Hermitian manifold

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 ${\displaystyle (E\otimes {\bar {E}})^{*}}$ such that for every point p in M,

$${\displaystyle h_{p}{\mathord {\left(\eta ,{\bar {\zeta }}\right)}}={\overline {h_{p}{\mathord {\left(\zeta ,{\bar {\eta }}\right)}}}}}$$

for all ζ, η in the fiber E_p and

$${\displaystyle h_{p}{\mathord {\left(\zeta ,{\bar {\zeta }}\right)}}>0}$$

for all nonzero ζ 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^α) as

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

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

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:

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

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

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

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:

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

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

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

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

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

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

$${\displaystyle h=g-i\omega .}$$

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

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

for all complex tangent vectors u and v.

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

1. a Hermitian metric h as above,
2. a Riemannian metric g that preserves the almost complex structure J, or
3. 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.

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:

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

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

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

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

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

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

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

$${\displaystyle d\omega =0\,.}$$

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.

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• 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.