We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary from a restriction of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here and are open sets in and the restriction corresponds to the case where the Dirichlet data is supported on and the Neumann data is measured on . In the novel case where , we show that determines the manifold uniquely, assuming that the wave equation is exactly controllable from the set of sources . Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is,
where are the Dirichlet eigenvalues and where is an orthonormal basis of the corresponding eigenfunctions.
"Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets." Duke Math. J. 163 (6) 1071 - 1103, 15 April 2014. https://doi.org/10.1215/00127094-2649534