Y. He

Z. Liu

S. Yau

For homotopy 9-spheres, positive sectional curvature is proved equivalent to vanishing Hitchin’s α-invariant, matching scalar curvature. 

A sphere’s smooth structure can obstruct positive curvature. For homotopy 9-spheres, Hitchin’s α-invariant, a Dirac index, is shown to determine exactly when positive sectional curvature is possible, just as it does for positive scalar curvature. Of the eight smooth types, four qualify. Metrics on the three exotic examples are constructed by gluing positively curved discs, giving a unified proof of the classification. 

We prove that a smooth homotopy 9-sphere admits a Riemannian metric of positive sectional curvature if and only if its Hitchin α-invariant vanishes, or equivalently if and only if it admits a Riemannian metric of positive scalar curvature.

Hitchin's α-invariant and positive sectional curvature on homotopy 9-spheres