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.

- Formal definition
- Riemannian metric and associated form
- Properties
- Kähler manifolds
- Integrability
- References

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.

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 written as a smooth section

such that

for all ζ, η in *E*_{p} and

for all nonzero ζ in *E*_{p}.

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.

On a Hermitian manifold the metric can be written in local holomorphic coordinates (*z*^{α}) as

where 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*:

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

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*:

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

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

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

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

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 - 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:

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

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

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:

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.

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,
- the holonomy group of ∇ is contained in the unitary group U(
*n*) associated to*J*.

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

**Differential geometry** is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

In mathematics, the **Hodge star operator** or **Hodge star** is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the **Hodge dual** of the element. This map was introduced by W. V. D. Hodge.

In mathematics and especially differential geometry, a **Kähler manifold** is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

In mathematics, **Hodge theory**, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold *M* using partial differential equations. The key observation is that, given a Riemannian metric on *M*, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called **harmonic**.

In differential geometry and complex geometry, a **complex manifold** is a manifold with an atlas of charts to the open unit disk in **C**^{n}, such that the transition maps are holomorphic.

In mathematics, an **almost complex manifold** is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry.

In mathematics, and specifically differential geometry, a **connection form** is a manner of organizing the data of a connection using the language of moving frames and differential forms.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, the **Fubini–Study metric** is a Kähler metric on projective Hilbert space, that is, on a complex projective space **CP**^{n} endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

In mathematics, a **CR manifold** is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.

In mathematics, in particular in algebraic geometry and differential geometry, **Dolbeault cohomology** is an analog of de Rham cohomology for complex manifolds. Let *M* be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers *p* and *q* and are realized as a subquotient of the space of complex differential forms of degree (*p*,*q*).

In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In differential geometry, a **Kähler–Einstein metric** on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be **Kähler–Einstein** if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.

In complex geometry, the term *positive form* refers to several classes of real differential forms of Hodge type *(p, p)*.

In heterotic string theory, the **Strominger's equations** are the set of equations that are necessary and sufficient conditions for spacetime supersymmetry. It is derived by requiring the 4-dimensional spacetime to be maximally symmetric, and adding a warp factor on the internal 6-dimensional manifold.

In mathematics, a **nearly Kähler manifold** is an almost Hermitian manifold , with almost complex structure , such that the (2,1)-tensor is skew-symmetric. So,

In mathematics, a **harmonic morphism** is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps i.e. those that are horizontally (weakly) conformal.

In algebraic geometry and differential geometry, the **Nonabelian Hodge correspondence** or **Corlette–Simpson correspondence** is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.

In mathematics and theoretical physics, and especially gauge theory, the **deformed Hermitian Yang–Mills (dHYM) equation** is a differential equation describing the equations of motion for a D-brane in the B-model of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger in the case of Abelian gauge group, and by Leung–Yau–Zaslow using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory.

In mathematics, and especially differential geometry and mathematical physics, **gauge theory** is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics *theory* means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a physical model of some natural phenomenon.

- Griffiths, Phillip; Joseph Harris (1994) [1978].
*Principles of Algebraic Geometry*. Wiley Classics Library. New York: Wiley-Interscience. ISBN 0-471-05059-8. - 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.

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