PlanetMath: Kähler manifold

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Kähler manifold

(Definition)

Suppose is also a Riemannian manifold with metric tensor such that $\forall_{X,Y} g(X,Y) = g(JX,JY)$ . We say that is an Hermitian metric tensor.

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

is a complex manifold with complex structure
is a Riemannian manifold with an Hermitian metric
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 $\omega(X,Y) = g(JX,Y)$

"Kähler manifold" is owned by cvalente.

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Other names:

kählerian manifold, kähler structure

Also defines:

Hermitian metric tensor

Keywords:

kähler, complex, hermitian, symplectic

Attachments:: $\mathbb{C}$ as a Kähler manifold (Example) by cvalente
a Kähler manifold is symplectic (Result) by cvalente

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Cross-references: symplectic form, Levi-Civita connection, metric, Hermitian, iff, differentiable manifold, metric tensor, Riemannian manifold, complex manifold
There are 3 references to this entry.

This is version 10 of Kähler manifold, born on 2006-03-04, modified 2006-09-27.
Object id is 7673, canonical name is KahlerManifold.
Accessed 2682 times total.

Classification:

53D99 (Differential geometry :: Symplectic geometry, contact geometry :: Miscellaneous)

Pending Errata and Addenda

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