Fractional anisotropic Calder\'on 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. This later is associated to a fractional power of the Laplace-Belrami operator $\Delta_g$. 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 \cite{FGKU}