as a Kähler manifold

can be interpreted as 2 with a complex structure ( J.

Parametrize 2 via the usual coordinates (x,y).

A point z in the complex plane can thus be written z=x+iy.

The tangent spacePlanetmathPlanetmath at each point is generated by the span{x,y} and the complex structure ( J is defined by11notice J acts as a counterclockwise rotation by π2, just as expected

J(x)=y (1)
J(y)=-x (2)

The metric can be the usual metric g=dxdx+dydy. This is a flat metric and therefore all the covariant derivativesMathworldPlanetmath are plain partial derivativesMathworldPlanetmath in the (x,y) coordinates22the Christoffel symbolsMathworldPlanetmath on these coordinates vanish.

So lets verify all the points in the definition.

  • g is Hermitian.

  • J is covariantly constant because its components in the (x,y) coordinates are constant and as previously stated, the covariant derivatives are just partial derivatives in this example.

is therefore a Kähler manifoldMathworldPlanetmath.

The symplectic formMathworldPlanetmath for this example is


This is the simplest example of a Kähler manifold and can be seen as a template for other less trivial examples. Those are generalizations of this example just as Riemannian manifolds are generalizations of n seen as a metric space.

Title as a Kähler manifold
Canonical name mathbbCAsAKahlerManifold
Date of creation 2013-03-22 15:46:32
Last modified on 2013-03-22 15:46:32
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 16
Author cvalente (11260)
Entry type Example
Classification msc 53D99
Related topic KahlerManifold
Related topic AlmostComplexStructure
Related topic SymplecticManifold