{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0274",
  "title": "Parallel Nilpotent Endomorphisms Without Parallel Null Vectors",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct a closed complete flat pseudo-Riemannian manifold of dimension 16 and signature (8,8) carrying a nonzero parallel self-adjoint endomorphism N with N² = 0, although neither the manifold nor any connected double cover admits a nonzero parallel vector field. This gives a negative answer, as stated, to Question 12.0.3 in the Burns–Matveev list of open problems about geodesics. The construction starts with a compact flat Riemannian manifold whose holonomy has odd order and zero fixed space, doubles its Euclidean representation, equips the double with the split metric, and sets N(u,v) = (0,u). Odd-order holonomy survives every index-two subgroup. An explicit input is the free (ℤ/3ℤ)² quotient of an Eisenstein four-torus constructed by Bauer and Gleissner. We also show that when the top image of a parallel nilpotent has rank one, the manifold or its orientation double cover does carry a parallel null vector; in particular, the original question is affirmative in Lorentzian and co-Lorentzian signature.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.DG"
  ],
  "keywords": [
    "flat pseudo-Riemannian manifolds",
    "parallel nilpotent endomorphisms",
    "holonomy",
    "Bieberbach manifolds",
    "parallel vector fields",
    "pseudo-Riemannian manifold",
    "parallel endomorphism",
    "nilpotent",
    "Bieberbach manifold",
    "AIM-GEOMETRY-0274",
    "open mathematics",
    "mathematical proof",
    "math.DG"
  ],
  "manuscript_version_date": "2026-08-30",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0274/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0274/paper.pdf?v=0ef1138a1de6",
  "doi": "10.5281/zenodo.22245515",
  "zenodo_record_url": "https://zenodo.org/records/22245515",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
      "sha256": "1f32789cb763e6fffd0010dd1c3bea36039d80b12b26a756d691459fd71ac9f8"
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}