The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds

We consider the fractional anisotropic Calderón problem for the nonlocal parabolic equation $({\mathrm{\partial }}_t-{\mathrm{\Delta }}_g{)}^{s}u=f$ ($0<s<1$) on closed Riemannian manifolds. More concretely, we can determine the Riemannian manifold $(M,g)$ up to isometry by using the local source-to-solution map in an arbitrarily small open cylinder in the spacetime domain. This can be regarded as a nonlocal analog of the anisotropic Calderón problem in the parabolic setting. We also study several useful properties for nonlocal parabolic operators by using comprehensive spectrum analysis with semigroup theory.