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 (holomorphic) 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 (U(n) 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** (or **cosymplectic structure**) 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 (i.e., it is a symplectic form), we get an **almost Kähler structure**. If both the almost complex structure and the fundamental form are integrable, then we have a Kähler structure.

## 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

such that

for all ζ, η in *E*_{p} and

for all nonzero ζ in *E*_{p}.

A **Hermitian manifold** is a complex manifold with a Hermitian metric on its holomorphic tangent space. Likewise, an **almost Hermitian manifold** is an almost complex manifold with a Hermitian metric on its holomorphic tangent space.

On a Hermitian manifold the metric can be written in local holomorphic coordinates (*z*^{α}) as

where are the components of a positive-definite Hermitian matrix.

## Riemannian metric and associated form

A Hermitian metric *h* on an (almost) complex manifold *M* defines a Riemannian metric *g* on the underlying smooth manifold. The metric *g* is defined to be the real part of *h*:

The form *g* is a symmetric bilinear form on *TM*^{C}, the complexified tangent bundle. 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*. The form ω is called variously the **associated (1,1) form**, the **fundamental form**, or the **Hermitian form**. 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*. That is,

for all complex tangent vectors *u* and *v*.

A Hermitian structure on an (almost) complex manifold *M* can therefore be specified by either

- a Hermitian metric
*h*as above, - a Riemannian metric
*g*that preserves the almost complex structure*J*, or - a nondegenerate 2-form ω which preserves
*J*and is positive-definite in the sense that ω(*u*,*Ju*) > 0 for all nonzero real tangent vectors*u*.

Note that many authors call *g* itself the Hermitian metric.

## Properties

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. The unitary frame bundle of *M* is the principal U(*n*)-bundle of all unitary frames.

Every almost Hermitian manifold *M* has a canonical volume form which is just the Riemannian volume form determined by *g*. This form is given in terms of the associated (1,1)-form ω by

where ω^{n} is the wedge product of ω with itself *n* times. The volume form is therefore a real (*n*,*n*)-form on *M*. In local holomorphic coordinates the volume form is given by

One can also consider a hermitian metric on a holomorphic vector bundle.

## Kähler manifolds

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

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 2*n* 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.

## References

- Griffiths, Phillip; Joseph Harris (1994) [1978].
*Principles of Algebraic Geometry*. Wiley Classics Library. New York: Wiley-Interscience. ISBN 0-471-05059-8. - Kobayashi, Shoshichi; Katsumi Nomizu (1996) [1963].
*Foundations of Differential Geometry, Vol. 2*. Wiley Classics Library. New York: Wiley Interscience. ISBN 0-471-15732-5. - Kodaira, Kunihiko (1986).
*Complex Manifolds and Deformation of Complex Structures*. Classics in Mathematics. New York: Springer. ISBN 3-540-22614-1.