The anisotropic Calderón problem on 3-dimensional conformally Stäckel manifolds

Abstract

Conformally Stäckel manifolds can be characterized as the class of $n$-dimensional pseudo-Riemannian manifolds $(M,G)$ on which the Hamilton–Jacobi equation

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 $n-1$ 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.