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Encyclopedia > Kähler manifold

In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. Kähler manifolds can thus be thought of as Riemannian manifolds and symplectic manifolds in a natural way. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...


Kähler manifolds are named for the mathematician Erich Kähler and are important in algebraic geometry. 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 abstract algebra, especially commutative algebra, with geometry. ...

Contents

Definition

A Kähler metric on a complex manifold M is a hermitian metric on the complexified tangent bundle satisfying a condition that has several equivalent characterizations (the most geometric being that parallel transport gives rise to complex-linear mappings on the tangent spaces). In terms of local coordinates it is specified in this way: if A hermitian metric on a complex vector bundle E over a smooth manifold M, is a positive-definite, hermitian inner product on each fiber Ep, that varies smoothly with the point p in M. An important special case is that of a hermitian metric on the complexified tangent bundle of... In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ... In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

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

is closed: that is, dω = 0. If M carries such a metric it is called a Kähler manifold. In mathematics, closed form can mean: a finitary expression, rather than one involving (for example) an infinite series, or use of recursion - this meaning usually occurs in a phrase like solution in closed form and one also says closed formula; a closed differential form: see Closed and exact differential forms. ...


Examples

  1. Complex Euclidean space Cn with the standard Hermitian metric is a Kähler manifold.
  2. A torus Cn/Λ (Λ a lattice) inherits a flat metric from the Euclidean metric on Cn, and is therefore a compact Kähler manifold.
  3. Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 (real) dimensions.
  4. Complex projective space CPn admits a homogeneous Kähler metric called the Fubini-Study metric. 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.
  5. 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. This is fundamental to their analytic theory.

An important subclass of Kähler manifolds are Calabi-Yau manifolds. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... See lattice for other meanings of this term, both within and without mathematics. ... Several specialized usages of the terms compact and compactness exist. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a Calabi-Yau manifold is a compact K hler manifold with a vanishing first Chern class. ...


See also

In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ...

References

  • Alan Huckleberry and Tilman Wurzbacher, eds. Infinite Dimensional Kähler Manifolds (2001), Birkhauser Verlag, Basel ISBN 3-7643-6602-8.

 
 

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