Rolling Stiefel manifolds equipped with $\alpha$-metrics

TL;DR

This paper investigates the intrinsic and extrinsic rolling of Stiefel manifolds with $\alpha$-metrics, deriving explicit ODEs and solutions, and exploring the limitations of generalizing rolling to normal naturally reductive homogeneous spaces.

Contribution

It introduces a comprehensive framework for rolling Stiefel manifolds with $\alpha$-metrics, including explicit equations and solutions, and discusses the generalization challenges to other homogeneous spaces.

Findings

Derived explicit time-variant ODEs for rolling

Obtained explicit solutions for specific development curves

Highlighted limitations in extending intrinsic rolling to certain spaces

Abstract

We discuss the rolling, without slip and without twist, of Stiefel manifolds equipped with $\alpha$-metrics, from an intrinsic and an extrinsic point of view. We, however, start with a more general perspective, namely by investigating intrinsic rolling of normal naturally reductive homogeneous spaces. This gives evidence why a seemingly straightforward generalization of intrinsic rolling of symmetric spaces to normal naturally reductive homogeneous spaces is not possible, in general. For a given control curve, we derive a system of explicit time-variant ODEs whose solutions describe the desired rolling. These findings are applied to obtain the intrinsic rolling of Stiefel manifolds, which is then extended to an extrinsic one. Moreover, explicit solutions of the kinematic equations are obtained provided that the development curve is the projection of a not necessarily horizontal one-parameter subgroup. In addition, our results are put into perspective with examples of rolling Stiefel manifolds known from the literature.