An integrability theorem for almost Kähler manifolds

In this paper we prove a new geometric integrability theorem for almost complex manifolds. Let M be a connected, oriented, smooth 2m real dimensional manifold with an U(m) structure on its tangent bundle. Moreover let ∇ be a compatible connection on it. Assume the complexified curvature of the connection has vanishing (0, 2) component. Then we claim that M can be given the structure of an m complex dimensional Kähler manifold and ∇ is the unique Levi–Civita connection on it. The proof has a Yang–Mills theoretic flavour and is based on Uhlenbeck’s theorem on the existence of local Coulomb gauges.