibly admit a contracting system, which accounts for the underlying geometry of the invariant metric. Applied to the sphere, this condition implies that no great circle can be contained in a contraction region. Finally, we apply our results to compute reachable sets for an attitude control problem.