  We consider, on a trivial vector bundle over a Riemannian manifold with
boundary, the inverse problem of uniquely recovering time- and space-dependent
coefficients of the dynamic, vector-valued Schr\"odinger equation from the
knowledge of the Dirichlet-to-Neumann map. We show that the D-to-N map uniquely
determines both the connection form and the potential appearing in the
Schr\"odinger equation, under the assumption that the manifold is either a)
two-dimensional and simple, or b) of higher dimension with strictly convex
boundary and admits a smooth, strictly convex function.
