Almost hermitian manifolds with closed symplectic form

Question

An hermitian manifold $M$ is a manifold equipped with a Riemannian metric $g$, a non-degenerate two-form $\omega$, and an almost complex structure $J$ such that

$\omega(\cdot,\cdot) = g(J\cdot,\cdot)$

$(M,g,J,\omega)$ is a Kahler manifold if and only if $J$ is integrable, namely it is a complex structure on $M$, and $\omega$ is closed, namely it is a symplectic form on $M$. What happens if we relax the condition of $J$ being integrable but we still demand $\omega$ to be closed? Does this kind of manifolds have a name? Do they enjoy interesting geometric properties as the more constained Kahler manifolds do?

Thanks.

Answer 1

The naming system is as follows:

• $J$ not integrable, $\omega$ not closed: Almost Hermitian
• $J$ integrable, $\omega$ not closed: Hermitian
• $J$ not integrable, $\omega$ closed: Almost Kahler
• $J$ integrable, $\omega$ closed: Kahler

Answer 2

Every symplectic manifold has a contractible (in particular, non-empty) space of compatible almost complex structures, so this is exactly a strong a condition on a manifold as being symplectic. These almost complex structures are used in symplectic geometry e.g. to define pseudoholomorphic curves.