$\u2102$ as a Kähler manifold
$\u2102$ can be interpreted as ${\mathbb{R}}^{2}$ with a complex structure (http://planetmath.org/AlmostComplexStructure) $J$.
Parametrize ${\mathbb{R}}^{2}$ via the usual coordinates $(x,y)$.
A point $z$ in the complex plane can thus be written $z=x+iy$.
The tangent space^{} at each point is generated by the ${\mathrm{span}}_{\mathbb{R}}\{\frac{\partial}{\partial x},\frac{\partial}{\partial y}\}$ and the complex structure (http://planetmath.org/AlmostComplexStructure) $J$ is defined by^{1}^{1}notice $J$ acts as a counterclockwise rotation by $\frac{\pi}{2}$, just as expected
$J\left({\displaystyle \frac{\partial}{\partial x}}\right)={\displaystyle \frac{\partial}{\partial y}}$  (1)  
$J\left({\displaystyle \frac{\partial}{\partial y}}\right)={\displaystyle \frac{\partial}{\partial x}}$  (2) 
The metric can be the usual metric $g=dx\otimes dx+dy\otimes dy$. This is a flat metric and therefore all the covariant derivatives^{} are plain partial derivatives^{} in the $(x,y)$ coordinates^{2}^{2}the Christoffel symbols^{} on these coordinates vanish.
So lets verify all the points in the definition.

•
$\u2102$ is a Riemannian Manifold^{}

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$g$ is Hermitian.
$$g(J\frac{\partial}{\partial x},J\frac{\partial}{\partial y})=g(\frac{\partial}{\partial y},\frac{\partial}{\partial x})=0=g(\frac{\partial}{\partial x},\frac{\partial}{\partial y})$$ $$g(J\frac{\partial}{\partial x},J\frac{\partial}{\partial x})=g(\frac{\partial}{\partial y},\frac{\partial}{\partial y})=1=g(\frac{\partial}{\partial x},\frac{\partial}{\partial x})$$ $$g(J\frac{\partial}{\partial y},J\frac{\partial}{\partial y})=g(\frac{\partial}{\partial x},\frac{\partial}{\partial x})=1=g(\frac{\partial}{\partial y},\frac{\partial}{\partial y})$$ 
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$J$ 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.
$\u2102$ is therefore a Kähler manifold^{}.
The symplectic form^{} for this example is
$$\omega =dx\wedge dy$$ 
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 ${\mathbb{R}}^{n}$ seen as a metric space.
Title  $\u2102$ as a Kähler manifold 

Canonical name  mathbbCAsAKahlerManifold 
Date of creation  20130322 15:46:32 
Last modified on  20130322 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 