Global Metrizability on Sprays of Scalar Curvature

@misc{duanmu2026global,
  title = {{Global Metrizability on Sprays of Scalar Curvature}},
  author = {Xingzhi Duanmu and Guojun Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18292},
  eprint = {2609.18292},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18292},
  note = {CC0},
}

TY  - UNPB
TI  - Global Metrizability on Sprays of Scalar Curvature
AU  - Xingzhi Duanmu
AU  - Guojun Yang
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18292
UR  - https://arxiv.org/abs/2609.18292
AB  - Sprays of scalar curvature constitute an important class of sprays, and such a class includes all two-dimensional sprays. In this paper, we consider the global metrizability of certain sprays on a manifold under some curvature and topological conditions. We prove that, for a regular spray {\bf G} of scalar curvature with almost everywhere nonzero Ricci curvature on a manifold M of dimension n≥ 3, {\bf G} is globally metrizable if and only if it is locally metrizable, provided that M has trivial first de Rham cohomology group. Further, we characterize the global metizability of a class of two-dimensional singular Berwald sprays on a manifold with trivial first de Rham cohomology group. Meanwhile, on a cylinder S^1× bf R (with non-trivial first de Rham cohomology group), we construct a family of singular Berwald sprays which are locally metrizable but not globally metrizable. Finally, we construct some examples of two-dimensional sprays on cylinders or spheres with special properties.
ER  -