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We prove that a compact 4‐manifold which supports a circle‐invariant fat    SO  (  3  )   ‐bundle is diffeomorphic to either    S  4    orC  P   ¯   2   . The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6‐manifold. Applying our result to the twistor bundle of Riemannian 4‐manifolds shows that    S  4    andC  P   ¯   2    are the only 4‐manifolds admitting circle‐invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang–Kleiner's theorem that only    S  4    and    C    P   2admit circle‐invariant metrics of positive sectional curvature.

Circle‐invariant fat bundles and symplectic Fano 6‐manifolds