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