Rotations, Shears, and Two-Generation on Spheres
Abstract
For every round sphere Sn, n≥2, we exhibit two explicit divergence-free vector fields whose flows generate a C∞-dense subgroup of SDiff0(Sn). One generator is an elementary planar rotation, A=R12, while the other combines a linearly modulated planar rotation with a chain of adjacent planar rotations: B=x3R12+∑k=2n Rk,k+1. 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 R12,R23,…,Rn,n+1 together with the shear x3R12 generate, through their flows, a C∞-dense subgroup of SDiff0(Sn). The infinitesimal argument is organized by transverse vector spherical harmonics. For n≥4 the relevant harmonic spaces are irreducible SO(n+1)-modules; on S2 the argument reduces to the Hamiltonian/Poisson formalism; and on S3 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∞ density in SDiff0(Sn).
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BibTeX
@misc{deturck2026rotations,
title = {{Rotations, Shears, and Two-Generation on Spheres}},
author = {Dennis DeTurck},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27507},
eprint = {2610.27507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27507},
}
RIS
TY - UNPB
TI - Rotations, Shears, and Two-Generation on Spheres
AU - Dennis DeTurck
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27507
UR - https://arxiv.org/abs/2610.27507
AB - For every round sphere S^n, n≥2, we exhibit two explicit divergence-free vector fields whose flows generate a C^∞-dense subgroup of 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_3R_12+∑_k=2^n R_k,k+1. 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,…,R_n,n+1 together with the shear x_3R_12 generate, through their flows, a C^∞-dense subgroup of SDiff_0(S^n). The infinitesimal argument is organized by transverse vector spherical harmonics. For n≥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^∞ density in SDiff_0(S^n).
ER -