Background: A scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any kind of tensor field. Which is the basic idea/intuition behind the extension from scalar fields to tensor ones?
This concept 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?
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 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.
