Rolling Maps in Riemannian Manifolds

Abstract

We study rolling of one submanifold upon another submanifold, both isometrically embedded in a Riemannian manifold.We generalise the definition of rolling in Sharpe (1997). In this new definition, the Euclidean group of motions is replaced by the Lie group of orientation preserving isometries. We show that rolling in this general situation is 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 these 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 serves as a motivation to roll other manifolds.