We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S<sup>4</sup> or ℂℙ<sup>2</sup>. 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<sup>4</sup> and ℂℙ<sup>2</sup> are the only 4-manifolds admitting circleinvariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kleiner's theorem that only S<sup>4</sup> and ℂℙ<sup>2</sup> admit circle-invariant metrics of positive sectional curvature.
|Number of pages||22|
|Journal||JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES|
|Publication status||Published - 2015|