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 {{math|h}} on an (almost) complex manifold {{math|M}} defines a Riemannian metric {{math|g}} on the underlying smooth manifold. The metric {{math|g}} is defined to be the real part of {{math|h}}:

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

The form {{math|g}} is a symmetric bilinear form on {{math|TM^C}}, the complexified tangent bundle. Since {{math|g}} is equal to its conjugate it is the complexification of a real form on {{math|TM}}. The symmetry and positive-definiteness of {{math|g}} on {{math|TM}} follow from the corresponding properties of {{math|h}}. In local holomorphic coordinates the metric {{math|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 {{math|h}} a complex differential form {{math|ω}} of degree (1,1). The form {{math|ω}} is defined as minus the imaginary part of {{math|h}}:

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

Again since {{math|ω}} is equal to its conjugate it is the complexification of a real form on {{math|TM}}. The form {{math|ω}} is called variously the associated (1,1) form, the fundamental form, or the Hermitian form. In local holomorphic coordinates {{math|ω}} can be written

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

where $dz^{\alpha}\wedge d\bar z^{\beta}=dz^{\alpha}\otimes d\bar z^{\beta}-d\bar z^{\beta}\otimes dz^{\alpha}.$

It is clear from the coordinate representations that any one of the three forms {{math|h}}, {{math|g}}, and {{math|ω}} uniquely determine the other two. The Riemannian metric {{math|g}} and associated (1,1) form {{math|ω}} are related by the almost complex structure {{math|J}} as follows

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

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

$h = g - i\omega.$

All three forms {{math|h}}, {{math|g}}, and {{math|ω}} preserve the almost complex structure {{math|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 {{mvar|u}} and {{mvar|v}}.

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

a Hermitian metric {{math|h}} as above,
a Riemannian metric {{math|g}} that preserves the almost complex structure {{math|J}}, or
a nondegenerate 2-form {{math|ω}} which preserves {{math|J}} and is positive-definite in the sense that {{math|ω(u, Ju) > 0}} for all nonzero real tangent vectors {{math|u}}.

Note that many authors call {{math|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′ 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 {{math|ω}} by

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

where {{math|ωⁿ}} is the wedge product of {{math|ω}} with itself {{mvar|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 {{math|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 {{math|(M, g, ω, J)}} be an almost Hermitian manifold of real dimension {{math|2n}} and let {{math|∇}} be the Levi-Civita connection of {{math|g}}. The following are equivalent conditions for {{math|M}} to be Kähler:

{{math|ω}} is closed and {{math|J}} is integrable,
{{math|1=∇J = 0}},
{{math|1=∇ω = 0}},
the holonomy group of {{math|∇}} is contained in the unitary group {{math|U(n)}} associated to {{math|J}},

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

In particular, if {{math|M}} is a Hermitian manifold, the condition dω = 0 is equivalent to the apparently much stronger conditions {{math|1=∇ω = ∇J = 0}}. The richness of Kähler theory is due in part to these properties.

References

{{cite book | first = Phillip | last = Griffiths |author2=Joseph Harris | title = Principles of Algebraic Geometry | series = Wiley Classics Library | publisher = Wiley-Interscience | location = New York | year = 1994 | orig-year = 1978 | isbn = 0-471-05059-8}}
{{cite book | first = Shoshichi | last = Kobayashi |author2=Katsumi Nomizu | title = Foundations of Differential Geometry, Vol. 2 | series = Wiley Classics Library | publisher = Wiley Interscience | location = New York | year = 1996 | orig-year = 1963 | isbn = 0-471-15732-5}}
{{cite book | first = Kunihiko | last = Kodaira | title = Complex Manifolds and Deformation of Complex Structures | series = Classics in Mathematics | publisher = Springer | location = New York | year = 1986| isbn = 3-540-22614-1}}

Category:Complex manifolds

Category:Differential geometry

Category:Riemannian geometry

Category:Riemannian manifolds

Category:Structures on manifolds