Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres

@misc{galazgarcia2026positive,
  title = {{Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres}},
  author = {Fernando Galaz-García},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.38126},
  eprint = {2609.38126},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38126},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres
AU  - Fernando Galaz-García
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.38126
UR  - https://arxiv.org/abs/2609.38126
AB  - We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Sperança's special S^3-S^3 bundle models with the compatible-disk construction of He, Liu and Yau. We show that the relevant bundles admit equivariant polar normal forms, with transition functions that are constant along meridians and conjugation-equivariant. We also show that the representation-dependent part of the He--Liu--Yau construction requires only a uniform bound on the infinitesimal action fields. Sperança's $8$- and $10$-dimensional examples satisfy this bound. The resulting northern and southern metrics have matching boundary metrics and compatible second fundamental forms, so the Reiser--Wraith gluing theorem gives the required positively curved metrics.
ER  -