Rotations, Shears, and Two-Generation on Spheres

For every round sphere $S^n$, $n\ge 2$, we exhibit two explicit divergence-free vector fields whose flows generate a $C^{\mathrm{\infty }}$-dense subgroup of ${\text{SDiff}}_0(S^n)$. One generator is an elementary planar rotation, $$A=R_{12},$$ while the other combines a linearly modulated planar rotation with a chain of adjacent planar rotations: $$B=x_3{R}_{12}+\sum _{k=2}^n{R}_{k,k+1}\mathrm{.}$$ Thus two one-parameter groups suffice, and two is the smallest possible number. The proof separates naturally into two steps. We first show that the $n$ adjacent rotations $$R_{12},R_{23},\dots ,R_{n,n+1}$$ together with the shear $x_3{R}_{12}$ generate, through their flows, a $C^{\mathrm{\infty }}$-dense subgroup of ${\text{SDiff}}_0(S^n)$. The infinitesimal argument is organized by transverse vector spherical harmonics. For $n\ge 4$ the relevant harmonic spaces are irreducible $SO(n+1)$-modules; on $S^2$ the argument reduces to the Hamiltonian/Poisson formalism; and on $S^3$ the transverse harmonic spaces split into two curl eigenspaces. We then show that all $n+1$ of these vector fields lie in the Lie algebra generated by $A$ and $B$. A Casimir operator separates the quadratic shear from the linear rotational terms, after which the remaining rotations can be recovered successively. Finally, standard approximation formulas for flows convert the Lie-algebraic generation result into $C^{\mathrm{\infty }}$ density in ${\text{SDiff}}_0(S^n)$.