Global uniqueness in an inverse problem for time fractional diffusion equations

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Given (M,g), a compact connected Riemannian manifold of dimension d ≥ 2, with boundary ∂ M, we consider an initial boundary value problem for a fractional diffusion equation on (0,T) × M, T>0, with time-fractional Caputo derivative of order α∈ (0,1) ∪ (1,2). We prove uniqueness in the inverse problem of determining the smooth manifold (M,g) (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of ∂ M at fixed time. In the "flat" case where M is a compact subset of Rᵈ, two out the three coefficients ρ (weight), a (conductivity) and q (potential) appearing in the equation ρ∂ₜαu-div(a ∇ u)+ q u=0 on (0,T)× Ω are recovered simultaneously.