  In [14] Duca and Nersesyan proved a small-time controllability property of
nonlinear Schr\"odinger equations on a d-dimensional torus $\mathbb{T}^d$. In
this paper we study a similar property, in the linear setting, starting from a
closed Riemannian manifold. We then focus on the 2-dimensional sphere $S^2$,
which models the bilinear control of a rotating linear top: as a corollary, we
obtain the approximate controllability in arbitrarily small times among
particular eigenfunctions of the Laplacian of $S^2$.
