Hermitian manifold

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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.

1 Formal definition
2 Riemannian metric and associated form
3 Properties
4 Kähler manifolds
5 References

[edit] 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

$h \in \Gamma(E\otimes\bar E)^*$

such that

$h_p(\eta, \bar\zeta) = \overline{h_p(\zeta, \bar\eta)}$

for all ζ, η in E_p and

$h_p(\zeta,\bar\zeta) > 0$

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

$h = h_{\alpha\bar\beta}\,dz^\alpha\otimes d\bar z^\beta$

where $h_{\alpha\bar\beta}$ are the components of a positive-definite Hermitian matrix.

[edit] 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:

$g = {1\over 2}(h+\bar h).$

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

$g = {1\over 2}h_{\alpha\bar\beta}\,(dz^\alpha\otimes d\bar z^\beta + d\bar z^\beta\otimes dz^\alpha).$

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:

$\omega = {i\over 2}(h-\bar h).$

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

$\omega = {i\over 2}h_{\alpha\bar\beta}\,dz^\alpha\wedge d\bar z^\beta.$

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

$\begin{align}\omega(u,v) &= g(Ju,v)\\ g(u,v) &= \omega(u,Jv)\end{align}$

for all complex tangent vectors u and v. The Hermitian metric h can be recovered from g and ω via the identity

$h = g - i\omega.\,$

All three forms h, g, and ω preserve the almost complex structure J. That is,

$\begin{align} h(Ju,Jv) &= h(u,v) \\ g(Ju,Jv) &= g(u,v) \\ \omega(Ju,Jv) &= \omega(u,v)\end{align}$

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.

[edit] 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:

$g'(u,v) = {1\over 2}\left(g(u,v) + g(Ju,Jv)\right).$

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 ω by

$\mathrm{vol}_M = \frac{\omega^n}{n!} \in \Omega^{n,n}(M)$

where ωⁿ 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

$\mathrm{vol}_M = \left(\frac{i}{2}\right)^n \det(h_{\alpha\bar\beta})\, dz^1\wedge d\bar z^1\wedge \cdots \wedge dz^n\wedge d\bar z^n.$

[edit] Kähler manifolds

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

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:

M is Kähler (i.e. 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.

So 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 is due in part to these properties.

[edit] References

Griffiths, Phillip; Joseph Harris [1978] (1994). Principles of Algebraic Geometry, Wiley Classics Library. New York: Wiley-Interscience. ISBN 0-471-05059-8.
Kobayashi, Shoshichi; Katsumi Nomizu [1963] (1996). 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.