Study of topis related to almost complex structures

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A smooth manifold M admitting a (1,1)-tensor field J satisfying J=・I is called an almost complex manifold and the tensor field J is called the almost complex structure. The concept of almost complex manifold is a generalization of complex manifold.. Almost complex manifold (M, J) is said to be integrable if M admits a complex structure and the derived almost complex structure coincides with the almost complex structure J. Any 2-dimensional almost complex manifold is always integrable. However, this is not true for higher dimensional cases in general. An almost complex manifold (M, J) equipped with a compatible (pseudo) Riemannian metric g is called an almost Hermitian manifold. A Kaehler manifold is the most typical one. In this research project, we study mainly the following topics related to the almost complex structures :(1) Integrability of almost complex structure(2) Submanifolds in almost complex manifolds(3) Intermediate and Related topics to (1),(2)Concerning (1), we study the integrability of almost Kaehler manifolds, for example Goldberg conjecture. Y.Matsushita et al. constructed an 8-dimensional counter example with a neutral Walker metric to the conjecture in the pseuo-Riemannian case. However, the conjecture itself is still remaining open in the case where the scalar curvature is negative. Concerning (2), H.Hashimoto studied several topics related to J-holomorphic curves in the nearly Kaehler 6-sphere S6 from the viewpoint of the Grassmann geometry and obtained interesting results on the deformations of super-minimal J-holomorphic curves and on some tubes around J-holomorphic curves in S6. Recently, the head investigator and H.Hashimoto et al. began to study 6-dimensional oriented submanifolds in the Octonions.and succeed to classify all extrinsic homogeneous almost hermitian 6-manifolds. Concerning (3), for example, K.Hasegawa gave an affirmative answer to the generalized Benson-Gordon conjecture.