Hermitian manifold

Template:Short description Template:Use American English 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 $(E\otimes \bar{E}{)}^{*}$ such that for every point $p$ in $M$, $$h_p(\eta ,\overline{\zeta })=\overline{h_p(\zeta ,\overline{\eta })}$$ for all $\zeta$, $\eta$ in the fiber $E_p$ and $$h_p(\zeta ,\overline{\zeta })>0$$ for all nonzero $\zeta$ in $E_p$.

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 $$h=h_{\alpha \overline{\beta }}d{z}^{\alpha }\otimes d{\overline{z}}^{\beta }$$ where $h_{\alpha \overline{\beta }}$ 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: $$g=\frac{1}{2}(h+\overline{h}).$$

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 $$g=\frac{1}{2}{h}_{\alpha \overline{\beta }}(d{z}^{\alpha }\otimes d{\overline{z}}^{\beta }+d{\overline{z}}^{\beta }\otimes d{z}^{\alpha }).$$

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: $$\omega =\frac{i}{2}(h-\overline{h}).$$

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 $$\omega =\frac{i}{2}{h}_{\alpha \overline{\beta }}d{z}^{\alpha }\wedge d{\overline{z}}^{\beta }.$$

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

All three forms h, g, and ω preserve the almost complex structure Template:Math. That is, $$\begin{array}{cc}h(Ju,Jv)& =h(u,v)\\ g(Ju,Jv)& =g(u,v)\\ \omega (Ju,Jv)& =\omega (u,v)\end{array}$$ for all complex tangent vectors Template:Mvar and Template:Mvar.

A Hermitian structure on an (almost) complex manifold Template:Math can therefore be specified by either

1. a Hermitian metric Template:Math as above,
2. a Riemannian metric Template:Math that preserves the almost complex structure Template:Math, or
3. a nondegenerate 2-form Template:Math which preserves Template:Math and is positive-definite in the sense that Template:Math for all nonzero real tangent vectors Template:Math.

Note that many authors call Template:Math 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: $$g^{\prime }(u,v)=\frac{1}{2}(g(u,v)+g(Ju,Jv)).$$

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 Template:Math by $${\mathrm{v}\mathrm{o}\mathrm{l}}_M=\frac{{\omega }^n}{n!}\in {\mathrm{\Omega }}^{n,n}(M)$$ where Template:Math is the wedge product of Template:Math with itself Template:Mvar times. The volume form is therefore a real (n,n)-form on M. In local holomorphic coordinates the volume form is given by $${\mathrm{v}\mathrm{o}\mathrm{l}}_M={\left(\frac{i}{2}\right)}^n\det \left(h_{\alpha \overline{\beta }}\right)d{z}^1\wedge d{\overline{z}}^1\wedge \cdots \wedge d{z}^n\wedge d{\overline{z}}^n.$$

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 Template:Math 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 Template:Math be an almost Hermitian manifold of real dimension Template:Math and let Template:Math be the Levi-Civita connection of Template:Math. The following are equivalent conditions for Template:Math to be Kähler:

• Template:Math is closed and Template:Math is integrable,
• Template:Math,
• Template:Math,
• the holonomy group of Template:Math is contained in the unitary group Template:Math associated to Template:Math,

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

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

References

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