  We prove that a compact 4-manifold which supports a circle-invariant fat
SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. 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 and CP^2-bar are the only 4-manifolds admitting circle-invariant
metrics solving a certain curvature inequality. This can be seen as an analogue
of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant
metrics of positive sectional curvature.
