Controllability of rolling without twisting or slipping in higher dimensions

We describe how the dynamical system of rolling two $n$-dimensional connected, oriented Riemannian manifolds $M$ and $\hat M$ without twisting or slipping, can be lifted to a nonholonomic system of elements in the product of the oriented orthonormal frame bundles belonging to the manifolds. By considering the lifted problem and using properties of the elements in the respective principal Ehresmann connections, we obtain sufficient conditions for the local controllability of the system in terms of the curvature tensors and the sectional curvatures of the manifolds involved. We also give some results for the particular cases when $M$ and $\hat M$ are locally symmetric or complete.

💡 Research Summary

The paper investigates the controllability of the non‑holonomic system that arises when two n‑dimensional connected, oriented Riemannian manifolds (M) and (\widehat M) roll on each other without twisting or slipping. The authors first formalize the configuration space as the bundle
(Q = { q : T_m M \to T_{\widehat m}\widehat M \mid q \text{ is an orientation‑preserving linear isometry} }),
which can be identified with the product of the oriented orthonormal frame bundles (F(M)\times F(\widehat M)) modulo the diagonal action of (SO(n)). A rolling motion is an absolutely continuous curve (q(t)) in (Q) that satisfies two constraints: the no‑slip condition (\dot{\widehat m}(t)=q(t)\dot m(t)) and the no‑twist condition that a vector field parallel along (M) is carried to a parallel field along (\widehat M). These constraints define an n‑dimensional distribution (D\subset TQ), called the rolling distribution.

To analyse (D), the authors lift the problem to the frame bundles and use the Levi‑Civita connection on each manifold. The Ehresmann connection associated with the Levi‑Civita connection yields a horizontal distribution (E) on each frame bundle, together with the canonical solder form (\theta) and the connection 1‑form (\omega). The Cartan structure equations
(d\theta^j + \omega^j_i\wedge\theta^i = 0) and
(d\omega^j_i + \omega^j_k\wedge\omega^k_i = \Omega^j_i)
relate the curvature 2‑form (\Omega) to the Riemann curvature tensor (R).

In local coordinates obtained from sections of the frame bundles, the authors write explicit generators of (D): \