Almost complex submanifolds of nearly Kähler manifolds

We study almost complex submanifolds of pseudo nearly Kahler manifolds. We show in particular that a 6 dimensional strict nearly Kahler manifold does not admit any 4 dimensional almost complex submanifolds. This generalises results obtained by Gray (Proc Am Math Soc 20:277–279, 1969) for the nearly Kahler 6-sphere and by Podesta and Spiro (J Geom Phys 60:156–164, 2010) in the Riemannian case.