  Given a fixed $\alpha \in (0,1)$, we study the inverse problem of recovering
the isometry class of a smooth closed and connected Riemannian manifold
$(M,g)$, given the knowledge of a source-to-solution map for the fractional
Laplace equation $(-\Delta_g)^\alpha u=f$ on the manifold subject to an
arbitrarily small observation region $\mathcal O$ where sources can be placed
and solutions can be measured. This can be viewed as a non-local analogue of
the well known anisotropic Calder\'{o}n problem that is concerned with the
limiting case $\alpha=1$. While the latter problem is widely open in dimensions
three and higher, we solve the non-local problem in broad geometric generality,
assuming only a local property on the a priori known observation region
$\mathcal O$ while making no geometric assumptions on the inaccessible region
of the manifold, namely $M\setminus \mathcal O$. Our proof is based on
discovering a hidden connection to a variant of Carlson's theorem in complex
analysis that allows us to reduce the non-local inverse problem to the Gel'fand
inverse spectral problem.
