Rolling Maps in a Riemannian Framework

Abstract

We study rolling of one connected submanifold upon another connected submanifold, both isometrically embedded into one and the same Riemannian manifold. We generalise the definition of rolling wellknown from the literature. By this new definition, the Euclidean group of motions is replaced by the connected component of the Lie group of isometries of the embedding manifold. We show that rolling in this more general situation is again unique. We prove a theorem that enables us to learn how to roll non-Euclidean manifolds that result from deformations of Euclidean submanifolds from the knowledge of the kinematic equations of rolling the Euclidean submanifolds. Taking into account that the ellipsoid is a deformed sphere, we apply the above mentioned theorem and the kinematic equations for the rolling sphere to derive the kinematic equations for rolling the ellipsoid. This example might serve as a motivation to roll other manifolds as well.