More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space.
On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure) that depends only on the chosen metric and the almost complex structure.
With the extra integrability condition that it is closed (i.e., it is a symplectic form), we get an almost Kähler structure.
A Hermitian metric on a complex vector bundle
is a smoothly varying positive-definite Hermitian form on each fiber.
Such a metric can be viewed as a smooth global section
Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent bundle.
On a Hermitian manifold the metric can be written in local holomorphic coordinates
A Hermitian metric h on an (almost) complex manifold M defines a Riemannian metric g on the underlying smooth manifold.
Since g is equal to its conjugate it is the complexification of a real form on TM.
The symmetry and positive-definiteness of g on TM follow from the corresponding properties of h. In local holomorphic coordinates the metric g can be written
One can also associate to h a complex differential form ω of degree (1,1).
The form ω is defined as minus the imaginary part of h:
Again since ω is equal to its conjugate it is the complexification of a real form on TM.
In local holomorphic coordinates ω can be written
It is clear from the coordinate representations that any one of the three forms h, g, and ω uniquely determine the other two.
The Riemannian metric g and associated (1,1) form ω are related by the almost complex structure J as follows
for all complex tangent vectors u and v. The Hermitian metric h can be recovered from g and ω via the identity
All three forms h, g, and ω preserve the almost complex structure J.
for all complex tangent vectors u and v. A Hermitian structure on an (almost) complex manifold M can therefore be specified by either Note that many authors call g itself the Hermitian metric.
Every (almost) complex manifold admits a Hermitian metric.
This follows directly from the analogous statement for Riemannian metric.
Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g′ compatible with the almost complex structure J in an obvious manner:
Choosing a Hermitian metric on an almost complex manifold M is equivalent to a choice of U(n)-structure on M; that is, a reduction of the structure group of the frame bundle of M from GL(n, C) to the unitary group U(n).
A unitary frame on an almost Hermitian manifold is complex linear frame which is orthonormal with respect to the Hermitian metric.
where ωn is the wedge product of ω with itself n times.
One can also consider a hermitian metric on a holomorphic vector bundle.
A Kähler form is a symplectic form, and so Kähler manifolds are naturally symplectic manifolds.
Let (M, g, ω, J) be an almost Hermitian manifold of real dimension 2n and let ∇ be the Levi-Civita connection of g. The following are equivalent conditions for M to be Kähler: The equivalence of these conditions corresponds to the "2 out of 3" property of the unitary group.
The richness of Kähler theory is due in part to these properties.