Positive sectional curvature on homotopy nine-spheres
Abstract
We determine which homotopy nine-spheres admit positive sectional curvature: they are precisely the four smooth types with vanishing real Dirac index. The other four admit no metric of nonnegative sectional curvature. In particular, we construct a strictly positively curved metric on the exotic Kervaire nine-sphere, answering the longstanding nonnegative-curvature existence question in this dimension. The proof keeps explicit attaching maps throughout. A quaternionic commutator realizes the Kervaire sphere, a mixed representation realizes the Hopf image of the exotic eight-sphere, and a four-periodic normal-framing formula identifies the remaining index-zero class. Following the compatible-disk method, we give the curvature and boundary estimates for the first two models, including an explicit lower bound for the Kervaire metric in terms of the curvature and first covariant-derivative norms of a fixed connection, with the base scale and fiber normalization fixed. For the third model, we construct a positive metric on a cylinder joining a quotient metric to its pullback by a marked right Hopf twist. A four-form modification controls the connection terms involving both left and right quaternionic multiplication. Two positive disk fillings close the cylinder with the prescribed attaching map. A local convolution lemma supplies the smooth gluing while preserving a strict sectional-curvature lower bound.
Related
- Hitchin's α-invariant and positive sectional curvature on homotopy $9$-spheres
- Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres
- Positive Sectional Curvature on All Smooth 7-spheres
- Positive sectional curvature and non-isometric circle actions on a family of eleven-spheres
- Positive sectional curvature on exotic spheres in infinitely many dimensions
BibTeX
@misc{chen2026positive,
title = {{Positive sectional curvature on homotopy nine-spheres}},
author = {Zhiqi Chen and Shaoqiang Deng and Junna Ni and Hui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03298},
eprint = {2610.03298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03298},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive sectional curvature on homotopy nine-spheres
AU - Zhiqi Chen
AU - Shaoqiang Deng
AU - Junna Ni
AU - Hui Zhang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.03298
UR - https://arxiv.org/abs/2610.03298
AB - We determine which homotopy nine-spheres admit positive sectional curvature: they are precisely the four smooth types with vanishing real Dirac index. The other four admit no metric of nonnegative sectional curvature. In particular, we construct a strictly positively curved metric on the exotic Kervaire nine-sphere, answering the longstanding nonnegative-curvature existence question in this dimension. The proof keeps explicit attaching maps throughout. A quaternionic commutator realizes the Kervaire sphere, a mixed representation realizes the Hopf image of the exotic eight-sphere, and a four-periodic normal-framing formula identifies the remaining index-zero class. Following the compatible-disk method, we give the curvature and boundary estimates for the first two models, including an explicit lower bound for the Kervaire metric in terms of the curvature and first covariant-derivative norms of a fixed connection, with the base scale and fiber normalization fixed. For the third model, we construct a positive metric on a cylinder joining a quotient metric to its pullback by a marked right Hopf twist. A four-form modification controls the connection terms involving both left and right quaternionic multiplication. Two positive disk fillings close the cylinder with the prescribed attaching map. A local convolution lemma supplies the smooth gluing while preserving a strict sectional-curvature lower bound.
ER -