Kähler manifold

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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 .


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

for all vector fields and

  • a symplectic form

is. The 2-form is then called the Kähler form of .

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 .

See also

Web links


  • Alan Huckleberry, Tilman Wurzbacher (eds.): Infinite Dimensional Kähler Manifolds (= DMV seminar. Vol. 31). Birkhäuser Verlag, Basel et al. 2001, ISBN 3-7643-6602-8 .
  • Andrei Moroianu: Lectures on Kähler Geometry (= London Mathematical Society Student Texts. Vol. 69). Cambridge University Press, Cambridge 2007, ISBN 978-0-521-68897-0 .