Kähler manifold - Citizendia

In mathematics, a Kähler manifold is a manifold with unitary structure (a $U (n)$ -structure) satisfying an integrability condition. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices In Differential geometry, a G -structure on an n - Manifold M, for a given Structure group G, is a G In Mathematics, certain systems of Partial differential equations are usefully formulated from the point of view of their underlying geometric and algebraic structure in terms In particular, it is a complex manifold, a Riemannian manifold, and a symplectic manifold, with these three structures all mutually compatible. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the

This 3-fold structure corresponds to the presentation of the unitary group as an intersection:

$U(n) = O(2n) \cap GL(n,\mathbf{C}) \cap Sp(2n)$

Without any integrability conditions, the analogous notion is an almost Hermitian manifold. In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. If the Sp-structure is integrable (but the complex structure need not be), the notion is an almost Kähler manifold; if the complex structure is integrable (but the Sp-structure need not be), the notion is a Hermitian manifold. In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold.

Kähler manifolds are named for the mathematician Erich Kähler and are important in algebraic geometry: they are a differential geometric generalization of complex algebraic varieties. Erich Kähler ( 16 January 1906 - 31 May 2000) was a German Mathematician with wide-ranging geometrical interests Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry

1 Definition
2 Examples
3 See also
4 References

Definition

A manifold with a Hermitian metric is an almost Hermitian manifold; a Kähler manifold is a manifold with a Hermitian metric that satisfies an integrability condition, which has several equivalent formulations. In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold.

Kähler manifolds can be characterized in many ways: they are often defined as a complex manifold with an additional structure (or a symplectic manifold with an additional structure, or a Riemannian manifold with an additional structure).

One can summarize the connection between the three structures via $h = g + i ω$ , where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure, and $ω$ is the almost symplectic structure. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. In Differential geometry, an almost symplectic structure on a Differentiable manifold M is a two-form ω on M which is everywhere

A Kähler metric on a complex manifold M is a hermitian metric on the tangent bundle $T M$ satisfying a condition that has several equivalent characterizations (the most geometric being that parallel transport induced by the metric gives rise to complex-linear mappings on the tangent spaces). In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In Geometry, parallel transport is a way of transporting geometrical data along smooth curves in a Manifold. In terms of local coordinates it is specified in this way: if

$h = \sum h_{i\bar j}\; dz^i \otimes d \bar z^j$

is the hermitian metric, then the associated Kähler form defined (up to a factor of i/2) by

$\omega = \sum h_{i\bar j}\; dz^i \wedge d \bar z^j$

is closed: that is, dω = 0. If M carries such a metric it is called a Kähler manifold.

The metric on a Kähler manifold locally satisfies

$g_{i\bar{j}} = \frac{\partial^2 K}{\partial z^i \partial \bar{z}^{j}}$

for some function K, called the Kähler potential.

A Kähler manifold, the associated Kähler form and metric are called Kähler-Einstein (or sometimes Einstein-Kähler) iff its Ricci tensor is proportional to the metric tensor, $R = λ g$ , for some constant λ. In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined This name is a reminder of Einstein's considerations about the cosmological constant. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification See the article on Einstein manifolds for more details. In Differential geometry and Mathematical physics, an Einstein manifold is a Riemannian or Pseudo-Riemannian manifold whose Ricci tensor

Examples

Complex Euclidean space Cⁿ with the standard Hermitian metric is a Kähler manifold.
A torus Cⁿ/Λ (Λ a full lattice) inherits a flat metric from the Euclidean metric on Cⁿ, and is therefore a compact Kähler manifold. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of
Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 (real) dimensions. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional
Complex projective space CPⁿ admits a homogeneous Kähler metric, the Fubini-Study metric. In Mathematics, complex projective space, P ( C n +1 P n ( C) or CP n In Mathematics, the Fubini-Study metric is a Kähler metric on Projective Hilbert space, that is Complex projective space CP An Hermitian form in (the vector space) Cⁿ⁺¹ defines a unitary subgroup U(n+1) in GL(n+1,C); a Fubini-Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action. By elementary linear algebra, any two Fubini-Study metrics are isometric under a projective automorphism of CPⁿ, so it is common to speak of "the" Fubini-Study metric.
The induced metric on a complex submanifold of a Kähler manifold is Kähler. In particular, any Stein manifold (embedded in Cⁿ) or algebraic variety (embedded in CPⁿ) is of Kähler type. In Mathematics, a Stein manifold in the theory of Several complex variables and Complex manifolds is a complex Submanifold of the Vector This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety This is fundamental to their analytic theory.
The unit complex ball Bⁿ admits a Kähler metric called the Bergman metric which has constant holomorphic sectional curvature. In Differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of Complex manifold.
Every K3 surface is Kähler (it follows by a theorem of Y. In Mathematics, in the field of Complex manifolds a K3 surface is an important and interesting example of a compact complex surface ( Complex dimension -T. Siu).

An important subclass of Kähler manifolds are Calabi-Yau manifolds. In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial

References

André Weil, Introduction à l'étude des variétés kählériennes (1958)
Alan Huckleberry and Tilman Wurzbacher, eds. In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. In Differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4 k and Holonomy group contained in Sp(''k'' In Differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian manifold whose Riemannian holonomy group is a In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions Infinite Dimensional Kähler Manifolds (2001), Birkhauser Verlag, Basel ISBN 3-7643-6602-8.
Andrei Moroianu, Lectures on Kähler Geometry (2007), London Mathematical Society Student Texts 69, Cambridge ISBN 978-0-521-68897-0.

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Contents

Definition

Examples

See also

References