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| In [[mathematics]], a '''Hermitian manifold''' is the complex analogue of a [[Riemannian manifold]]. Specifically, a Hermitian manifold is a [[complex manifold]] with a smoothly varying [[Hermitian form|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 manifold|complex structure]].
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| A complex structure is essentially an [[almost complex structure]] with an integrability condition, and this condition yields an unitary structure ([[G-structure|U(n) structure]]) on the manifold. By dropping this condition we get an '''almost Hermitian manifold'''.
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| On any almost Hermitian manifold we can introduce a '''fundamental 2-form''', or '''cosymplectic structure''', that depends only on the chosen metric and almost complex structure. This form is always non-degenerate, with the suitable integrability condition (of it also being closed and thus a [[symplectic form]]) we get an '''almost Kähler structure'''. If both almost complex structure and fundamental form are integrable, we have a [[Kähler structure]].
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| ==Formal definition==
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| A '''Hermitian metric''' on a [[complex vector bundle]] ''E'' over a [[smooth manifold]] ''M'' is a smoothly varying [[definite bilinear form|positive-definite]] [[Hermitian form]] on each fiber. Such a metric can be written as a smooth section
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| :<math>h \in \Gamma(E\otimes\bar E)^*</math> | |
| such that
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| :<math>h_p(\eta, \bar\zeta) = \overline{h_p(\zeta, \bar\eta)}</math>
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| for all ζ, η in ''E''<sub>''p''</sub> and
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| :<math>h_p(\zeta,\bar\zeta) > 0</math>
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| for all nonzero ζ in ''E''<sub>''p''</sub>.
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| 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.
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| On a Hermitian manifold the metric can be written in local holomorphic coordinates (''z''<sup>α</sub>) as
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| :<math>h = h_{\alpha\bar\beta}\,dz^\alpha\otimes d\bar z^\beta</math>
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| where <math>h_{\alpha\bar\beta}</math> are the components of a positive-definite [[Hermitian matrix]].
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| ==Riemannian metric and associated form==
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| 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'':
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| :<math>g = {1\over 2}(h+\bar h).</math>
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| The form ''g'' is a symmetric bilinear form on ''TM''<sup>'''C'''</sup>, 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
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| :<math>g = {1\over 2}h_{\alpha\bar\beta}\,(dz^\alpha\otimes d\bar z^\beta + d\bar z^\beta\otimes dz^\alpha).</math>
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| 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'':
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| :<math>\omega = {i\over 2}(h-\bar h).</math>
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| 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
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| :<math>\omega = {i\over 2}h_{\alpha\bar\beta}\,dz^\alpha\wedge d\bar z^\beta.</math>
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| 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
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| :<math>\begin{align}\omega(u,v) &= g(Ju,v)\\ g(u,v) &= \omega(u,Jv)\end{align}</math>
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| for all complex tangent vectors ''u'' and ''v''. The Hermitian metric ''h'' can be recovered from ''g'' and ω via the identity
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| :<math>h = g - i\omega.\,</math>
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| All three forms ''h'', ''g'', and ω preserve the [[almost complex structure]] ''J''. That is,
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| :<math>\begin{align}
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| h(Ju,Jv) &= h(u,v) \\
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| g(Ju,Jv) &= g(u,v) \\
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| \omega(Ju,Jv) &= \omega(u,v)\end{align}</math>
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| for all complex tangent vectors ''u'' and ''v''. | |
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| A Hermitian structure on an (almost) complex manifold ''M'' can therefore be specified by either
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| #a Hermitian metric ''h'' as above,
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| #a Riemannian metric ''g'' that preserves the almost complex structure ''J'', or
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| #a [[nondegenerate form|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''.
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| Note that many authors call ''g'' itself the Hermitian metric.
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| ==Properties==
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| 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:
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| :<math>g'(u,v) = {1\over 2}\left(g(u,v) + g(Ju,Jv)\right).</math>
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| Choosing a Hermitian metric on an almost complex manifold ''M'' is equivalent to a choice of [[G-structure|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 bundle|principal U(''n'')-bundle]] of all unitary frames.
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| 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
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| :<math>\mathrm{vol}_M = \frac{\omega^n}{n!} \in \Omega^{n,n}(M)</math>
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| where ω<sup>''n''</sup> 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
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| :<math>\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.</math>
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| ==Kähler manifolds==
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| The most important class of Hermitian manifolds are [[Kähler manifold]]s. These are Hermitian manifolds for which the Hermitian form ω is [[closed differential form|closed]]:
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| :<math>d\omega = 0\,.</math>
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| 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 manifold]]s.
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| 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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| ===Integrability===
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| A Kähler manifold is an almost Hermitian manifold satisfying an [[integrability condition]]. This can be stated in several equivalent ways.
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| Let (''M'', ''g'', ω, ''J'') be an almost Hermitian manifold of real dimension 2''n'' and let ∇ be the [[Levi-Civita connection]] of ''g''. The following are equivalent conditions for ''M'' to be Kähler:
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| * ω is closed and ''J'' is integrable
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| * ∇''J'' = 0,
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| * ∇ω = 0,
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| * the [[holonomy group]] of ∇ is contained in the [[unitary group]] U(''n'') associated to ''J''.
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| The equivalence of these conditions corresponds to the "[[Unitary_group#2-out-of-3_property|2 out of 3]]" property of the [[unitary group]].
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| In particular, 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 theory is due in part to these properties.
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| ==References== | |
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| *{{cite book | first = Phillip | last = Griffiths | coauthors = Joseph Harris | title = Principles of Algebraic Geometry | series = Wiley Classics Library | publisher = Wiley-Interscience | location = New York | year = 1994 | origyear = 1978 | isbn = 0-471-05059-8}}
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| *{{cite book | first = Shoshichi | last = Kobayashi | coauthors = Katsumi Nomizu | title = [[Foundations of Differential Geometry]], Vol. 2 | series = Wiley Classics Library | publisher = [[Wiley Interscience]] | location = New York | year = 1996 | origyear = 1963 | isbn = 0-471-15732-5}}
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| *{{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}}
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| [[Category:Complex manifolds]]
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| [[Category:Structures on manifolds]]
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| [[Category:Riemannian geometry]]
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