On the geometry of almost complex 6-manifolds (Q863782)

In the present paper the geometry of almost complex structure on 6-dimensional manifolds is investigated. First, the author introduces a 2-parameter familly of intrinsic first order functionals on such manifolds and obtains their Euler-Lagrange equations. Then he generalizes the holomorphic bundles over complex manifolds to almost comples manifolds, and finds a class of nonintegrable almost complex manifolds for which there exist such nontrivial bundles. In particular, the \(G_2\)-invariant almost complex structure on the 6-sphere admits such nontrivial bundles. The class of such almost complex structures in dimension 6 is called quasi-integrable. Finally, the author shows that quasi-integrability is not an involutive condition, and describes how general these structures can be.