(1, 2)-Symplectic Structures, nearly Kahler Structures and S6

The exact relations between the Hermitian, the Symplectic and the Nearly Kahler Structures are established. The standard complex structure J_s on SO(2n + l)/U(n) is shown to be the same as the almost complex structure J₁(one of the canonical almost complex structure on SO(2n+l)/U(n) considered as a twistor space over (S²ⁿ, g₀)). A corollary is that S⁶ does not allow any complex structure orthogonal to the standard metric g₀. On an almost complex manifold with an arbitrary metric a (1, 2)-tensor A is defined. If A is closed as a vector valued 2-form, then it is proved that constant sectional curvature implies zero curvature. This is related to Hsiung’s work, which is discussed at the end.