On the Controllability of the Rolling Problem onto the Hyperbolic $n$-space

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In the present paper, we study the controllability of the control system associated with rolling without slipping or spinning of a Riemannian manifold $(M,g)$ onto the hyperbolic $n$-space ${\mathbb H}^n$. Our main result states that the system is completely controllable if and only if $(M,g)$ is not isometric to a warped product of a special form, in analogy to the classical de Rham decomposition theorem for Riemannian manifolds. The proof is based on the observations that the controllability issue in this case reduces to determining whether $(M,g)$ admits a reducible action of a hyperbolic analog of the holonomy group and a well-known fact about connected subgroups of ${O}(n,1)$ acting irreducibly on the Lorentzian space ${\mathbb R}^{n,1}$.