Fundamental 2-form of an almost-Hermitian manifold

Let $(M,J,g)$ be an almost-Hermitian manifold. Its fundamental $2$ -form is the differential form $\omega$ defined by

\omega(X,Y)=g(JX,Y)

for tangent vectors $X,Y$ at the same point. Compatibility of $g$ with $J$ makes $\omega$ skew-symmetric, while positivity of $g$ makes it nondegenerate. Thus $\omega$ is a smooth, nondegenerate real $2$ -form, but it need not be closed. The displayed order fixes the sign convention: interchanging $X$ and $JY$ changes the sign.

Recovering the compatible data

The form and almost-complex structure recover the metric by

g(X,Y)=\omega(X,JY).

Conversely, compatible pairs among $J$ , $g$ , and $\omega$ , together with the appropriate positivity condition, determine the third object. This is pointwise Hermitian linear algebra applied smoothly to the tangent bundle.

Closedness and integrability

If $d\omega=0$ , the almost-Hermitian structure is called almost Kähler, and $\omega$ is a symplectic form. If $J$ is integrable, the structure is Hermitian. It is Kähler when both conditions hold. Neither closedness of $\omega$ nor integrability of $J$ follows from compatibility alone.

Conventions and scope

References

Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, vol. II, Wiley, 1969. Publisher record. Relevant: Chapter IX, §1, almost-Hermitian structures and their fundamental forms.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Chapter 3, Hermitian and Kähler geometry.