A four-dimensional sphere type theorem

This talk is based on a joint work with Joel Fine. A result of Kleiner and Wilking states that a positively curved Riemannian 4-manifold admitting an isometric S^1-action is diffeomorphic to S^4, RP^4 of CP^2. Using elementary symplectic geometry considerations we prove a similar result for geometric structures that are "softer" than metrics. Namely we consider S^1-equivariant definite connections on 4-manifolds and prove that a 4-manifold admitting such a structure is diffeomorphic to S^4 of CP^2.
A definite connection on a 4-manifold M is a metric connection on a rank 3 bundle over M whose curvature Q satisfy the following inequality: for any two vectors (u,v) in T_x(M) the matrix Q(u,v) is non-zero.