1 Answers
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure.
A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields a unitary structure structure] on the manifold. By dropping this condition, we get an almost Hermitian manifold.
On any almost Hermitian manifold, we can introduce a fundamental 2-form that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate. With the extra integrability condition that it is closed , we get an almost Kähler structure. If both the almost complex structure and the fundamental form are integrable, then we have a Kähler structure.