{
  "abstract": "By Chern--Weil theory, rational Pontrjagin classes are represented by curvature forms. Sprays, which require no metric and include all Finsler and Riemannian geodesic sprays, provide a natural setting for the vanishing problem. Shen and Zhao proved that if a manifold admits a locally projectively flat spray, then all of its rational Pontrjagin classes vanish. We ask whether this rigidity persists for the wider class of projectively affine sprays. We prove that if a smooth manifold admits a locally spherically symmetric projectively affine spray, then all rational Pontrjagin classes vanish. Stronger, the Pontrjagin forms of the associated Berwald spray vanish identically as differential forms, a genuine geometric rigidity. Hence a manifold with nonzero rational Pontrjagin class cannot carry such a spray; in particular, it admits no Riemannian metric whose geodesic spray is locally spherically symmetric. We also construct non-Berwald, non-projectively-flat examples with identically vanishing Pontrjagin forms, showing the phenomenon is not a byproduct of projective flatness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36841",
  "author_total": 2,
  "authors": [
    {
      "name": "Zuhuan Jia"
    },
    {
      "name": "Benling Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/avqp5g1ybyj1d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36841",
  "fulltext_url": null,
  "id": "avqp5g1ybyj1d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36841",
  "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",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7224083383",
  "pdf_url": "https://arxiv.org/pdf/2610.36841",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/avqp5g1ybyj1d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36841",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36841"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Pontrjagin Classes of Locally Spherically Symmetric Projectively Affine Sprays",
  "title_text": "Pontrjagin Classes of Locally Spherically Symmetric Projectively Affine Sprays",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/avqp5g1ybyj1d",
  "venue": "arXiv",
  "versions": []
}