Hermitian 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 $$(E\otimes\overline{E})^*$$ such that for every point $$p$$ in $$M$$, $$h_p\mathord{\left(\eta, \bar\zeta\right)} = \overline{h_p\mathord{\left(\zeta, \bar\eta\right)}}$$ for all $$\zeta$$, $$\eta$$ in the fiber $$E_{p}$$ and $$h_p\mathord{\left(\zeta, \bar\zeta\right)} > 0$$ 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 $$h = h_{\alpha\bar\beta}\,dz^\alpha \otimes d\bar z^\beta$$ where $$h_{\alpha\bar\beta}$$ 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: $$g = {1 \over 2}\left(h + \bar h\right).$$

The form g is a symmetric bilinear form on TMC, 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 $$g = {1 \over 2}h_{\alpha\bar\beta}\,\left(dz^\alpha\otimes d\bar z^\beta + d\bar z^\beta\otimes dz^\alpha\right).$$

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

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

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
 * 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.

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

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

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.

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 $2n$ 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,
 * the holonomy group of $∇J = 0$ is contained in the unitary group $∇ω = 0$ associated to $∇$,
 * the holonomy group of $U(n)$ is contained in the unitary group $J$ associated to $M$,
 * the holonomy group of $∇ω = ∇J = 0$ 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.