Pontrjagin Classes of Locally Spherically Symmetric Projectively Affine Sprays
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.
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BibTeX
@misc{jia2026pontrjagin,
title = {{Pontrjagin Classes of Locally Spherically Symmetric Projectively Affine Sprays}},
author = {Zuhuan Jia and Benling Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36841},
eprint = {2610.36841},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36841},
}
RIS
TY - UNPB
TI - Pontrjagin Classes of Locally Spherically Symmetric Projectively Affine Sprays
AU - Zuhuan Jia
AU - Benling Li
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36841
UR - https://arxiv.org/abs/2610.36841
AB - 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.
ER -