The anisotropic Calderón problem on 3-dimensional conformally Stäckel manifolds (Q2078944)

Summary: Conformally Stäckel manifolds can be characterized as the class of \(n\)-dimensional pseudo-Riemannian manifolds \((M,G)\) on which the Hamilton-Jacobi equation \[ G(\nabla u, \nabla u) = 0 \] for null geodesics and the Laplace equation \(-\Delta_G \, \psi = 0\) are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of \(n-1\) \textit{commuting conformal symmetry operators} for the Laplace-Beltrami operator \(\Delta_G\). In this paper, we solve the anisotropic Calderón problem on compact \(3\)-dimensional Riemannian manifolds with boundary which are conformally Stäckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to diffeomorphims that preserve the boundary.