{
  "abstract": "For every round sphere $S^n$, $n\\ge2$, we exhibit two explicit divergence-free vector fields whose flows generate a $C^\\infty$-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}+\\sum_{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},\\ldots,R_{n,n+1} \\] together with the shear $x_3R_{12}$ generate, through their flows, a $C^\\infty$-dense subgroup of $\\SDiff_0(S^n)$. The infinitesimal argument is organized by transverse vector spherical harmonics. For $n\\ge4$ 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 $C^\\infty$ density in $\\SDiff_0(S^n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27507",
  "author_total": 1,
  "authors": [
    {
      "name": "Dennis DeTurck"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6kavz48rfqfxe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27507",
  "fulltext_url": null,
  "id": "6kavz48rfqfxe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27507",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7223144839",
  "pdf_url": "https://arxiv.org/pdf/2610.27507",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6kavz48rfqfxe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27507",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27507"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rotations, Shears, and Two-Generation on Spheres",
  "title_text": "Rotations, Shears, and Two-Generation on Spheres",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/6kavz48rfqfxe",
  "venue": "arXiv",
  "versions": []
}