  Let $(\Omega^3,g)$ be a compact smooth Riemannian manifold with smooth
boundary and suppose that $U$ is a an open set in $\Omega$ such that $g|_U$ is
the Euclidean metric. Let $\Gamma= \overline{U} \cap \partial \Omega$ be
connected and suppose that $U$ is the convex hull of $\Gamma$. We will study
the uniqueness of an unknown potential for the Schr\"{o}dinger operator $
-\triangle_g + q $ from the associated Dirichlet to Neumann map, $\Lambda_q$.
We will prove that if the potential $q$ is a priori explicitly known in $U^c$,
then one can uniquely reconstruct $q$ over the convex hull of $\Gamma$ from
$\Lambda_q$. We will also outline a reconstruction algorithm. More generally we
will discuss the cases where $\Gamma$ is not connected or $g|_{U}$ is
conformally transversally anisotropic and derive the analogous result.
