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DEFINITION

A metric ''g'' is called a Hermitian metric if

:g(v,w) = g(Jv, Jw)

for all vector fields ''v'', ''w'' on ''M''. In index notation, one writes

:g_{pq} = J^r_p J^s_q g_{rs}

A manifold with such a metric is said to be a Hermitian manifold or to posses a Hermitian structure.


HERMITIAN FORM

The Differential Form

:\omega(\cdot, \cdot) = g(J\cdot, \cdot)

is called the Hermitian form of ''J''. The is, for all vector fields ''u'', ''v'' on ''M'', one has

:\omega(u,v) = g(Ju, v)

In index notation, one writes

:\omega_{rq} = J_r^p g_{pq}

The form \omega is a form of type-(1,1), indicating that it has a holomorphic and an anti-holomorphic part. The decomposition of differential forms into holomorphic and anti-holomorphic subspaces is discussed in the article on Almost-complex Manifold s.


KäHLER FORM

If the Hermitian form is closed, dω = 0, then the form is called a Kähler form, and the Hermitian manifold is called a Kähler Manifold . Conditions equivalent to the vanishing of dω = 0 are

:
abla J=0

and

:
abla \omega = 0

where
abla is the Levi-Civita Connection of ''g''. The last two conditions imply that ''J'' and ω are constant tensors on the manifold, and thus are preserved by the Holonomy Group of ''M''. In essence, this implies that the holonomy group of a Kähler manifold is a subgroup of U(m), where the dimension of ''M'' is 2''m''.

A Kähler form is a Symplectic Form , and thus, Kähler manifolds are Symplectic Manifold s endowed with a metric.