Kähler manifold
Let $M$ be a complex manifold with integrable complex structure^{} (http://planetmath.org/AlmostComplexStructure) $J$.
Suppose $M$ is also a Riemannian manifold^{} with metric tensor $g$ such that ${\forall}_{X,Y}g(X,Y)=g(JX,JY)$. 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:

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$M$ is a complex manifold with complex structure $J$

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$M$ is a Riemannian manifold with an Hermitian metric $g$

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$J$ is covariantly constant with regard to the LeviCivita connection^{} ($\nabla J=0$)
Kähler manifolds are symplectic in a natural way with symplectic form^{} defined by $\omega (X,Y)=g(JX,Y)$
Title  Kähler manifold 
Canonical name  KahlerManifold 
Date of creation  20130322 15:43:26 
Last modified on  20130322 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 