as a Kähler manifold
can be interpreted as with a complex structure (http://planetmath.org/AlmostComplexStructure) .
Parametrize via the usual coordinates .
A point in the complex plane can thus be written .
The tangent space at each point is generated by the and the complex structure (http://planetmath.org/AlmostComplexStructure) is defined by11notice acts as a counterclockwise rotation by , just as expected
The metric can be the usual metric . This is a flat metric and therefore all the covariant derivatives are plain partial derivatives in the coordinates22the Christoffel symbols on these coordinates vanish.
So lets verify all the points in the definition.
is therefore a Kähler manifold.
The symplectic form 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 seen as a metric space.
|Title||as a Kähler manifold|
|Date of creation||2013-03-22 15:46:32|
|Last modified on||2013-03-22 15:46:32|
|Last modified by||cvalente (11260)|