Rolling Reductive Homogeneous Spaces

TL;DR

This paper studies the geometric concept of rolling reductive homogeneous spaces without slip or twist, providing an intrinsic framework and explicit differential equations for describing such motions, with applications to Lie groups and Stiefel manifolds.

Contribution

It introduces an intrinsic approach to rolling reductive homogeneous spaces using principal bundles and connections, deriving explicit ODEs for the rolling motion.

Findings

Derived explicit kinematic equations for rolling without slip or twist.

Provided solutions for initial value problems in specific cases.

Applied the framework to Lie groups and Stiefel manifolds.

Abstract

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space $G / H$ with reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$, we consider rollings of $\mathfrak{m}$ over $G / H$ without slip and without twist, where $G / H$ is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space $Q$ which is tangent to a certain distribution. By considering a $H$-principal fiber bundle $\overline{\pi} \colon \overline{Q} \to Q$ over the configuration space equipped with a suitable principal connection, rollings of $\mathfrak{m}$ over $G / H$ can be expressed in terms of horizontally lifted curves on $\overline{Q}$. The total space of $\overline{\pi} \colon \overline{Q} \to Q$ is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of $\mathfrak{m}$ over $G / H$ as solutions of an explicit, time-variant ordinary differential equation (ODE) on $\overline{Q}$, the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in $G / H$ is the projection of a one-parameter subgroup in $G$. Lie groups and Stiefel manifolds are discussed as examples.