Given a connected compact Riemannian manifold (M,g) without boundary, dimM≥2, we consider a space–time fractional diffusion equation with an interior source that is supported on an open subset V of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order α∈(0,1], and the space fractional part by (−Δg)β, where β∈(0,1] and Δg is the Laplace–Beltrami operator on the manifold. The case α=β=1, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on V determines the manifold up to a Riemannian isometry.