{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0263",
  "title": "A Compactness Obstruction to Linear Growth Along Null Geodesics",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let (M,g) be a compact semi-Riemannian manifold of indefinite signature whose null geodesics are complete. We prove that no C¹ one-form η can have the property that, along every nonconstant affinely parametrized null geodesic γ: ℝ → M, the function η(γ̇) is affine with nonzero slope. This gives a negative answer to Question 9.2.1 in the Burns–Matveev list of open problems about geodesics. The proof normalizes the null cone by an auxiliary Riemannian metric. Compactness then gives a uniform lower bound for the quadratic form (∇_vη)(v) on normalized null vectors, while homogeneity forces the speed of any fixed null geodesic to remain bounded. Compactness also bounds the norm of η, contradicting the asserted nonzero linear growth. Only null completeness, rather than full geodesic completeness, is used.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DG"
  ],
  "keywords": [
    "semi-Riemannian geometry",
    "null geodesics",
    "compactness",
    "projective vector fields",
    "geodesic completeness",
    "null geodesic",
    "projective vector field",
    "AIM-GEOMETRY-0263",
    "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-0263/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0263/paper.pdf?v=c55a01ba729f",
  "doi": "10.5281/zenodo.22245396",
  "zenodo_record_url": "https://zenodo.org/records/22245396",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
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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."
}