a Kähler manifold is symplectic
. Here we used the fact that is an Hermitian tensor on a Kähler manifold ()
Due to anti-symmetry, we just need to check linearity on the second slot. Since is by definition linear, will also be linear.
is non degenerate
First note that
Here we used the fact that both and are covariantly constant ( and )
Since this is a tensorial identity, WLOG we can assume that at a specific point in the Kähler manifold and prove the indentity for these vector fields22in particular this works for the canonical base of associated with a local coordinate system.
The Levi-Civita connection is torsion-free, thus:
is therefore closed.
|Title||a Kähler manifold is symplectic|
|Date of creation||2013-03-22 16:07:54|
|Last modified on||2013-03-22 16:07:54|
|Last modified by||cvalente (11260)|