Exact solutions of diffusion equation on sphere

We examine the diffusion equation on the sphere. In this sense, we answer the question of the symmetry classification. We provide the algebra of symmetry and build the optimal system of Lie subalgebras. We prove for one-dimensional optimal systems of Eq. (1.4), space is expanding Ricci solitons. Reductions of similarities related to subalgebras are classified, and some exact invariant solutions of the diffusion equation on the sphere are presented.