Background: A scalar field on the unit sphere can be expanded in spherical harmonics, see this. This seems related to the multipole expansion for vector fields, but it's not exactly the same concept. Moreover, I know that something called vector spherical harmonics exists, and I am quite sure that this is the concept I have in mind for the case of vectors (see the example below and this question). For the rank-2 case, I do not understand if the notion of spherical tensor provides a solution to the question.
Question: Given my doubts reported above, the definition of Wikipedia of Vector spherical harmonics seems quite arbitrary to me: I am not able to grasp the "rationale" behind it. Which is the general concept we have to use when moving from scalar fields to tensor ones? What is the basic idea/intuition behind the extension from scalar fields to tensor ones?
Example: Imagine having a field theory defined on the sphere (the base manifold is the unit sphere) and that we want to expand the energy-momentum tensor in series: we should use a proper basis of orthogonal "matrix fields" on the sphere. The same if we have a fluid flowing on the surface of a sphere: we need a set of vector fields that live on the tangent bundle of the sphere that are "complete" and "orthogonal". I expect that this set will be "countable" (like usual spherical harmonics for scalars) because the sphere is a compact manifold.
