Y. He

Z. Liu

S. Yau

A strictly positive sectional curvature metric is explicitly constructed for all 28 smooth homotopy seven-spheres using a gluing theorem.

Determining which manifolds admit positive sectional curvature is a fundamental problem in geometry. The longstanding question of whether exotic seven-spheres admit such metrics is resolved: strictly positively curved metrics are constructed for all 28 oriented diffeomorphism classes. Compatible smooth metrics on two seven-disks are then glued, showing that positive curvature does not exclude exotic smooth structures. 

We construct a smooth Riemannian metric with strictly positive sectional curvature on every smooth homotopy seven-sphere. For each smooth structure, we construct positively curved metrics on two seven-dimensional disks whose induced boundary metrics agree under the prescribed attaching map. Under this identification, a gluing theorem then yields the required smooth metric on the closed manifold. The construction applies to all 28 oriented diffeomorphism classes.

Positive Sectional Curvature on All Smooth 7-spheres