- Hermitian manifold
In
mathematics , a Hermitian manifold is the complex analog of aRiemannian manifold . Specifically, a Hermitian manifold is acomplex manifold with a smoothly varying Hermitianinner product on each (holomorphic)tangent space . One can also define a Hermitian manifold as a complex manifold with aRiemannian metric that preserves thealmost complex structure "J".There are several related concepts coming from
integrability condition s:A Hermitian manifold has a unitary structure (a U(n)-structure), with thealmost complex structure integrable. Without the integrability condition, the notion is an almost Hermitian manifold. With an additional integrability condition on the Sp-structure, one obtains aKähler manifold . With integrability on the Sp-structure but not the almost complex structure, one obtains an almost Kähler manifold.Formal definition
A Hermitian metric on a
complex vector bundle "E" over asmooth manifold "M" is a smoothly varyingpositive-definite Hermitian form on each fiber. Such a metric can be written as a smooth section:h in Gamma(Eotimesar E)^*such that:h_p(eta, arzeta) = overline{h_p(zeta, areta)}for all ζ, η in "E""p" and:h_p(zeta,arzeta) > 0for all nonzero ζ in "E""p".A Hermitian manifold is a
complex manifold with a Hermitian metric on itsholomorphic tangent space . Likewise, an almost Hermitian manifold is analmost complex manifold with a Hermitian metric on its holomorphic tangent space.On a Hermitian manifold the metric can be written in local holomorphic coordinates ("z"α) as:h = h_{alphaareta},dz^alphaotimes dar z^etawhere h_{alphaareta} 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 = {1over 2}(h+ar h).The form "g" is a symmetric bilinear form on "TM"C, thecomplexified 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 = {1over 2}h_{alphaareta},(dz^alphaotimes dar z^eta + dar z^etaotimes dz^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 = {iover 2}(h-ar 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 = {iover 2}h_{alphaareta},dz^alphawedge dar z^eta.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:egin{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 - iomega.,All three forms "h", "g", and ω preserve thealmost complex structure "J". That is,:egin{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
#a Hermitian metric "h" as above,
#a Riemannian metric "g" that preserves the almost complex structure "J", or
#anondegenerate 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"′ compatible with the almost complex structure "J" in an obvious manner::g'(u,v) = {1over 2}left(g(u,v) + g(Ju,Jv) ight).
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 theframe bundle of "M" from GL("n",C) to theunitary group U("n"). A unitary frame on an almost Hermitian manifold is complex linear frame which isorthonormal with respect to the Hermitian metric. Theunitary 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 thewedge 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} ight)^n det(h_{alphaareta}), dz^1wedge dar z^1wedge cdots wedge dz^nwedge dar z^n.Kähler manifolds
The most important class of Hermitian manifolds are
Kähler manifold s. These are Hermitian manifolds for which the Hermitian form ω is closed::domega = 0,.In this case the form ω is called a Kähler form. A Kähler form is asymplectic form , and so Kähler manifolds are naturallysymplectic manifold s.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 2"n" 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
* ∇"J" = 0,
* ∇ω = 0,
* theholonomy group of ∇ is contained in theunitary group U("n") associated to "J".The equivalence of these conditions corresponds to the "2 out of 3" property of theunitary group .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.
References
*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
*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
*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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