# 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:

• $M$ 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 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 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