Kähler manifold

In mathematics designated Kählermannigfaltigkeit ( by Erich Kähler ) is a smooth manifold with a complex structure and a Riemannian metric ( in the sense of a Riemannian manifold) which are compatible with each other.

The concept of Kählermannigfaltigkeit is used in the representation theory of Lie groups and is a central concept of geometric quantization. An important also in string theory example of Kähler manifolds are Calabi -Yau manifolds.

Definition

Let be a smooth manifold, a complex structure and a Riemannian metric, where the space of smooth vector fields referred to. The triple is called Kählermannigfaltigkeit when

And

• Is a symplectic form

Valid for all vector fields.

If the Ricci tensor is proportional to the Riemannian metric, then one speaks of a Kähler -Einstein (or Einstein - Kähler ) manifold. For more details see the article Einstein manifold.