Kähler manifold

Let M be a complex manifold with integrable complex structureMathworldPlanetmath (http://planetmath.org/AlmostComplexStructure) J.

Suppose M is also a Riemannian manifoldMathworldPlanetmath with metric tensor g such that X,Yg(X,Y)=g(JX,JY). We say that g is an Hermitian metric tensor.

A differentiable manifold M is said to be a Kähler manifold iff all the following conditions are verified:

  • M is a complex manifold with complex structure J

  • M is a Riemannian manifold with an Hermitian metric g

  • J is covariantly constant with regard to the Levi-Civita connectionMathworldPlanetmath (J=0)

Kähler manifolds are symplectic in a natural way with symplectic formMathworldPlanetmath defined by ω(X,Y)=g(JX,Y)

Title Kähler manifold
Canonical name KahlerManifold
Date of creation 2013-03-22 15:43:26
Last modified on 2013-03-22 15:43:26
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 13
Author cvalente (11260)
Entry type Definition
Classification msc 53D99
Synonym kählerian manifold
Synonym kähler structure
Related topic almostcomplexstructure
Related topic RiemannianMetric
Related topic HyperkahlerManifold
Related topic MathbbCIsAKahlerManifold
Related topic SymplecticManifold
Related topic aKahlerManifoldIsSymplectic
Related topic AKahlerManifoldIsSymplectic
Related topic AlmostComplexStructure
Defines Hermitian metric tensor