Abstract
Abstract. We consider the inverse problem to determine a smooth compact Riemannian manifold [Formula: see text] from a restriction of the source-to-solution operator, [Formula: see text], for the wave equation on the manifold. Here, [Formula: see text] and [Formula: see text] are open sets on [Formula: see text], and [Formula: see text] represents the measurements of waves produced by smooth sources supported on [Formula: see text] and observed on [Formula: see text]. We emphasize that [Formula: see text] and [Formula: see text] could be disjoint. We demonstrate that [Formula: see text] determines the manifold [Formula: see text] uniquely under the following spectral bound condition for the set [Formula: see text]: There exists a constant [Formula: see text] such that any normalized eigenfunction [Formula: see text] of the Laplace–Beltrami operator on [Formula: see text] satisfies [Formula: see text]. We note that, for the Anosov surface, this spectral bound condition is fulfilled for any nonempty open subset [Formula: see text]. Moreover, we solve the analogue of this problem for the heat equation by showing that the source-to-solution maps for the heat and wave equations determine each other.