Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

In this paper, we deal with the inverse problem of determining simple metrics on a compact Riemannian manifold from boundary measurements. We take this information in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension \begin{document}$ n\geq 2 $\end{document} that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the simple metric (up to an admissible set). We also prove a Hölder-type stability estimate by the construction of geometrical optics solutions of the Schrödinger equation and the direct use of the invertibility of the geodesical X-ray transform.