Fractional anisotropic Calderón problem on complete Riemannian manifolds

We prove that the metric tensor $g$ of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator associated with a fractional power of the Laplace–Beltrami operator ${\mathrm{\Delta }}_g$. Our result holds under the condition that the metric tensor $g$ is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in [A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021); arXiv:2112.03480].