Compatible almost-complex structures and Kähler geometry

Every Kähler manifold is almost Kähler, but a compatible almost-complex structure need not be integrable. A Kähler manifold requires that $J$ come from a complex-manifold structure in addition to compatibility with the closed form $\omega$. Thus the existence of compatible $J$ on every symplectic manifold does not make every symplectic manifold Kähler.

For the sign convention used by the examples below, the Nijenhuis tensor is

$N_J(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]\mathrm{.}$

Direct substitution of the vector-field commutator shows that all derivatives of scalar coefficients cancel, so $N_J$ is $C^{\mathrm{\infty }}$-linear in $X$ and $Y$. If $J$ is integrable, take local real coordinates underlying holomorphic coordinates. On their coordinate frame $J$ has the constant standard matrix and all coordinate brackets vanish; hence the displayed formula is zero on every pair of frame vectors and therefore $N_J=0$. Thus nonvanishing of $N_J$ is a direct obstruction to integrability; the converse is the substantially deeper Newlander–Nirenberg theorem and is not used here.

When $J$ is integrable, the identities $g=\omega (\cdot ,J\cdot )$ and $\omega (\cdot ,\cdot )=g(J\cdot ,\cdot )$ connect the symplectic, complex, and Riemannian descriptions.

All assertions are local and apply in real dimension zero. Compatibility makes the associated metric positive definite, so degenerate forms are outside the hypotheses. There is no interval or endpoint assertion, and the coordinate test uses no choice principle.