  We study an analog of the anisotropic Calder\'on problem for fractional
Schr\"odinger operators $(-\Delta_g)^\alpha + V$ with $\alpha \in (0,1)$ on
closed Riemannian manifolds of dimensions two and higher. We prove that the
knowledge of a Cauchy data set of solutions of the fractional Schr\"odinger
equation, given on an open nonempty a priori known subset of the manifold
determines both the Riemannian manifold up to an isometry and the potential up
to the corresponding gauge transformation, under certain geometric assumptions
on the manifold as well as the observation set. Our method of proof is based
on: (i) studying a new variant of the Gel'fand inverse spectral problem without
the normalization assumption on the energy of eigenfunctions, and (ii) the
discovery of an entanglement principle for nonlocal equations involving two or
more compactly supported functions. Our solution to (i) makes connections to
antipodal sets as well as local control for eigenfunctions and quantum chaos,
while (ii) requires sharp interpolation results for holomorphic functions. We
believe that both of these results can find applications in other areas of
inverse problems.
