Hermitian manifold Totally Explained

Everything about Hermitian Metric totally explained

In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a complex manifold with a Riemannian metric that preserves the almost complex structure J. There are several related concepts coming from integrability conditions:
A Hermitian manifold has a unitary structure (a

U(n)

-structure), with the almost complex structure integrable. Without the integrability condition, the notion is an almost Hermitian manifold. With an additional integrability condition on the Sp-structure, one obtains a Kähler manifold. With integrability on the Sp-structure but not the almost complex structure, one obtains an almost Kähler manifold.

Formal definition

A Hermitian metric on a complex vector bundle E over a smooth manifold M is a smoothly varying positive-definite Hermitian form on each fiber. Such a metric can be written as a smooth section ť

h in Gamma(Eotimesar E)^*

such that ť

h_p(eta, arzeta) = overline), dz^1wedge dar z^1wedge cdots wedge dz^nwedge dar z^n.

Kähler manifolds

The most important class of Hermitian manifolds are Kähler manifolds. These are Hermitian manifolds for which the Hermitian form ω is closed: ť

domega = 0,.

In this case the form ω is called a Kähler form. A Kähler form is a symplectic form, and so Kähler manifolds are naturally symplectic manifolds.
An almost Hermitian manifold whose associated (1,1)-form is closed is naturally called an almost Kähler manifold. Any symplectic manifold admits a compatible almost complex structure making it into an almost Kähler manifold.

Integrability

A Kähler manifold is an almost Hermitian manifold satisfying an integrability condition. This can be stated in several equivalent ways.
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:

ω is closed and J is integrable

∇J = 0,

∇ω = 0,

the holonomy group of ∇ is contained in the unitary group U(n) associated to J. The equivalence of these conditions corresponds to the "2 out of 3" property of the unitary group. In particular, if M is a Hermitian manifold, the condition dω = 0 is equivalent to the apparently much stronger conditions ∇ω = ∇J = 0. The richness of Kähler theory is due in part to these properties.

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