On the Twistor Spaces of Almost Hermitian Manifolds

We study Grassmannian bundles G_k(M) of analytical 2k-planes over an almost Hermitian manifold M²ⁿ, from the point of view of the generalized twistor spaces of 13], and with the method of the moving frame 9]. G₁(M⁴) is the classical twistor space. We find four distinguished almost Hermitian structures, one of them being that of 13], and discuss their integrability and Kählerianity. For n=2, we compute the corresponding Hermitian connections, and derive consequences about the corresponding first Chern classes.