Rolling against a sphere: The non transitive case

TL;DR

This paper investigates the controllability of a rolling system of a Riemannian manifold on a sphere, linking it to holonomy groups and constructing special geometric structures in specific cases.

Contribution

It establishes a connection between the controllability of the rolling system and the holonomy of a related vector bundle, and explicitly constructs Sasakian and 3-Sasakian structures for particular holonomy groups.

Findings

Controllability linked to the holonomy of a vector bundle connection.

Explicit construction of Sasakian structures when holonomy is unitary.

Explicit construction of 3-Sasakian structures when holonomy is symplectic.

Abstract

We study the control system of a Riemannian manifold $M$ of dimension $n$ rolling on the sphere $S^n$. The controllability of this system is described in terms of the holonomy of a vector bundle connection which, we prove, is isomorphic to the Riemannian holonomy group of the cone $C(M)$ of $M$. Using Berger's list, we reduce the possible holonomies to a few families. In particular, we focus on the cases where the holonomy is the unitary and the symplectic group. In the first case, using the rolling formalism, we construct explicitly a Sasakian structure on $M$; and in the second case, we construct a 3-Sasakian structure on $M$.