He–Liu–Yau report positive curvature on all smooth seven-spheres
Positive sectional curvature on all smooth seven-spheres
- Mathematics
Smooth manifolds can be topologically the same seven-sphere while having different differentiable structures.
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Understanding the problem
Read the background
Smooth manifolds can be topologically the same seven-sphere while having different differentiable structures. This question asks whether every such smooth structure admits a metric with positive curvature on every tangent two-plane.
The oriented smooth structures on the seven-sphere form 28 diffeomorphism classes, including the standard sphere. Positive sectional curvature means that every tangent two-plane has positive curvature.
Nonnegative curvature was already known for every exotic seven-sphere. Strict positivity is a stronger existence question; strong quarter-pinching is a separate rigidity condition.
Why it matters
Separates the existence of positive curvature from stronger curvature-pinching conditions that constrain smooth structure.
What counts as progress
A sourced advance on strict positive sectional curvature for the stated smooth classes, with the prior nonnegative-curvature baseline, human and AI roles, and checking scope kept explicit.
What to watch for
- Specialist assessment of the prescribed attaching maps, global disk metrics and gluing hypotheses.
- Changes to the manuscript, reconciliation with the v12 computational source, or a correction or withdrawal.
- More detailed human–AI provenance and checking of the dependencies outside the computational package.
Background review: 2026-10-16. Selected sources:
- Original seven-spheres claim, authors and submission date
- Theorem 1.1, construction, prior work and AI acknowledgments
- Author SageMath companion: v12 verification scope and limits
- v12 check results: 80 PASS and seven UNVERIFIED groups
- Goette–Kerin–Shankar: nonnegative curvature on all exotic seven-spheres
- Reiser–Wraith gluing theorem used in the construction
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