Rolling Connections

This is an in-progress set of notes describing the often-neglected geometrical, dynamical interpretation of a connection on a manifold. The essential idea is that, while the Levi-Civita connection is associated to the sliding action of parallel transport, it is perhaps more natural to think of a tangent space to a surface being rolled around as we move from point to point. The difference manifests in the corresponding structure groups: the orthogonal group so(n) for the Levi-Civita connection, versus the Euclidean group aso(n) for the rolling connection. These notes aim to describe the relationship between these two approaches. 1 A Geometric Model for Rolling Let f : M −→ R define an immersed surface. We aim to formalize the notion of “rolling a plane along a curve in f(M)”. I claim that the following is a reasonable definition of “rolling a plane along a surface”: Definition 1.1. Begin with a plane tangent to the surface S at a point p. Rolling this plane from p to q along a curve γ yields an isometry ργ : TpS −→ TqS. This rolling map is uniquely characterized by the requirement that points in any plane TxS move in the normal direction under an infinitesimal rolling. Equivalently, we ask that all points in the plane move as little as possible as we roll.