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 .

See also

• Quaternionic Kähler manifold

Web links

• AL Onishchik: Kähler manifold . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• 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 .