Almost Kähler Manifold

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 viewed as a smooth global section $h$ of the vector bundle $(E\otimes\overline)^*$ such that for every point $p$ in $M$,
$h_p\mathord = \overline$
for all $\zeta$, $\eta$ in the fiber $E_$ and
$h_p\mathord > 0$
for all nonzero $\zeta$ in $E_$.

A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent bundle. 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 $(z^\alpha)$ as
$h = h_\,dz^\alpha \otimes d\bar z^\beta$
where $h_$ are the components of a positive-definite Hermitian matrix.

Riemannian metric and associated form

A Hermitian metric on an (almost) complex manifold defines a Riemannian metric on the underlying smooth manifold. The metric is defined to be the real part of :
$g = \left(h + \bar h\right).$

The form is a symmetric bilinear form on , the complexified tangent bundle. Since is equal to its conjugate it is the complexification of a real form on . The symmetry and positive-definiteness of on follow from the corresponding properties of . In local holomorphic coordinates the metric can be written
$g = h_\,\left(dz^\alpha\otimes d\bar z^\beta + d\bar z^\beta\otimes dz^\alpha\right).$

One can also associate to a complex differential form of degree (1,1). The form is defined as minus the imaginary part of :
$\omega = \left(h - \bar h\right).$

Again since is equal to its conjugate it is the complexification of a real form on . The form is called variously the associated (1,1) form, the fundamental form, or the Hermitian form. In local holomorphic coordinates can be written
$\omega = h_\,dz^\alpha\wedge d\bar z^\beta,$
where $dz^\wedge d\bar z^=dz^\otimes d\bar z^-d\bar z^\otimes dz^.$

It is clear from the coordinate representations that any one of the three forms , , and uniquely determine the other two. The Riemannian metric and associated (1,1) form are related by the almost complex structure as follows
$\begin \omega(u, v) &= g(Ju, v)\\ g(u, v) &= \omega(u, Jv) \end$
for all complex tangent vectors and . The Hermitian metric can be recovered from and via the identity
$h = g - i\omega.$

All three forms , , and preserve the almost complex structure . That is,
$\begin h(Ju, Jv) &= h(u, v) \\ g(Ju, Jv) &= g(u, v) \\ \omega(Ju, Jv) &= \omega(u, v) \end$
for all complex tangent vectors and .

A Hermitian structure on an (almost) complex manifold can therefore be specified by either
# a Hermitian metric as above,
# a Riemannian metric that preserves the almost complex structure , or
# a nondegenerate 2-form which preserves and is positive-definite in the sense that for all nonzero real tangent vectors .

Note that many authors call 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:
$g'(u, v) = \left(g(u, v) + g(Ju, Jv)\right).$

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 by
$\mathrm_M = \frac \in \Omega^(M)$
where is the wedge product of with itself times. The volume form is therefore a real (''n'',''n'')-form on ''M''. In local holomorphic coordinates the volume form is given by
$\mathrm_M = \left(\frac\right)^n \det\left(h_\right)\, dz^1 \wedge d\bar z^1 \wedge \dotsb \wedge dz^n \wedge d\bar z^n.$

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 is closed:
$d\omega = 0\,.$
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 be an almost Hermitian manifold of real dimension and let be the Levi-Civita connection of . The following are equivalent conditions for to be Kähler:
* is closed and is integrable,
* ,
* ,
* the holonomy group of is contained in the unitary group associated to ,

The equivalence of these conditions corresponds to the " 2 out of 3" property of the unitary group.

In particular, if is a Hermitian manifold, the condition dω = 0 is equivalent to the apparently much stronger conditions . The richness of Kähler theory is due in part to these properties.

References

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Complex manifolds
Differential geometry
Riemannian geometry
Riemannian manifolds
Structures on manifolds