Algebraic derivation of some theorems concerning contact structures on the 3-sphere

Two theorems involving curl eigenfields on the 3–sphere are obtained using angular momentum theory. Spinor hyperspherical harmonics are shown to form an explicit, convenient basis. In particular, a spin–one vector calculus is reviewed. An easy proof of the vanishing of `odd’ eigenfields is given and related to the sign change of fermionic spinors under 2$\pi$ rotations. The theorem that curl eigenfields with constant norm have to be proportional to a fundamental eigenfield (Hopf field) is also rapidly obtained. Attention is drawn to the relevance of early work of Schrödinger on Maxwell theory in an expanding universe.