# Kähler manifold

In mathematics , the Kähler manifold (after Erich Kähler ) is a smooth manifold together with a complex structure and a Riemannian metric (in the sense of a Riemannian manifold ) that are compatible with one another.

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

## definition

Is a smooth manifold , has a complex structure, that is a smooth mapping with and a Riemannian metric , wherein the space of smooth vector fields on indicated. The triple is called the Kähler manifold if ${\ displaystyle M}$${\ displaystyle J \ colon TM \ to TM}$${\ displaystyle J \ colon TM \ to TM}$${\ displaystyle J ^ {2} = - Id}$${\ displaystyle g \ colon {\ mathcal {V}} (M) \ times {\ mathcal {V}} (M) \ to C ^ {\ infty} (M; \ mathbb {R})}$${\ displaystyle {\ mathcal {V}} (M)}$${\ displaystyle M}$${\ displaystyle (M, J, g)}$

• ${\ displaystyle g (JX, JY) = g (X, Y)}$

for all vector fields and ${\ displaystyle X, Y \ in {\ mathcal {V}} (M)}$

• ${\ displaystyle \ omega (X, Y): = g (JX, Y)}$a symplectic form

is. The 2-form is then called the Kähler form of . ${\ displaystyle \ omega}$${\ displaystyle M}$

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